phizikis-saphudzvlebi1

devid holidei robert rezniki jerli uolkeri fizikis safuZvlebi ilias saxelmwifo universiteti ))%)!' 2010 ფიზიკის საფუძვლები დევიდ ჰოლიდეი, რობერტ რეზნიკი, ჯერლი უოლკერი Fundamentals of Physics, David Halliday, Robert Resnick, Jearl Walker ვერც ერთი სხვა წიგნი ვერ შეედრება ჰოლიდეის, რეზნიკისა და უოლკერის ‘ფიზიკის საფუძვლების’ ოცდაათწლიან წარმატებას ბაზარზე! სახალისო, ადვილად გასაგები სტილი ფიზიკის საფუძვლიან გაგებას გვთავაზობს და მკითხველს ამ კონცეპტუალური ცოდნის რაოდენობრივი ამოცანების ამოხსნისას გამოყენებაში ეხმარება. გამოცემა უკანასკნელი 100 წლის განმავლობაში ფიზიკის განვითარებას მოიცავს. სახელმძღვანელო თეორიული შინაარსისა და სახალისო პრაქტიკის უნიკალური კომბინაციაა. 2002 წელს ამერიკის ფიზიკოსთა საზოგადოებამ ‘ფიზიკის საფუძვლები მე-20 საუკუნის გამორჩეულ სახელმძღვანელოდ დაასახელა. წიგნის თარგმანი დაფინანსდა საქართველოს მეცნიერებისა და განათლების სამინისტროს პროგრამით - ’საუნივერსიტეტო სახელმძღვანელოების განვითარება’. წიგნი მომზადდა ილიას სახელმწიფო უნივერსიტეტის მიერ. ილიას სახელმწიფო უნივერსიტეტის გამომცემლობა ქაქუცა ჩოლოყაშვილის 3/5, თბილისი, 0162, საქართველო Ilia State University Press 3/5 K. Cholokashvili Ave, Tbilisi, 0162, Georgia sarCevi nawili I Tavi 5. Zala da moZraoba................................131 Tavi 1. erTeulTa sistema..................................9 5.1. ra aris fizika?.......................................................131 5.2. niutonis meqanika...................................................131 5.3. niutonis pirveli kanoni.....................................132 5.4. Zala............................................................................132 5.5. masa.............................................................................134 5.6. niutonis meore kanoni.........................................135 5.7. zogierTi konkretuli Zala...............................139 5.8. niutonis mesame kanoni.........................................144 5.9. niutonis kanonebis gamoyeneba..........................145 mimoxilva da Sejameba..................................................152 SEekiTxvebi........................................................................153 amocanebi..........................................................................155 1.1. ra aris fizika?...........................................................9 1.2. sidideebis gazomva....................................................9 1.3. erTeulebis saerTaSoriso sistema...................10 1.4. erTeulTa Secvla....................................................11 1.5. sigrZe..........................................................................13 1.6. dro...............................................................................14 1.7. masa...............................................................................17 mimoxilva da Sejameba....................................................18 amocanebi............................................................................18 Tavi 2. wrfivi moZraoba....................................26 2.1. ra aris fizika?.........................................................26 2.2 moZraoba......................................................................26 Tavi 6. Zala da moZraoba................................168 2.3. mdebareoba da gadaadgileba................................27 6.1. ra aris fizika?.......................................................168 6.2. xaxuni.........................................................................168 6.3. xaxunis Tvisebebi...................................................170 6.4. wevis Zala da saboloo siCqare..........................174 2.4. saSualo manZili da saSualo siCqare.................28 2.5. myisi manZili da myisi siCqare .............................31 2.6. aCqareba.......................................................................33 2.7. Tanabari aCqareba: gansakuTrebuli SemTxveva...35 2.8. kidev erTxel Tanabari aCqarebis Sesaxeb.........38 2.9. Tavisufali vardnis aCqareba..............................39 2.10. grafikuli integrali moZraobis SeswavlaSi...42 mimoxilva da Sejameba....................................................44 SekiTxvebi .........................................................................45 amocanebi . ........................................................................ 46 Tavi 3. veqtorebi.................................................60 3.1. ra aris fizika?.........................................................60 3.2. veqtorebi da skalarebi.........................................60 3.3. veqtorebis geometriuli Sekreba......................61 3.4. veqtorebis komponentebi......................................64 3.5. erTeulovani veqtorebi........................................68 3.6. veqtorebis Sekreba komponentebis saSualebiT.................................................................69 3.7. veqtorebi da fizikis kanonebi............................73 3.8. veqtorebis gamravleba..........................................73 mimoxilva da Sejameba....................................................78 SekiTxvebi..........................................................................79 amocanebi............................................................................80 Tavi 4. organzomilebiani da samganzomilebiani moZraoba...........88 4.1. ra aris fizika?.........................................................88 4.2. mdebareoba da gadaadgileba................................88 4.3. saSualo siCqare da myisi siCqare........................91 4.4. saSualo aCqareba da myisi aCqareba....................94 4.5. Wurvisebri moZraoba..............................................96 4.6. Wurvisebri moZraobis analizi............................98 4.7. Tanabari wriuli moZraoba..................................104 4.8. erTganzomilebiani fardobiTi moZraoba......106 4.9. organzomilebiani fardobiTi moZraoba........108 mimoxilva da Sejameba..................................................110 SekiTxvebi........................................................................111 amocanebi..........................................................................113 6.5. Tanabari wriuli moZraoba..................................177 mimoxilva da Sejameba..................................................183 SekiTxvebi........................................................................184 amocanebi..........................................................................185 Tavi 7. kinetikuri energia da muSaoba......199 7.1. ra aris fizika?.......................................................199 7.2. ra aris energia?.....................................................199 7.3. kinetikuri energia................................................200 7.4. muSaoba......................................................................201 7.5. muSaoba da kinetikuri energia..........................201 7.6. mizidulobis Zalis mier Sesrulebuli muSaoba......................................................................205 7.7. drekadobis Zalis mier Sesrulebuli muSaoba......................................................................209 7.8. cvladi Zalis mier Sesrulebuli muSaoba.....212 7.9. simZlavre.................................................................216 mimoxilva da Sejameba..................................................218 SekiTxvebi........................................................................219 amocanebi..........................................................................220 Tavi 8. potenciuri energia da energiis Senaxva.....................................................231 8.1. ra aris fizika?.......................................................231 8.2. muSaoba da potenciuri energia........................232 8.3. Semnaxveli Zalebis traeqtoriisgan damoukidebloba.....................................................233 8.4. potenciuri energiis mniSvnelobebis gansazRvra...............................................................235 8.5. meqanikuri energiis Senaxva................................237 8.6. potenciuri energiis grafikis wakiTxva........242 8.7. gare Zalis mier sistemaze Sesrulebuli muSaoba.......................................................................245 8.8. energiis Senaxva......................................................248 mimoxilva da Sejameba..................................................253 SekiTxvebi........................................................................254 amocanebi..........................................................................256 Tavi 9. masis centri da impulsi..................275 9.1. ra aris fizika?.......................................................275 9.2. masis centri............................................................276 9.3. niutonis meore kanoni nivTier wertilTa sistemisTvis............................................................280 9.4. impulsi.....................................................................284 9.5. nivTier wertilTa sistemis impulsi...............285 9.6. Sejaxeba da impulsi..............................................285 9.7. impulsis mudmivoba...............................................289 9.8. impulsi da kinetikuri energia Sejaxebebis dros...........................................................................293 9.9. araelastiuri Sejaxebebi erT ganzomilebaSi.........................................................293 9.10. elastiuri Sejaxebebi erT ganzomilebaSi..297 9.11. Sejaxebebi or ganzomilebaSi...........................300 9.12. cvladi masis mqone sistemebi: raketa...........301 mimoxilva da Sejameba..................................................303 SekiTxvebi........................................................................304 amocanebi..........................................................................307 Tavi 10. brunva....................................................324 10.1. ra aris fizika?.....................................................324 10.2. brunvis parametrebi...........................................324 10.3. arian Tu ara kuTxuri sidideebi veqtorebi?.............................................................329 10.4. brunva Tanabari kuTxuri aCqarebiT..............330 10.5. wrfivi da kuTxuri cvladebis erTmaneTTan dakavSireba............................................................332 10.6. brunvis kinetikuri energia..............................335 nawili II Tavi 12. wonasworoba da elastiuroba.....406 12.1. ra aris fizika?.....................................................406 12.2. wonasworoba..........................................................346 12.3. wonasworobisTvis saWiro pirobebi...............408 12.4. mizidulobis centri..........................................409 12.5. statikuri wonasworobis ramdenime magaliTi.................................................................411 12.6. statikurad ganusazRvreli konstruqcia.........................................................417 12.7. elastiuroba.........................................................419 mimoxilva da Sejameba..................................................423 kiTxvebi............................................................................424 amocanebi..........................................................................425 Tavi 13. miziduloba.........................................439 13.1. ra aris fizika?.....................................................439 13.2. niutonis mizidulobis kanoni.........................440 13.3. miziduloba da zeddebis principi.................442 13.4. miziduloba dedamiwis zedapirTan axlos...444 13.5. miziduloba dedamiwis SigniT.........................447 13.6. mizidulobis potenciuli energia.................449 13.7. planetebi da Tanamgzavrebi: kepleris kanonebi....................................................................453 13.8. Tanamgzavrebi: orbitebi da energia..............457 13.9. ainStaini da miziduloba..................................459 mimoxilva da Sejameba..................................................461 SekiTxvebi........................................................................462 amocanebi..........................................................................464 Tavi 14. siTxeebi................................................476 14.1. ra aris fizika?.....................................................476 14.2. ra aris siTxe?.......................................................476 14.3. simkvrive da wneva.................................................477 14.4. siTxeebi uZrav mdgomareobaSi.........................479 10.7. brunviTi inerciis gamoTvla...........................336 10.8. mabruni momenti...................................................340 10.9. niutonis meore kanoni brunvisTvis..............342 10.10. muSaoba da brunvis kinetikuri energia .....345 mimoxilva da Sejameba..................................................349 SekiTxvebi........................................................................350 amocanebi..........................................................................352 Tavi 11. gorva, mabruni momenti da kuTxuri impulsi...............................367 11.1. ra aris fizika?.....................................................367 11.2. gorva, rogorc gadataniTi moZraobis da brunvis gaerTianeba............................................368 11.3. gorvis kinetikuri energia...............................368 11.4. gorvis Zalebi........................................................370 11.5. io-io........................................................................373 11.6. mabruni momentis xelaxla ganxilva...............374 11.7. kuTxuri impulsi..................................................376 11.8. niutonis meore kanonis kuTxuri forma.......379 11.9. nivTier wertilTa sistemis kuTxuri impulsi....................................................................381 11.10. fiqsirebuli RerZis garSemo mbrunavi myari sxeulis kuTxuri impulsi....................382 11.11. kuTxuri impulsis mudmivoba.........................385 11.12. giroskopis precesia.........................................390 mimoxilva da Sejameba..................................................391 kiTxvebi............................................................................392 amocanebi..........................................................................394 14.5. wnevis gazomva.......................................................482 14.6. paskalis principi................................................483 14.7. arqimedes principi..............................................485 14.8. moZravi idealuri siTxeebi..............................488 14.9. uwyvetobis gantoleba.......................................489 14.10. bernulis gantoleba........................................491 mimoxilva da Sejameba..................................................496 kiTxvebi............................................................................497 amocanebi..........................................................................498 Tavi 15. rxevebi..................................................510 15.1. ra aris fizika?.....................................................510 15.2. martivi harmoniuli moZraoba.........................510 15.3. Zalis kanoni martivi harmoniuli moZraobisTvis.......................................................514 15.4. energia martiv harmoniul moZraobaSi.........516 15.5. kuTxuri martivi harmoniuli oscilatori...........................................................518 15.6. qanqarebi.................................................................519 15.7. martivi harmoniuli moZraoba da erTgvarovani wriuli moZraoba.......................524 15.8. milevadi martivi harmoniuli moZraoba......525 15.9. iZulebiTi rxevebi da rezonansi.....................527 mimoxilva da Sejameba..................................................528 SekiTxvebi........................................................................529 amocanebi..........................................................................531 Tavi 16. talRebi...............................................544 16.1. ra aris fizika?.....................................................544 16.2. talRebis saxeobebi.............................................545 16.3. ganivi da gaswvrivi talRebi.............................545 16.4. talRis sigrZe da sixSire.................................547 16.5. moZravi talRis siCqare.....................................550 16.6. talRis siCqare daWimul Tokze......................553 16.7. Tokis gaswvriv moZravi talRis energia da simZlavre.................................................................555 16.8. talRis gantoleba..............................................557 16.9. talRebis superpoziciis principi................559 16.10. talRebis interferencia...............................560 16.11. veqtorebi.............................................................563 16.12. mdgari talRebi..................................................564 16.13. mdgari talRebi da rezonansi........................567 mimoxilva da Sejameba..................................................569 SekiTxvebi........................................................................571 amocanebi..........................................................................572 Tavi 17. talRebi............................................... 583 17.1. ra aris fizika?.....................................................583 17.2. bgeris talRebi....................................................584 17.3. bgeris siCqare.......................................................584 17.4. moZravi bgeriTi talRebi..................................587 17.5. interferencia......................................................590 17.6. intensivoba da xmamaRlobis done...................592 17.7. musikaluri bgerebis wyaroebi........................596 17.8. ritmebi....................................................................598 17.9. dopleris efeqti.................................................600 17.10. zebgeriTi siCqareebi, dartymiTi talRebi...................................................................604 mimoxilva da Sejameba..................................................605 SekiTxvebi........................................................................607 amocanebi..........................................................................608 Tavi 18. temperatura, siTbo da Termodinamikis pirveli kanoni ...............622 18.1. ra aris fizika?.....................................................622 18.2. temperatura..........................................................623 18.3. Termodinamikis nulovani kanoni....................623 18.4. temperaturis gazomva.......................................624 18.5. celsiusis da farengeitis Skalebi................626 18.6. Tburi gafarToeba...............................................628 18.7. temperatura da siTbo.......................................631 18.8. myari sxeulebis da siTxeebis mier siTbos STanTqma...................................................................632 18.9. kidev erTxel siTbosa da muSaobis Sesaxeb.......................................................................637 18.10. Termodinamikis pirveli kanoni....................639 18.11. Termodinamikis pirveli kanonis gansakuTrebuli SemTxvevebi.............................640 18.12. siTbos gadacemis meqanizmebi........................642 mimoxilva da Sejameba..................................................647 SekiTxvebi........................................................................649 amocanebi..........................................................................650 Tavi 19. airebis kinetikuri Teoria............ 661 19.1. ra aris fizika?.....................................................661 19.2. avogadros ricxvi................................................662 19.3. idealuri airebi...................................................662 19.4. wneva, temperatura da saSualo kvadratuli siCqare......................................................................666 19.5. winsvliTi kinetikuri energia.........................668 19.6. saSualo Tavisufali garbeni...........................669 19.7. molekuluri siCqareebis ganawileba.............671 19.8. idealuri airis moluri kuTri siTbotevadoba.......................................................674 19.9. Tavisuflebis xarisxis da moluri kuTri siTbotevadoba.......................................................678 19.10. miniSneba qvantur Teoriaze...........................680 19.11. idealuri airis adiabaturi gafarToeba.............................................................681 mimoxilva da Sejameba..................................................685 SekiTxvebi........................................................................686 amocanebi..........................................................................687 Tavi 20. entropia da Termodinamikis meore kanoni........................................696 20.1. ra aris fizika?.....................................................696 20.2. Seuqcevadi procesebi da entropia................696 20.3. entropiis cvlileba...........................................697 20.4. Termodinamikis meore kanoni...........................703 20.5. entropia realur samyaroSi: Zravebi............704 20.6. entropia realur samyaroSi: macivrebi........709 20.7. realuri Zravebis margi qmedebis koeficientebi........................................................711 20.8. entropiis statistikuri ganxilva.................712 mimoxilva da Sejameba..................................................716 SekiTxvebi........................................................................717 amocanebi..........................................................................718 nawili III Tavi 21. eleqtruli muxti.............................727 21.1. ra aris fizika?.....................................................727 21.2. eleqtruli muxti................................................728 21.3. gamtarebi da dieleqtrikebi............................729 21.4. kulonis kanoni.....................................................732 21.5. muxti kvantirdeba...............................................738 21.6. muxti inaxeba.........................................................739 mimoxilva da Sejameba..................................................740 SekiTxvebi........................................................................741 amocanebi..........................................................................743 Tavi 22. eleqtruli velebi......................... 7451 22.1. ra aris fizika?.....................................................751 22.2. eleqtruli veli..................................................752 22.3. eleqtruli velis Zalwirebi...........................753 22.4. wertilovani muxtiT gamowveuli eleqtruli veli...................................................755 22.5. eleqtruli dipoliT gamowveuli eleqtruli veli...................................................757 22.6. damuxtuli wiriT gamowveuli eleqtruli veli...................................................759 22.7. damuxtuli diskos Sedegad warmoqmnili eleqtruli veli...................................................764 22.8. wertilovani muxti eleqtrul velSi...........765 22.9. dipoli eleqtrul velSi..................................767 mimoxilva da Sejameba..................................................770 SekiTxvebi........................................................................771 amocanebi..........................................................................773 Tavi 23. gausis kanoni......................................783 23.1. ra aris fizika?.....................................................783 23.2. nakadi.......................................................................784 23.3. nakadi eleqtrul velSi.....................................785 23.4. gausis kanoni.........................................................787 23.5. gausis kanoni da kulonis kanoni.....................789 23.6. damuxtuli izolirebuli gamtari.................791 23.7. gausis kanonis gamoyeneba: cilindruli simetria...................................................................794 23.8. gausis kanonis gamoyeneba: planaruli simetria...................................................................795 23.9. gausis kanonis gamoyeneba: sferuli simetria...................................................................798 mimoxilva da Sejameba..................................................800 SekiTxvebi........................................................................800 amocanebi..........................................................................802 Tavi 24. eleqtruli potenciali................812 24.1. ra aris fizika?.....................................................812 24.2. eleqtruli potenciuri energia....................812 24.3. eleqtruli potenciali....................................814 24.4. ekvipotenciuri zedapirebi.............................816 24.5. potencialis gamoTvla velis saSualebiT..............................................................817 24.6. wertilovani muxtiT gamowveuli potenciali.............................................................819 24.7. wertilovani muxtebis jgufiT gamowveuli potenciali.............................................................821 24.8. eleqtruli dipoliT gamowveuli potenciali.............................................................822 24.9. muxtebis uwyveti ganawilebiT gamowveuli potenciali.............................................................824 24.10. velis gamoTvla potencialis saSualebiT..............................................................826 24.11. wertilovani muxtebis sistemis eleqtruli potenciuri energia.............................................828 24.12. damuxtuli izolirebuli gamtaris potenciali.............................................................830 mimoxilva da Sejameba..................................................832 SekiTxvebi........................................................................833 amocanebi..........................................................................835 Tavi 25. tevadoba...............................................848 25.1. ra aris fizika?.....................................................848 25.2. tevadoba.................................................................849 25.3. tevadobis gamoTvla...........................................851 25.4. mimdevrobiTi da paraleluri kondensatorebi.....................................................855 25.5. eleqtrul velSi Senaxuli energia................860 25.6. kondensatori dieleqtrikTan erTad...........863 25.7. dieleqtrikebi: atomisturi Sexeduleba.....865 25.8. dieleqtrikebi da gausis kanoni.....................866 mimoxilva da Sejameba..................................................869 SekiTxvebi........................................................................870 amocanebi..........................................................................871 Tavi 26. deni da winaRoba................................881 26.1. ra aris fizika?.....................................................881 26.2. eleqtruli deni...................................................882 26.3. denis simkvrive.....................................................884 26.4. winaRoba da kuTri winaRoba.............................887 26.5. omis kanoni.............................................................891 26.6. omis kanonis mikroskopuli ganxilva............893 26.7. simZlavre eleqtrul wredebSi.......................895 26.8. naxevargamtarebi.................................................896 26.9. supergamtarebi....................................................898 mimoxilva da Sejameba..................................................899 SekiTxvebi........................................................................900 amocanebi..........................................................................901 Tavi 27. wredebi..................................................909 27.1. ra aris fizika?.....................................................909 27.2. àmotumbuli~ muxtebi.......................................909 27.3. muSaoba, energia da emZ.......................................910 27.4. denis gamoTvla erTkonturian wredSi.........912 27.5. sxva erTkonturiani wredebi............................914 27.6. or wertils Soris potencialTa sxvaoba.....915 27.7. mravalkonturiani wredebi..............................919 27.8. ampermetri da voltmetri................................925 27.9. RC wredebi.............................................................926 mimoxilva da Sejameba..................................................930 SekiTxvebi........................................................................931 amocanebi..........................................................................933 Tavi 28. magnituri velebi..............................946 28.1. ra aris fizika?.....................................................946 28.2. ra warmoqmnis magnitur vels?.........................947 28.3. -s gansazRvreba.................................................947 28.4. mkveTi velebi: eleqtronis aRmoCena.............951 28.5. mkveTi velebi: holis efeqti............................953 28.6. wriulad moZravi damuxtuli nawilaki.........956 28.7. ciklotroni da sinqrotroni..........................960 28.8. denis gamtar mavTulze moqmedi magnituri Zala...........................................................................962 28.9. brunvis momenti denis konturze...................965 28.10. magnituri dipolis impulsi...........................967 mimoxilva da Sejameba..................................................969 SekiTxvebi........................................................................970 amocanebi..........................................................................971 Tavi 29. deniT gamowveuli magnituri veli.................................................................982 29.1. ra aris fizika?.....................................................982 29.2. deniT gamowveuli magnituri velis gamoTvla.................................................................983 29.3. or paralelur dens Soris moqmedi Zala......989 29.4. amperis kanoni.......................................................991 29.5. solenoidebi da toroidebi.............................995 29.6. denis gamtari koWa, rogorc magnituri dipoli......................................................................998 mimoxilva da Sejameba................................................1000 SekiTxvebi......................................................................1001 amocanebi........................................................................1003 Tavi 30. induqcia da induqciuroba.........1015 30.1 .ra aris fizika?...................................................1015 30.2. ori eqsperimenti................................................1016 30.3. faradeis induqciis kanoni.............................1016 30.4. lencis kanoni......................................................1019 30.5. induqcia da energiis gadacema......................1023 30.6. inducirebuli eleqtruli velebi...............1027 30.7. induqtorebi da induqciuroba.....................1031 30.8. TviTinduqciuroba...........................................1033 30.9. RL wredebi..............................................................1034 30.10. magnitur velSi Senaxuli energia...............1038 30.11. magnituri velis energiis simkvrive..........1039 30.12. urTierTinduqcia...........................................1041 mimoxilva da Sejameba................................................1044 SekiTxvebi......................................................................1045 amocanebi........................................................................1047 Tavi 31. eleqtromagnituri rxevebi da cvladi deni...............................................1060 Tavi 32. maqsvelis gantolebebi: materiis magnetizmi.........................................................1093 31.1. ra aris fizika?...................................................1060 31.2. LLC rxevebi xarisxobrivad...............................1060 31.3. eleqtrul-meqanikuri analogia...................1064 31.4. LLC rxevebi raodenobrivad..............................1064 31.5. milevadi rxevebi RLC wredSi.........................1067 31.6. cvladi deni.........................................................1069 31.7. iZulebiTi rxevebi.............................................1070 31.8. sami martivi wredi.............................................1070 31.9. mimdevrobiTi RLC wredi....................................1077 31.10. simZlavre cvladi denis wredebSi..............1081 31.11. transformatorebi.........................................1084 mimoxilva da Sejameba................................................1088 SekiTxvebi......................................................................1089 amocanebi........................................................................1090 32.1. ra aris fizika?...................................................1093 32.2. gausis kanoni magnituri velebisaTvis.......1094 32.3. aRZruli magnituri velebi............................1096 32.4. gadanacvlebis deni ..........................................1099 32.5. maqsvelis gantolebebi....................................1101 32.6. magnitebi..............................................................1102 32.7. magnetizmi da eleqtronebi............................1103 32.8. magnituri masalebi...........................................1108 32.9. diamagnetizmi.....................................................1108 32.10. paramagnetizmi.................................................1110 32.11. feromagnetizmi................................................1112 mimoxilva da Sejameba................................................1117 kiTxvebi..........................................................................1119 amocanebi........................................................................1121 nawili IV Tavi 33. eleqtromagnituri talRebi......1138 33.1. ra aris fizika?...................................................1138 33.2. maqsvelis cisartyela......................................1138 33.3. moZravi eleqtromagnituri talRa..............1140 33.4. moZravi eleqtromagnituri talRa..............1143 33.5. energiis gadatana da pointingis veqtori..................................................................1146 33.6. gamosxivebis wneva .............................................1149 33.7. polarizacia........................................................1152 33.8. arekvla da gardatexa.......................................1156 33.9. sruli Sida arekvla..........................................1162 33.10. polarizacia arekvliT..................................1164 mimoxilva da Sejameba................................................1165 SekiTxvebi......................................................................1166 amocanebi........................................................................1168 Tavi 34. gamosaxulebebi...............................1183 34.1. ra aris fizika?...................................................1183 34.2. gamosaxulebis ori tipi...................................1185 34.3. brtyeli sarkeebi...............................................1187 34.4. sferuli sarkeebi..............................................1189 34.5. gamosaxuleba sferul sarkeSi.......................1193 34.6. sferuli gardamtexi zedapirebi..................1188 34.7. Txeli linzebi....................................................1196 34.8. optikuri xelsawyoebi......................................1202 34.9. sami mtkicebuleba.............................................1206 mimoxilva da Sejameba................................................1208 SekiTxvebi......................................................................1209 amocanebi........................................................................1211 Tavi 35. interferencia.................................1225 35.1. ra aris fizika?...................................................1225 35.2. sinaTle, rogorc talRa.................................1226 35.3. difraqcia............................................................1231 35.4. iangis interferenciis eqsperimenti...........1231 35.5. Tanmimdevruloba..............................................1236 35.6. daZabuloba ori xvreliT interferenciaSi.................................................1237 35.7. interferencia Txeli fenebidan..................1241 35.8. mixelsonis interferometri.........................1249 mimoxilva da Sejameba................................................1250 SekiTxvebi......................................................................1251 amocanebi........................................................................1253 Tavi 36. difraqcia..........................................1267 36.1. ra aris fizika?...................................................1267 36.2. difraqcia da sinaTlis talRuri buneba...1268 36.3. erTi xvreliT difraqcia: minimumis mdebareobis dadgena..........................................1269 36.4. xarisxobrivi intensivoba erTi xvreliT difraqciaSi..........................................................1273 36.5. raodenobrivi intensivoba erTi xvreliT difraqciaSi..........................................................1274 36.6. wriuli xvreliT difraqcia...........................1277 36.7. ori xvreliT difraqcia..................................1280 36.8. difraqciuli meseri.........................................1283 36.9. mesrebi: dispersia da gadawyvetunarianoba.........................................1288 36.10. rentgenis sxivebis difraqcia.....................1290 mimoxilva da Sejameba................................................1293 kiTxvebi..........................................................................1294 amocanebi........................................................................1295 Tavi 37. fardobiTobis Teoria..................1308 37.1. ra aris fizika?...................................................1308 37.2. postulatebi ......................................................1309 37.3. movlenis gazomva...............................................1310 37.4. erTdroulobis fardobiToba.......................1312 37.5. drois fardobiToba.........................................1313 37.6. sigrZis farTobiToba......................................1319 37.7. lorencis gardaqmna.........................................1322 37.8. lorencis formulebis zogierTi Sedegi..1324 37.9. siCqareebis fardobiToba...............................1326 37.10. dopleris efeqti sinaTlisTvis.................1327 37.11. axali Sexeduleba impulsze.........................1331 37.12. axali Sexeduleba energiaze........................1332 mimoxilva da Sejameba................................................1337 SekiTxvebi......................................................................1339 amocanebi........................................................................1340 1 erTeulTa sistema 1885 wlis ivnisSi GrdiloeT naxevarsferos caze adreuli RamiT Txeli, molurjo-movercxlisfro Rrublebi gamoGnda. momdevno aTwleulebSi aseTi Rrublebi kidev ramdenjerme SeamGnies. ukanasknel wlebSi maT gacilebiT ufro xSirad vxedavT, rac imaze metyvelebs, rom dedamiwis atmosferoSi mniSvnelovani cvlilebebi xdeba. rogor anaTebs mze am Rrublebs daisis Semdeg? pasuxs am TavSi SeityobT 1.1 ra aris fizika? mecniereba sxvadasxva sidideebis gazomvasa da Sedarebas efuZneba. amisTvis garkveuli wesebi gvWirdeba. aseve gvWirdeba eqsperimentebi, romlebiTac am sidideebis erTeulebs davadgenT. fizikis erT-erTi amocana swored am eqsperimentebis dagegmva da ganxorcielebaa. magaliTad, fizikosebi cdiloben zezusti saaTis Seqmnas, raTa nebismieri dro an drois Sualedi zustad iqnas dadgenili. ai, erT-erTi magaliTi imisa, Tu risTvisaa saWiro zusti saaTebi: maT gareSe ver imuSavebda mdebareobis dadgenis msoflio sistema GPS, romelsac udidesi mniSvneloba aqvs msoflio navigaciisTvis. 1.2 sidideebis gazomva fizikis gacnoba imis SeswavliT iwyeba, Tu rogor izomeba sxvadasxva sidideebi, maga liTad: sigrZe, dro, masa, temperatura, wneva da denis Zala. TiToeul fizikur sidides Tavisi zomis erTeuli aqvs, romelsac etalonTan vadarebT. zomis erTeuli sididis raodenobrivad gamoxatvis saSualebaa. magaliTad, sigrZis erTeulia metri (m). etaloni sididis zustad erT erTeuls Seesabameba. SemdgomSi vnaxavT, rom sigrZis erTeulis etaloni, romlis sigrZe zustad erTi metris tolia, aris manZili, 9 10 erTeulTa sistema Tavi 1 romelsac sinaTle gaivlis vakuumSi drois garkveul monakveTSi. zomis erTeuli da misi eta loni CvenTvis xelmisawvdomi nebismieri gziT SegviZlia ganvsazRvroT. Tumca, es ise unda gavakeToT, rom am erTeulis mizanSewonilebasa da moxerxebulobas msoflios yvela mecnie ri daeTanxmos. mas Semdeg, rac etaloni, magaliTad, sigrZis etaloni ganvsazRvreT, unda SevimuSavoT procedura, romliTac SesaZlebeli iqneba etalonis saSualebiT nebismieri sigrZis gamoxatva, wyalbadis atomis diametri iqneba es, skeitbordis RerZis sigrZe Tu manZili romelime varskvlavamde. sigrZis gazomvis erT-erT aseT proceduras uzrunvelyofs saxazavi, romelic sizustiT sigrZis etalons uaxlovdeba. Tumca Cveni gazomvebis didi nawili advili Sesasrulebeli ar iqneba. SeuZlebelia, magaliTad, atomis radiusisa da or varskvlavs Soris manZilis saxazaviT gazomva. fizikuri sidideebis didi raodenobis gamo maTi organizeba sakmaod rTulia. sabednierod, yvela maTgani ar aris damoukidebeli. magaliTad, siCqare manZilisa da drois fardobaa. amitom, saerTaSoriso SeTanxmebiT, virCevT mxolod mcire raodenobis fizikur sidideebs, magaliTad, manZilsa da dros, da ganvsazRvravT etalonebs am sidideebisTvis. Semdeg am sabaziso sidideebisa da maTi etalonebis (e.w. sabaziso etalonebis) safuZvelze ganvsazRvravT sxva fizikur sidideebsac. sabaziso etalonebi unda iyos xelmisawvdomi da ucvleli. Tu manZilis etalons gansazRvravT, rogorc manZils adamianis cxvirsa da saCvenebeli TiTis wvers Soris, es etaloni namdvilad xelmisawvdomi, Tumca sxvadasxva adamianisTvis sxvadasxvanairi iqneba. mecnie reba maqsimalur sizustes moiTxovs, magram gansakuTrebuli yuradReba unda davuTmoT ucvlelobasac. amis Semdeg unda Seiqmnas sabaziso etalonis aslebi, romlebic yvela msurveli saTvis xelmisawvdomi iqneba. 1.3 erTeulebis saerTaSoriso sistema 1971 wels zoma-wonis me-14 saerTaSoriso konferenciaze dadginda Svidi sabaziso sidide, romlebic erTeulTa saerTaSoriso sistemis (SemoklebiT SI) safuZvels Seadgens. 1.1 cxrilSi naCvenebia sami ZiriTadi sididis – sigrZis, masisa da drois erTeulebi, romlebsac wignis pirvel TavebSi gamoviyenebT. SI sistemis bevri Sedgenili erTeuli am sabaziso erTeulebiT ganisazRvreba. magaliTad, simZlavris erTeuli vati (vt) ganisazRvreba masis, sigrZis da drois erTeulebiT, rasac me-7 TavSi gavecnobiT. 1 vati = 1 vt = 1kg·m2/w3 cxrili 1.1 SI sistemis ZiriTadi sidideebis erTeulebi sidide manZili (1.1) simboloebis es jgufi ase ikiTxeba: kilogram metr kvadrati wam kubze. Zalian didi da Zalian mcire sidi erTeulis erTeulis saxeli simbolo metri m vis meTods, romelic 10-is xarisxebs efuZneba. am meTodis mixedviT, dro wami wm masa kilogrami kg deebis gamosaxatavad viyenebT ricxvebis eqsponencialuri 3560000000 m = 3,56 × 109 m gamoxat- (1.2) xolo 0,000000492 m = 4,92 × 10-7 m (1.3) 11 1.4 erTeulTa Secvla am meTods kompiuterebSi zogjer sxvagvari saxe aqvs, magaliTad 3,56E9 da 4,92E–7, sadac E àTis xarisxs~ niSnavs. zog kalkulatorze E-c ki cxrili SI sistemis erTeulebis prefiqsebi gamotovebulia. Zalian did da Zalian patara ricxvebze muSao- xarisxi prefiqsi a simbolo 24 10 iota- YY 1021 zeta- Z 10 eqsa- E bis kidev ufro gasamartiveblad gamoiyeneba prefiqsebi, romlebic 1-2 cxrilSia mocemuli. ro- 18 gorc xedavT, TiToeuli prefiqsi 10-is romelime xarisxs warmoadgens. SI sistemis erTeulisTvis prefiqsis mierTeba am erTeulis Sesabamis sidideze gamravlebas gulisxmobs. magaliTad, denis simZlavris konkretuli sidide SegviZlia ase warmovadginoT: 1,27×109 vati = 1,27 gigavati = 1,27 gvt (1.5) 10 peta- P 1012 tera- T 109 giga- G (g) 106 mega- M (mg) 103 kilo- k (k) 102 heqto- h 1 deka- da 10–1 deci- d (p) 10–2 santi- c (s) 10–3 mili- m (m) 10–6 mikro- μ (mk) 10 zogierT prefiqss albaT ukve kargad icnobT iseTi erTeulebis saSualebiT, rogorebicaa milimetri, santimetri, kilogrami da megabaiti. a 1.4 15 10 (1.4) xolo drois konkretuli monakveTi – ase: 2,35×10-9 wami = 2,35 nanowami = 2,35 nwm 1.2 –9 nano- n (n) 10–12 piko- p (p) 10–15 femto- f 10–18 ato- a 10–21 zepto- z 10–24 ioqto- y muqi SriftiT mocemulia yvelaze xSirad gamoyenebuli prefiqsebi da maTi simboloebi qarTulad erTeulTa Secvla xSirad gvWirdeba im erTeulis Secvla, romliTac gamoxatulia esa Tu is konkretuli fizikuri sidide. amas jaWvuri gadayvanis meTodiT vakeTebT. am meTodis mixedviT, sawyis sidides vamravlebT gadayvanis koeficientze, anu sidideTa fardobaze, romelic erTis tolia. magaliTad, 1 wT da 60 wm drois erTi da igive monakveTebia, amitom 1 wT = 1 60 wm da 60 wm = 1. 1 wT amgvarad, wiladebi 1wT/60wm da 60wm/1wT SeiZleba gadayvanis koeficientebad gamoviyenoT. es tolobebi araviTar SemTxvevaSi ar unda davweroT ase: 1/60 = 1 an 60 = 1. yvela ricxvs Sesabamis erTeuli unda eweros. imis gamo, rom nebismieri sididis erTze gamravleba Sedegs ar cvlis, gadayvanis koeficientebis gamoyeneba SegviZlia nebismier, CvenTvis saWiro dros. jaWvuri gadayvanis dros koeficientebi arasasurveli erTeulebis gamosaricxad gamoiyeneba. magaliTad, 2 wuTis wamebSi gadayvanisTvis: 2 wT = (2 wT) (1) = (2 wT) 60 wm 1 wT = 120 wm (1.6) Tu gadayvanis koeficientis Semotanis Sedegad arasasurveli erTeuli ar gabaTilda, maSin koeficienti unda SecvaloT da xelaxla scadoT. gadayvanis dros erTeulebi imave algebrul kanonebs emorCileba, rasac cvladebi da ricxvebi. 12 Tavi 1 erTeulTa sistema danarTSi mocemulia gadayvanis koeficientebi SI sistemasa da erTeulTa sxva sistemebs, maT Soris, aSS-Si moqmed erTeulTa sistemas Soris. Tumca es koeficientebi mocemulia ara wiladis, aramed tolobis saxiT, magaliTad, `1 wT = 60 wm~. qvemoT moyvanili amocana aseTi wiladis miRebis magaliTia. amocanis amoxsnis nimuSi 1.1 Zv.w. 490 wels, saberZneTis sparseTze amoxsna: jaWvuri gadayvanis meTodis mi gamarjvebis Sesatyobineblad, fidipide ma xedviT, gamoviyenoT iseTi gadayvanis koe raTonidan aTenamde sirbiliT mivida. misi ficientebi, rom arasasurveli erTeulebi siCqare saaTSi daaxloebiT 23 raidi iyo. ra gabaTildes: idi manZilis ZvelberZnuli erTeulia, iseve, rogorc stadiumi da pleTroni. 1 raidSi 23 raidi/sT = 4 stadiumia, xolo 1 stadiumSi _ 6 pleTroni. Tanamedrove erTeulebSi 1 pleTroni 30,8 m-is tolia. gamoTvale fidipides siC- 23 raidi h × 4 stadiumi 1 raidi 30,8 m 1 pleTroni 6 pleTroni 1 stadiumi 1 km 1000 m 1h 3600 w = 4,7227 × 10–3 km/wm ≈ 4,7 × 10–3 km/wm qare km/wm-ebSi. (pasuxi) amocanis amoxsnis nimuSi 1.2 krani axlad daWerili qaSayis asawyavi amoxsna: D danarTidan viciT, rom 1 litri = Zveli britanuli erTeulia. 1 krani 170,464 1000 sm3. imisTvis, rom kuburi santimetrebi litr Tevzs anu daaxloebiT 750 qaSays ud kubur kovidoebSi gadaviyvanoT, santime- rida. davuSvaT, britanelTa gemi, romlis trebisa da kovidoebis fardoba kubSi unda gembanzec 1255 krani Tevzia, saudis ara- aviyvanoT. amgvarad, miviRebT: beTis sabaJoze dgas, sadac Tavisi nadavli kubur kovidoebSi unda warmoadginos. kovido sigrZis arabuli erTeulia, 1 kovido = 48,26 sm. ra unda daweron britanelebma sabaJo deklaraciaSi? 1255 krani = (1255 krani) 170.474 l 1 krani 1000 sm3 1l 1 kovido 48.26 sm 3 = 1.903 × 103 kovido 3. (pasuxi) amocanis amoxsnis taqtika taqtika 1: mniSvnelovani cifrebi da aTwiladebi Tu 1.1 amocanaSi pasuxi damrgvalebis ga- mxolod minimaluri raodenobaa mocemuli reSe gamoiTvale, maSin Sens kalkulatorze (Tumca zogjer SenarCunebulia erTi damate- gamoCndeboda ricxvi 4,72266666667×10-3. ric biTi mniSvnelovani cifri). roca ukiduresi xvis aseTi sizuste gamousadegaria, amitom marcxena mosacilebeli cifri 5-is tolia an pasuxi 4,7×10 km/wm-mde davamrgvaleT. ase 5-ze metia, ukanaskneli SenarCunebuli cifri rom, siCqare 23 raidi/wm SI sistemaSi gaday- erTiT izrdeba, winaaRmdeg SemTxvevaSi ki vanisas ori niSniT gamoixateba, romlebsac ucvleli rCeba. magaliTad, 11,3516 damrgval- mniSvnelovani cifrebi ewodeba. ase rom, deba, rogorc 11,4, xolo 11,3279 – rogorc Cveni pasuxi or mniSvnelovan ciframde davamrgvaleT. am wignSi gamoTvlebis Sedegi xSirad 11,3 (amocanebis pasuxebSi tolobis niSnad viyenebT “=” da ara “≈” simbolos, maSinac ki, damrgvalebulia da mniSvnelovani cifrebis roca pasuxi damrgvalebulia). -3 1.5 sigrZe rodesac amocanaSi mocemulia ricxvi sxvagan ar gaakeToT. 3,15 an 3,15×10 , mniSvnelovani cifrebis ra- araviTar SemTxvevaSi ar unda agverios odenoba aSkaraa, magram rogoria es raode- mniSvnelovani cifrebi da aTwiladebi. 3 noba ricxvSi 3000? aris Tu ara es raodenoba ganvixiloT sidideebi 35,6 mm; 3,56 mm da erTis toli (3×103)? an iqneb aq oTxi mniS- 0,00356 mm. samive ricxvSi mniSvnelovani ci- vnelovani cifria (3,000×10 )? am wignSi da- frebis raodenoba samis tolia, magram mZi- vuSviT, rom aseT ricxvebSi yvela nuli mniS- mis Semdeg arsebuli niSnebis raodenoba, Se- vnelovania, Tumca girCevT, aseTi daSveba sabamisad, erTs, ors da xuTs udris. 3 1.5 sigrZe 1792 wels safrangeTis axlad gamocxadebul respublikaSi SemoiRes erTeulTa axali sistema. am sistemis qvakuTxedi iyo metri, romelic ganisazRvra, rogorc CrdiloeT polusidan ekvatoramde manZilis erTi meaTedmemilionedi. mogvianebiT, praqtikuli mosazrebebis gamo, es etaloni Seicvala da metri ganisazRvra, rogorc manZili or xazs Soris, romlebic amoWrilia platinsa da iridi- cxrili 1.3 umis Senadnobisgan damzadebul zogierTi miaxloebiTi manZili Zelakze. es metris etaloni safrangeTSi, zoma-wonis saerTaSoriso biuroSi inaxeba. misi sidide sigrZe metrebSi manZili pirvel galaqtikebamde 2×1026 manZili andromedas galaqtikamde 2×1022 manZili uaxloes varskvlav proqsima centavramde 4×1016 manZili plutonamde 6×1012 ufro xelmisawvdomi etalone- dedamiwis radiusi 6×106 bi, ase rom, yvela Tanamedrove mwverval everestis simaRle 9×103 sazomi am gverdis sisqe 1×10-4 tipuri virusis sigrZe 1×10-8 wyalbadis atomis radiusi 5×10-11 protonis radiusi 1×10-15 zusti aslebi mTeli msoflios standartizaciis riebSi anu laborato- daagzavnes.am meoradi aslebis etalonebis safuZvelze Seqmnes sxva, kidev xelsawyo grZeli da rTuli jaWviT metris etalons ukavSirdeba. mecnierebisa da teqnikis ganviTareba TandaTan ufro zust etalons iTxovda. 1960 wels metrisTvis SemuSavda axali, sinaTlis talRis sigrZeze dafuZnebuli etaloni. am etalonis mixedviT, metri gazganmuxtvis kameraSi kripton-86-is izotopis mier gamosxivebuli narinjisfer-wiTeli talRis sirZeze zustad 1650763,73-jer metia. es rTuli ricxvi ise iqna arCeuli, rom axali etaloni ZvelTan maqsimalurad axlos yofiliyo. 1983 wels sizusteze moTxovnam iseT dones miaRwia, rom mas kripton-86-is etalonic veRar akmayofilebda. amitom imave wels mecnierebma Zalian Tamami nabiji gadadges. metri ganisazRvra, rogorc sinaTlis mier ganvlili manZili drois konkretul monakveTSi. zomisa da wonis me-17 saerTaSoriso konferenciaze damtkicda: `metri aris 1/299792458 wamSi sinaTlis mier vakuumSi ganvlili manZili.~ 13 14 Tavi 1 erTeulTa sistema drois es monakveTi ise iqna arCeuli, rom sinaTlis siCqared miCneuli iyo: C = 299 792 458 m/wm. sinaTlis siCqaris gazomva sakmaod didi sizustiTaa SesaZlebeli, amitom sinaTlis siCqaris gansazRvrul sidided miCneva da misi gamoyeneba metris gansazRvrisTvis sakmaod gonivruli gadawyvetileba iyo. 1.3 cxrilSi mocemulia sxvadasxva manZili da sigrZei, samyaros saplanetaTaSoriso sivrciT dawyebuli da umciresi sagnebiT damTavrebuli. amocanis amoxsnis taqtika taqtika 2 : sididis TAanrigi sididis Tanrigi aris aTis xarisxi, ro gis mixedviT. Cvens SemTxvevaSi A sididis desac ricxvi mocemulia eqsponencialu uaxloesi Tanrigia 4, xolo B-si – 5. aseTi ri gamoxatvis meTodiT. magaliTad, Tu Sefaseba miRebulia, rodesac gamoTvlebis A=2,3×104 da B=7,8×104, maSin A da B meoTxe dros zusti monacemebi cnobili ar aris an Tanrigis ricxvebia. Znelad sapovnelia. magaliTi mocemulia amo inJinrebi da mecnierebi gamoTvlebis canis amoxsnis 1.3 nimuSSi. Sedegebs xSirad afaseben uaxloesi Tanri- amocanis amoxsnis nimuSi msoflios udidesi Zafis burTis radiu- 1.3 vebs, iqneba: si 2 m-ia. risi tolia burTSi Zafis mTliani V = (ganivkveTis farTobi)× (sigrZe)=d 2 L sigrZe sididis uaxloesi Tanrigis TvalsazrisiT? amoxsna: ra Tqma unda, SegviZlia bur- es sidide daaxloebiT burTis moculobis tolia, romelic 4/3πR3 formuliT Ti davSaloT da ise gavigoT Zafis sigrZe, gamoiTvleba, Tumca, vinaidan π daaxloebiT magram es did Zalisxmevas moiTxovs da Tan sams udris, es gamosaxuleba SegviZlia ga- burTis Semqmnelsac gavanawyenebT. vinaidan vamartivoT da davweroT 4R3. ase rom, mivi- mxolod uaxloesi Tanrigi gvWirdeba, ami- RebT: d 2L = 4R3 tom nebismieri saWiro sididis daaxloebiT gamoTvla SegviZlia. davuSvaT, burTi sferuli formisaa da misi radiusi R = 2 m. burTSi moTavsebuli Zafi ar aris mWidrod Sekruli (e.i. Zafebs an L= 4(2 m)3 4 R3 = d2 (4 × 10–3 m)2 = 2 × 106 m ≈ 106 m = 103 km. (pasuxi) Soris gvaqvs uricxvi RriWo). es RriWoebi rom gaviTvaliswinoT, davuSvaT, rom Zafis (rogorc xedavT, aseTi martivi gamoTv- ganivkveTi kvadratia da misi gverdis sigrZe lisTvis kalkulatori ar gvWirdeba). uax- d = 4 mm ganivkveTis d 2 farTobiT da L sigrZ- loesi Tanrigis mixedviT, burTi daaxloe- iT, mTliani moculoba, romelsac Zafi ika- biT 1000 km sigrZis Zafs Seicavs! 1.6 dro dros ori aspeqti aqvs. samoqalaqo da, zogjer, mecnieruli TvalsazrisiT, drois codna imisTvis gvWirdeba, rom movlenebi garkveuli TanmimdevrobiT davalagoT. seriozuli sa- 1.6 dro mecniero muSaobisas gvainteresebs, ramden xans grZeldeba konkretuli movlena. maSasadame, drois nebismieri etaloni or SekiTxvas unda pasuxobdes: `rodis moxda?~ da `ra drois manZilze?~. 1.4 cxrilSi drois zogierTi monakveTia mocemuli. cxrili 1.4 drois zogierTi miaxloebiTi monakveTi drois Sualedi sidide a wamebSi protonis sicocxlis xangrZlivoba (savaraudo) 1×1039 samyaros asaki 5×1017 xeofsis piramidis asaki 1×1011 adamianis sicocxlis saSualo xangrZlivoba 2×109 dRis xangrZlivoba 9×104 dro adamianis or guliscemas Soris 8×10-1 miu-mezonis sicocxlis xangrZlivoba 2×10-6 sinaTlis umoklesi impulsi 1×10-16 yvelaze aramyari nawilakis sicocxlis xangrZlivoba 1×10-23 plankis droa 1×10-43 es aris dro didi afeTqebis Semdeg, romlidan moyolebuli SesaZlebelia fizikis CvenTvis cnobili kanonebis gamoyeneba. nebismieri ganmeorebadi movlena SeiZleba miCneul iqnas drois etalonad. magaliTad, saukuneebis manZilze dRis xangrZlivobis ganmsazRvrelad dedamiwis brunvis princips iyenebdnen. 1.1 suraTze naCvenebia dedamiwis brunvis principze Seqmnili kvarcis saaTi. kvarcis saa Ti, romelSic kvarcis rgoli mudmivad irxeva, dedamiwis brunvis mixedviTaa dagraduirebuli. aseTi saaTebi la boratoriebSi gamoiyeneba drois gasazomad. Tumca, maTi sizuste ar aris iseTi, rogorsac Tanamedrove samecnie ro da sainJinro teqnologiebi moiTxovs. drois ukeTesi etalonis dadgenisTvis Seiqmna ato muri saaTi. is moTavsebulia amerikis kolorados Statis qalaq bulderis etalonebisa da teqnologiebis nacionalur institutSi (NIST) da miCneulia universaluri globaluri drois (UTC) etalonad aSS-Si. misi droiTi signalebi ismis mokletalRovan radiosadgurebSi (WWV da WWVH) da telefonSi (303-499-7111). drois signalebi (da drosTan dakavSirebuli sxva informacia) mocemulia, agreTve, aSS-is sazRvao observatoriis veb-saitze http:// tycho.usno.navy.mil/time.html (saaTis zustad gasasworeblad unda gaiTvaliswino dro, romelic am signalebis Cvenamde mosaRwevad aris saWiro). 1.2 suraTze naCvenebia 4 wlis ganmavlobaSi dedamiwaze sur. 1.1 rodesac 1792 wels metruli sistema SemoiRes, saaTi Tavidan gansazRvres, ise, rom dReSi 10 saaTi yofiliyo. es idea xalxma Znelad aRiqva, amitom am saaTis Semqmneli Wkvianurad moiqca da patara ciferblatic daamata, romelic, Cveulebriv, 12-saaTian dros aCvenebda. aCvenebs Tu ara es ori ciferblati erTsa da imave dros? 15 erTeulTa sistema Tavi 1 dRis xangrZlivobis cvalebadoba ceziumis atomuri saaTis mixedviT. aq naCvenebi cvalebadoba sezonuri da ganmeorebadia. mas mTvaris zemoqmedeba da Zlieri qarebi iwvevs. 1967 wels zoma-wonis me-13 konferenciaze ceziumis saaTze dayrdnobiT damtkicda wamis etaloni: èrTi wami aris dro, romelic sWirdeba cezium-133-is atomis mier gamosxivebuli sinaTlis talRas 9 192 631 770 rxevis Sesasruleblad.~ atomuri saaTebi Zalian stabiluria. ori aseTi saaTis maCvenebels Soris sxvaoba 1 wamze meti rom gaxdes, amisaTvis daaxloebiT 6000 welia saWiro. Tumca es sizustec ki uferuldeba Tanamedrove saaTebis sizustesTan SedarebiT – 1 wm cdomileba 1×1018 wm-Si (anu daaxloebiT 3×1010 weliwadSi). sur. 1.2 4 wlis ganmavlobaSi dRis xangrZlivobis cvlileba. aRsaniSnavia, rom vertikaluri RerZis sigrZe mxolod 3 mwm-ia (0,003 wm). gansxvaveba dRis 24-saaTian da realur xangrZlivobebs Soris (mwm) 16 amocanis amoxsnis nimuSi 1.4 am Tavis dasawyisSi fotoze gamosaxu- a.S. meTani atmosferoSi did simaRleze adis li Rrublebi pirvelad 1883 wels kunZul da qimiur cvlilebas ganicdis, rac mezosf- krakatauze samxreT-aR- eruli Rrublebis formirebisTvis saWiro mosavleTiT, iavasTan axlos) vulkanis amo- wylis molekulebisa da yinulis nawilakebis frqvevis Semdeg gamoCnda. amofrqveva imde- raodenobis zrdas iwvevs. (wynari okeanis nad Zlieri iyo, rom ferfli mezosferos mezosferuli Rrublebi mzis Casvlis (atmosferos civi fena stratosferos ze- Semdeg moCans, vinaidan atmosferos mziT viT) miswvda. iq dagrovili wyali ferflTan ganaTebul zeda fenebSia. dRisiT maT ver erTad gaiyina da gaCnda nawilakebi, romleb- vxedavT, radgan atmosferos qveda fene- mac pirveli aseTi Rrublebi Seqmna. mas Sem- bi ufro mkveTradaa ganaTebuli da maTi deg Rrublebi xSirad Cndeba ara vulkanebis amofrqvevis, aramed atmosferoSi meTanis raodenobis zrdis gamo, rac, Tavis mxriv, gamowveulia mrewvelobis A da B wertilebSi dacemuli mzis ukanaskneli sxivebi Rrublebi ganviTarebiT, nagavsayrelebiT, gamonabolqvi gazebiT da sur. 1.3. mzis ukanaskneli sxivi A wertils ecema. mas Semdeg, rac mze θ kuTxiT mobrundeba, sxivi damkvirveblidan H simaRleze myof Rrubels ecema. meti sicxadisaTvis suraTze H simaRle H da θ kuTxe gadidebulia. dedamiwis centri 17 1.7 masa garCeva SeuZlebelia. Tu Rrublebi mzis CasgauCinarda, ra simaRlezea isini dedamiwis θ kuTxe aris, agreTve, kuTxe dedamiwis A da B wertilebSi dedamiwis r radiusebs zedapiridan? Soris. suraTze mocemulia marTkuTxa sam- vlidan 38 wuTSi gamoCnda da Semdeg swrafad kuTxedi, romlis kaTetia r, xolo hipot- amoxsna: 1.3 suraTze naCvenebia damkvir vebeli, romelic dedamiwis zedapiris A enuza – r+H. trigonometriidan viciT, rom kuTxis kosinusi cos θ = wertilSi imyofeba da mezosferuli Rrublebidan H manZiliT aris daSorebuli. mzis ukanaskneli sxivebi A wertilSi ecema. mzis r r+H (1-7) C danarTSi vxedavT, rom dedamiwis ra- Casvlidan 38 wuTis Semdeg sxivebi, romlebic diusi r = 6,37×106 m. CavsvaT radiusisa da mezosferul Rrublebs anaTebs, dedamiwis kuTxis es mniSvnelobebi 1.7 formulaSi. mi- zedapirs B wertilSi ecema. viRebT cos 9,5° = 1.3 suraTze mzis sxivebis am or traeqtorias Soris kuTxea θ. es is kuTxea, romliTac mze dedamiwis mimarT gadaadgilda 38 6.37 × 10 6 m (6.37 × 10 6 m) + H saidanac H = 8,86 × 104 m ≈ 89 km wuTis ganmavlobaSi. sruli dRis manZilze (pasuxi) (daaxloebiT 24sT), mze dedamiwis mimarT gadaadgildeba 360°-iT, maSasadame t = 38 wuT Rrublebis gamoCenis gaxSireba imaze metyvelebs, rom ukanaskneli aTwleulebis Si mze gadaadgildeba kuTxiT manZilze dedamiwis zedapirze meTanis ra- θ = (38 wT) 1.7 1 sT 360° 60 wT 24 sT = 9,5°. odenobis zrdam mezosferozec ki moaxdina gavlena. masa kilogramis etaloni 1.5 cxrili SI sistemis masis etaloni – platina- zogierTi miaxloebiTi masebi iridiumis cilindri (sur. 1.4) inaxeba sxeuli masa kilogramebSi CvenTvis cnobili samyaro 1×1053 Cveni galaqtika 2×1041 mze 2×1030 mTvare 7×1022 asteroidi erosi 5×1015 patara mTa 1×1012 saokeano laineri 7×107 spilo 5×103 yurZeni 3×10-3 mtvris nawilaki 7×10-10 penicilinis molekula 5×10-17 uranis atomi 4×10-25 protoni 2×10-27 eleqtroni 9×10-31 parizis zoma-wonis saerTaSoriso biuroSi da sayovelTao SeTanxmebiT aRiarebulia 1 kilogrami masis etalonad. masis etalonis zusti aslebi dagzavnilia sxvadasxva qveynebis standartizaciis laboratoriebSi. nebismieri sxeulis masa am etalonTan SedarebiT ganisazRvreba. 1.5 cxrilSi mocemulia zogierTi sxeulis masa kilogramebSi. kilogramis etalonis amerikuli as li inaxeba NIST-is sacavSi. weliwadSi erTxel mas sxva aslebis Sesamowmeblad iyeneben. 1889 wlis Semdeg etaloni orjer waiRes safrangeTSi pirvelad etalonTan Sesadareblad. 18 erTeulTa sistema Tavi 1 sur. 1.4. 1 kg masis saerTaSoriso etaloni – 3,9 sm simaRlisa da diametris platina-iridiumis cilindri. masis meore etaloni atomebis masebis erTmaneTTan Sedareba ufro zustad SeiZleba, vidre kilogramis etalonTan. amis gamo arsebobs masis meore etaloni. es aris naxSirbad-12-is atomi, romelic saerTaSoriso SeTanxmebiT miCneulia 12 atomuri masis erTeulad (u). kavSiri or etalons Soris aseTia: 1 u = 1,66 054 02×10-27 kg, sadac aTwiladis bolo ori cifris cdomileba ±10-is tolia. mecnierebs SeuZliaT maRali sizustiT gansazRvron sxva atomebis masa naxSirbad-12-is atomis masis saSualebiT. samwuxarod, aseTi sizuste dRemde miuRwevelia masis ufro gavrcelebuli erTeulebis, maga liTad, kilogramis SemTxvevaSi. mimoxilva da Sejameba sidideebi fizikaSi. fizika efuZneba erTeulTa Secvla. erTeulTa Secv- fizikuri sidideebis gazomvas. garkveuli la xdeba jaWvuri gadayvanis meTodiT. am fizikuri sidideebi miCneulia sabazisod SemTxvevaSi, sawyisi monacemebi mravldeba (magaliTad, sigrZe, dro da masa), gansazRv gadayvanis specialur koeficientebze da rulia yvela maTganis etaloni da mocemu- saTanado algebruli gamoTvlebis Sedegad lia zomis erTeuli (magaliTad metri, wami miiReba sasurveli erTeulebi. da kilogrami). sxva fizikuri sidideebi ganisazRvreba sabaziso sidideebis, maTi eta lonebisa da erTeulebis meSveobiT. SI sistemis erTeulebi. am wignSi ZiriTadad gamoyenebulia erTeulTa saer- (SI) sistemis erTeulebi. 1.1 TaSoriso sigrZe. metri gansazRvrulia, rogorc sinaTlis mier ganvlili manZili drois zustad dadgenil SualedSi. dro. wami gansazRvrulia atomuri (cezium-133) wyaros mier gamosxivebuli sidideebi sinaTlis talRis rxevebis mixedviT. atomur gamoiyeneba pirvel sam TavSi. sabaziso saaTze naCvenebi zusti dro mTels msofli- sideebisTvis oSi igzavneba radiosignalebis saxiT. cxrilSi mocemuli fizikuri saerTaSoriso SeTanxmebiT dadgenilia xelmisawvdomi da ucvleli et- masa. kilogrami gansazRvrulia pariz- alonebi. es etalonebi gamoiyeneba rogorc Si Senaxuli platina-iridiumis etalonis ZiriTadi, aseve meoradi sidideebis gazom- mixedviT. atomuri masStabis gazomvebis vis dros. 1.2 cxrilSi mocemuli mecnieruli Casatareblad gamoiyeneba naxSirbad-12-is Canawerebi da prefiqsebi gamoiyeneba gazom- atomis safuZvelze gansazRvruli atomuri vis Sedegis Caweris gasamartiveblad. masa. amocanebi – amoxsnebi mocemulia Student Solutions Manual-Si. – amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday ILW – Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday. • – ••• – wertilebis raodenoba amocanis sirTules gviCvenebs. SSM WWW 19 Tavi 1-5 •1 sigrZe doRis dros cxenebma 4 farlongi sigrZis trasa unda gairbinon. ra sigrZisaa trasa a) rodebSi da b) CeinebSi? (1 farlon- sm-iT gadaadgilda, risi tolia es gadaadgileba a) pikebSi da b) pointebSi? •5 dedamiwis sferos radiusi daax- gi = 201,168 m, 1 rodi = 5,0292 m da 1 Ceini = loebiT 6,37×106 metria. ra iqneba misi: 20,117 m) SSM WWW. •2 grai Zveli inglisuri sigrZis erTeulia da lainis 1/10-s udris. lainic Zveli a) garSemoweriloba kilometrebSi? b) far- inglisuri sigrZis erTeulia da udris dui- erTeulebis gadayvana iolad Tobi kvadratul kilometrebSi? g) moculoba kubur kilometrebSi? •• 6 SSM. mis 1/12-s. sagamomcemlo saqmeSi sigrZis er- SeiZleba kalkulatorisa Tu kompiuteris Teulad gamoiyeneba pointi, romelic, Tavis daxmarebiT, magram D danarTSi mocemuli mxriv, duimis 1/72-ia. gadaiyvane 0,5 grai2 gadayvanis cxrilis gamoyeneba mainc unda farTobi kvadratul pointebSi (pointi2). SegveZlos. 1.6 cxrilSi mocemulia mocu- •3 mikrometrs (1 mkm) xSirad mikrons lobis erTeulebis Zveli espanuri sistema. uwodeben. a) ramdeni mikronia 1 km-Si? 2) ram- 1 fanega = 55,501 dm3. cxrilis Sesavsebad, ra deni santimetria 1 mkm-Si? g) ramdeni mikro- ricxvebi (sami mniSvnelovani cifriT) unda nia 1 iardSi ? iqnas Setanili a) kahizis svetSi, * •4 am wignis beWdvisas simboloebs Soris manZili gamoiTvales pointebsa da pikebSi. 12 pointi = 1 pikas, xolo 6 pika = 1 duims**. Tu gverdis koreqtirebis dros simbolo 0,8 cxrili b) fanegas svetSi, g) kvartilas svetSi da d) almudes svetSi? gamosaxe 7 almude e) me dioebSi; v) kahizebSi da z) kubur santimetrebSi (sm3). 1.6 amocana 6 1 kahizi kahizi fanega kvartila almude medio 1 12 48 144 288 1 4 12 24 1 3 6 1 2 1 fanega 1 kvartila 1 almude 1 medio 1 antarqtika 2000km radiusis mqone es erTeuli 1962 wlis kursdamTavrebulis, naxevarwrea (sur. 1.5). misi yinulis safaris oliver rid smut umcrosis simaRlis to- saSualo lia. swored misi sxeuliT gazomes misma amx- •• 7 sisqe 3000 m-ia. ramden kubur santimetr 2000 km 3000 m anagebma xidis sigrZe, Tan gzad 1 smutis sigrZis monakveTebi moniSnes. mas Semdeg, yovel or weliwadSi erTxel studentebi am niSnebs yinuls Seicavs antarqtika (dedamiwis zedapiris ganaaxleben xolme, rogorc wesi, gadat- amozneqiloba ugulebelyofilia). virTuli saavtomobilo moZraobis dros, SSM. •• 8 mdinare Carlzze agebuli harvard- roca policia maT xels ver SeuSlis. (poli- is xidi, romelic miCiganis teqnologiis cia albaT imitom iyo ukmayofilo, rom institutis Senobas studentur sacxovre- smuti SI sistemis erTeuli ar aris, Tumca, belTan akavSirebs, 364,4 smuti sigrZisaa. rogorc Cans, dRes am erTeulis arsebobas * 1 iardi ≈ 0,914 m 1 duimi ≈ 25,4×10–3 m = 2,54 sm ** 20 erTeulTa sistema Tavi 1 isic Seegua). 1.6 suraTze naCvenebia smutebSi (S), uilebSi (W) da zeldebSi (Z) gazomili sami paraleluri xazi. gamoxate 50 smuti a) uilebSi da b) zeldebSi. 0 32 •• 9 212 258 0 60 216 Si S amerikis SeerTebul Stateb- inJiner-hidroteqnikosebi wylis moculobis sazom erTeulad xSirad W Z iyeneben akr-futs. akr-futi ganisazRvreba, rogorc im wylis moculoba, sur. 1.6 amocana 8. romelic 1 akr miwas 1 fut simaRleze dafaravs. kokipirulma wvimam 26km farTobis qalaqi 30 wuTSi 2 duimiT datbora. gamoTvale 2 qalaqSi wylis moculoba akr-futebSi. Tavi 1-6 •10 ILW. dro leqciis xangrZlivoba (50wT) daax- loebiT 1 mikrosaukunea. a) ramdeni wuTia procentuli sxvaoba = •11 mikrosaukuneSi? b) ipove procentuli sxva oba Semdegi formulis gamoyenebiT: faqtobrivi – miaxloebiTi fortnaiti (fourteen nights – ToTx- meti Rame) inglisuri drois sazomi erTeulia da 2 kviras udris. drois es monakveTi, faqtobrivi 100, gaizarda. gamoTvale misi zrdis tempi mikrometr/wamebSi. •13 laboratoriaSi xuT saaTs amow- Tu mas sasiamovno sazogadoebaSi atareb, meben. ramdenime dRis manZilze, zustad TavisTavad, sasiamovnoa, magram usiamovno SuadRisas, romlis dadgomasac WWV drois kompaniaSi mikrowamebis mtanjvel wyebad signali gvauwyebs, saaTebi cxrilSi moce- iqceva. ramdeni mikrowamia fortnaitSi? mul dros aCvenebs. daalage saaTebi maRa- •12 yvelaze swrafad mzardi mcenare hesperoika uiplei 14 dReSi 3,7 metriT li sizustidan yvelaze dabal sizustemde. daamtkice Seni arCevani. SSM. saaTi kv. orS. samS. oTx. xuT. par. Sab. A B C D E 12:36:40 12:36:56 12:37:12 12:37:27 12:37:44 12:37:59 12:38:14 11:59:59 12:00:02 11:59:57 12:00:07 12:00:02 11:59:56 12:00:03 15:50:45 15:51:43 15:52:41 15:53:39 15:54:37 15:55:35 15:56:33 12:03:59 12:02:52 12:01:45 12:00:38 11:59:31 11:58:24 11:57:17 12:03:59 12:02:49 12:01:54 12:01:52 12:01:32 12:01:22 12:01:12 •14 1883 wlamde amerikis SeerTebuli – 10 saaTisgan, erTi saaTi – 100 wuTisagan, Statebis yvela qalaqs sakuTari, adgilo- xolo erTi wuTi – 100 wamisagan. risi tolia brivi dro hqonda. dRes mogzaurebi saaTebs a) franguli aTobiTi kviris Sefardeba eta- mxolod maSin asworeben, rodesac gansx- lonur kvirasTan? b) franguli aTobiTi wu- vaveba 1 saaTi an metia. daaxloebiT grZedis Tis Sefardeba etalonur wuTTan? ramdeni gradusi unda gaiaro, rom saaTis •16 gasworeba 1 saaTiT mogixdes? (dedamiwa 24 atomur saaTSi 360°-ian bruns asrulebs.) varaudod, •15 drois etalonebi dafuZnebulia saaTze. momavali dafuZnebuli etaloni, iqneba sa- mbrunav safrangeTis revoluciidan 10 wlis neitronul varskvlavebze – pulsarebze. Semdeg safrangeTis xelisuflebam drois zogierTi pulsari, romlis brunvis siC- ZiriTad sazomebad daadgina aTis jeradebi: qare Zalian stabiluria, yoveli Semobrune- erTi kvira Sedgeboda 10 dRisagan, erTi dRe bisas dedamiwaze radiosignals gzavnis. 21 magaliTad, pulsari PSR 1937+21 yovel •21 dedamiwis masa 5,98×1024 kilogramia. 1,55780644887275±3 mwm-Si erTxel Semobrun- dedamiwis Semadgeneli atomebis saSualo deba. a) ramden bruns gaakeTebs es pulsari masaa 40 u. ramdeni atomia dedamiwaSi? •22 7 dReSi? b) ra dro dasWirdeba mas milioni SuSis boTlis damzadebis rekordi vadasxva tempSi muSaobs da sxvadasxva dros 1992 wels niu-jersis Statis qalaq melvilSi damyarda. boTlis moculoba 193 aSS galoni iyo. a) ramdeniT naklebia es moculoba 1 aCvenebs. 1.7 suraTze mocemulia saaTebis milion sm3-ze? b) Tu boTls wyliT aavseben brunis gasakeTeblad? •• 17 sami cifruli saaTi, A, B da C, sx- Cvenebebi (magaliTad, pirvel SemTxvevaSi B aCvenebs 25 wm-s, xolo C – 92 wm-s). Tu A saaTze or movlenas Soris dro 600 wm-ia, ra dro iqneba am movlenebs Soris a) B saaTze da b) C saaTze? g) rodesac A saaTi aCvenebs 400 wm-s, ras aCvenebs B saaTi? d) rodesac C saaTi aCvenebs 15 wm-s, ras aCvenebs B saa- 1,8 g/wT siCqariT, ra dro dasWirdeba boTlis avsebas? 1 kuburi metri moculobis wylis masa 1000 kilogramia. •23 1 kuburi santimetri moculobis rkinis masa 7,87 gramia, xolo rkinis atomis masa – 9,27×10-26 kg. Tu atomebi sferulia da mWidrod ganlagebuli, maSin a) risi tolia Ti? (dros nulovan wertilamde uaryofiTi rkinis atomis moculoba? b) ra manZilia Cveneba mianiWe). gverdigverd ganlagebuli atomebis cen- 312 25.0 125 512 200 142 92.0 290 A (wm) B (wm) C (wm) sur. 1.7 amocana 17. trebs Soris? ••24 erTi kuburi santimetri Rrubeli Seicavs 50-dan 500-mde wylis wveTs, TiToeulis radiusi 10 mkm-ia. a) ramdeni kuburi santimetri wyalia 3 km simaRlis da 1 km ra- dedamiwis brunvis TandaTano- diusis cilindruli formis RrubelSi? b) biT Senelebis gamo yoveli momdevno dRis ramden erTlitrian boTls aavsebs es wya- xangrZlivoba izrdeba: yoveli saukunis li? g) erTi kuburi metri moculobis wylis •••18 ukanaskneli dRe 1 mwm-iT ufro grZelia, masa 1000 kilogramia. ra masa aqvs RrubelSi vidre saukunis dasawyisSi. 20 saukunis man- Semaval wyals? Zilze sul ra droiT gaizarda dRis xangr- damatebiTi amocanebi 25 Zveli inglisuri sabavSvo leqsi am- Zlivoba? Tavi 1-7 •19 masa a) davuSvaT, wylis kuburi santimetris masa zustad 1 gramia. gamoTvale wylis erTi kuburi metris masa kilogramebSi; b) davuSvaT, wyliT savse 5700m3 moculobis rezervuaris daclas 10 saaTi sWirdeba. risi tolia wylis masis Semcirebis siCqare kilogram/wamebSi? •20 oqros kuburi santimetris masa 19,32 gramia. oqro yvelaze rbili liTonia da SeiZleba misi furceliviT gabrtyeleba an boWkosaviT gawelva. a) Tu 27,63 gr oqrosgan gavakeTebT 1 mkm sisqis furcels, risi toli iqneba misi farTobi? b) Tu imave raodenobis oqrosgan gavakeTebT 2,5 mkm radiusis cilindruli formis boWkos, ra iqneba boWkos sigrZe? bobs: `patara mis mafeti Camojda da xaWos Wama daiwyo. oboba movida da gverdiT miujda...~ oboba Camojda ara xaWos gulisTvis, aramed imitom, rom mis mafets Tan hqonda 11 tafeti gamomSrali buzebis maragi. 1 tafeti = 2 peki = 0,5 imperiuli buSeli, xolo 1 imperiuli buSeli = 36,3687 litri (L). risi tolia mis mafetis mwerebis maragi a) pekebSi, b) imperiul buSelebSi da g) litrebSi? 26 atomur molSi 6,02×1023 atomia. daaxloebiT ramdeni moli atomia did Sinaur kataSi? wyalbadis, Jangbadis da naxSirbadis atomebis masebia Sesabamisad 1 u, 16 u da 12 u. 27 Zvelsa da axals, didsa da mcires Soris kontrastis dasanaxad waikiTxe Semdegi: Zveli inglisis soflebSi 1 haidi (100 da 120 22 erTeulTa sistema Tavi 1 akrs Soris) iyo miwis farTobi, romelic erTeulebi gamoiyeneboda. ramdenime maT- daakmayofilebda 1 guTnis mqone 1 ojaxs (1 gani mocemulia 1.7 cxrilSi. cxrilis Ses- akri = 4047 m2). 1 vapenteiki farTobis miwa avsebad ra ricxvebi (meaTasedis sizustiT) daakmayofilebda 100 aseT ojaxs. kvantur unda Caiweros a) veis svetSi, b) Caldronis fizikaSi birTvis ganivi kveTis farTobi svetSi, g) begis svetSi, d) potlis svetSi, izomeba barnebSi, sadac 1 barni = 1×10 m . e) gilis svetSi. 1 begi = 0,1091 m3. Zvel ing- risi tolia 25 vapenteikis fardoba 11 barn- lisur moTxrobaSi jadoqari saZagel siTx- sTan? es 1,5 Caldroni moculobis qvabSi xarSavs. -28 2 28 metruli sistemis SemoRebamde aSS- ras udris es moculoba m3-Si? Si siTxis moculobis sxvadasxva sazomi cxrili 1. 7 amocana 28 vei 1 vei = Caldroni 1 10/9 begi potli gili 40/3 640 120 240 1 Caldroni = 1 begi = 1 potli = 1 gili = 29 turistma manqana inglisSi SeiZina da desertis kovzi raodenobis bulioni. Sem- aSS-Si miemgzavreba. manqanis instruqciaSi deg naTqvamia, rom swor gzaze manqana 40 milSi Reroebi, sanam 0,5 kvarti ar dagrCeba. Re- xelTaTmanebiT daacalkeve WinWris erT galon sawvavs moixmars. turisti ver roebi bulionSi Cayare da daamate 1 magidis mixvda, rom inglisuri galoni amerikulis- kovzi moxarSuli brinji da 1 marilis kovzi gan gansxvavdeba: marili. aduRe 15 wuTis manZilze~. cxril- 1 inglisuri galoni = 4,5459631 litri 1 amerikuli galoni = 3,7853060 litri. 750 milis* manZilze (aSS-Si) a) ramdeni galoni sawvavi dasWirdeba manqanas am turistis azriT? b) ramdeni galoni dasWirdeba manqanas sinamdvileSi? 30 Zveli inglisuri receptebis wignSi mocemulia WinWris supis recepti: moxarSe 1 sauzmis Wiqa, plius 1 Cais Wiqa, plius 1 Zveli inglisuri erTeulebi Si mocemulia Zvel britanul da amerikul sazomebs Soris sxvaobebi. siTxeebisTvis 1 britanuli Cais kovzi = 1 amerikul Cais kovzs. mSrali saWmelisTvis 1 britanuli Cais kovzi = 2 amerikul Cais kovzs, xolo 1 britanuli kvarti = 1 amerikul kvarts. ramdeni a) bulioni, b) WinWris Rero, g) brinji da d) marilia saWiro am receptiT amerikul sazomebSi? amerikuli erTeulebi Cais kovzi = 2 marilis kovzi magidis kovzi = 3 Cais kovzi desertis kovzi = 2 Cais kovzi naxevari Wiqa = 8 magidis kovzi magidis kovzi = 2 desertis kovzi Wiqa = 2 naxevari Wiqa Cais Wiqa = 8 magidis kovzi sauzmis Wiqa = 2 Cais Wiqa 31 Zvel xelnawerSi naTqvamia, rom mefe bels gaaCnda 3 akri mosaxnavi miwa da piru- arturis mmarTvelobis dros miwismflo- tyvisTvis damatebiTi farTobi, 25 perCi * 1 mili = 1,609 km 23 4 perCze. risi toli iyo mTeli farTobi kvadratuli futis SeRebva SeiZleba. a) ga- a) Zvel erTeulebSi – rudzebSi da b) Taname- mosaxe es sidide m2/litrebiT. b) gamosaxe es drove erTeulebSi – m -ebSi? am SemTxvevaSi, sidide SI sistemis erTeulebiT (ixile dan- 1 akri = 40×4 perCi, 1 rudi = 40×1 perCi, arTebi A da D). g) rogori iqneba sawyisi raode- xolo 1 perCi = 16,5 futi**. nobis sididis Sebrunebuli mniSvneloba? 2 * 32 XX saukunis 20-iani wlebis aSS-Si d) ra fizikuri mniSvneloba aqvs mas? gamoiyeneboda ori tipis bareli. vaSlis 36 warmoidgine, rom ekvatorTan axlos bareli 7056 kuburi duimis toli iyo, xolo plaJze wevxar da wammzoms rTav, raTa mzis StoSis*** bareli – 5826 kuburi duimisa. Tu Casvlamde darCenili dro daiTvalo. amis vaWari StoSis 20 barel saqonels mihyidis Semdeg dgebi, ris Sedegadac Seni Tvalebi klients, romelsac hgonia, rom saqonels H=1,70 m simaRleze aRmoCndeba, da kvlav elo- vaSlis barelebSi yidulobs, ra gansxvaveba debi mzis wveris gauCinarebas. Tu gasuli iqneba saqonlis raodenobebs Soris li- dro t = 11,1 wm, risi tolia dedamiwis r ra- trebSi? diusi? (miniSneba: ixile amocanis amoxsnis 33 kaliforniis plaJis qviSis marcvlebi daaxloebiT 50 mkm radiusis sferoebia da siliciumis orJangisgan Sedgeba. 1 m3 mocu- nimuSi 1.4). 37 kordi aris moWrili xe-tyis moculoba da 8 futi sigrZis, 4 futi siganisa da lobis siliciumis orJangis kubis masa 2600 4 futi simaRlis xe-tyis moculobis tolia. kg-ia. ra masis qviSis marcvlebs eqneba zeda- ramdeni kordia 1m3-Si? piris iseTive farTobi (cal-calke sferoebis farTobebis jami), romelic 1 m wibos 38 iaponiis tradiciuli sigrZis sazomi erTeulia keni (1 keni = 1,97 m). risi tolia mqone kubis zedapiris farTobis tolia? a) kvadratuli kenis Sefardeba kvadratul 34 amerikuli Tojinas saxli namdvili metrTan da b) kuburi kenis Sefardeba ku- saxlis 1/12-ia, xolo miniaturuli saxli bur metrTan? risi tolia 5 keni siRrmisa da (kidev ufro patara saxli, romelic Tojinis 3 keni radiusis cilindruli formis wylis saxlSi eteva) namdvili saxlis 1/144-ia. da- WurWlis moculoba g) kubur kenebSi da d) ku vuSvaT, namdvili saxlis (sur. 1.8) sigrZe bur metrebSi? 20 m-ia, sigane – 12 m, simaRle – 6 m, xolo 39 wyrTa uZvelesi sazomi erTeulia da daxrili Weris simaRle – 3 m. risi tolia utoldeba mzomelis idayvidan Sua TiTis a) Tojinas saxlis da b) miniaturuli saxlis wveramde manZils. davuSvaT, rom es sidide moculobebi kubur metrebSi? 43-dan 53 sm-mde icvleba, da warmovidginoT, rom uZveles naxatze asaxulia 9 wyrTa sigrZisa da 2 wyrTa diametris cilindruli boZi. rogori iqneba minimaluri da maqsima 3.0 m luri sidideebi, Sesabamisad, a) cilindris sigrZisa metrebSi, b) cilindris sigrZisa milimetrebSi da g) cilindris moculobisa 6.0 m kubur metrebSi? 20 m 40 wylis erTi molekula (H2O) Seicavs wyalbadis or da Jangbadis erT atoms. wyal sur. 1.8 amocana 34. 35 erT-erTi saxeobis saRebavis aRweraSi weria, rom am saRebavis erTi galoniT 460 * 1 akri = 4040 m2 1 futi ≈ 30,48 sm ** badis atomis masaa daaxloebiT 1 u, xolo Jangbadis atomis masa – daaxloebiT 16 u. a) ra iqneba wylis erTi molekulis masa kilogramebSi? b) wylis ramdeni molekulaa *** StoSi – tanmorCili buCqbalaxa mcenare. isxams wiTeli feris wvrilsa da mJave nayofs. ixmareba sakvebad, saWmlis Sesakazmad. samurabed, wvnisa da sirofis dasamzadeblad (qarTuli enis ganmartebiTi leqsikoni, tomi VII). 24 Tavi 1 erTeulTa sistema dedamiwis wylebSi, Tu maTi mTliani masa minis WurWelSi asxamen, romlis zomebia 1,4×10 kg-ia? 41 zRvaosnobaSi xSirad gamoiyeneba mo 40 sm × 40 sm × 30 sm. WurWeli pirTamde unda culobis sazomi erTeuli – tona. Tumca misi mocemuli zomis boTlebSi. rac ufro didia gamoyeneba garkveul sifrTxiles moiTxovs, boTli, miT ufro iafi jdeba Rvino. mini vinaidan arsebobs tonis, sul cota, sami maluri fasis misaRebad, a) ra zomis da ram- nairsaxeoba: gadaadgilebis tona 7 bare- deni boTli unda iqnas SeZenili? rodesac li moculobis tolia, tvirTis tona – 8 WurWeli gaivseba, ramdeni Rvino darCeba bareli moculobisa, xolo registris tona b) standartul boTlebSi da g) litrebSi. 21 iyos savse. Rvinis SeZena SeiZleba cxrilSi – 20 bareli moculobisa. bareli moculo- 1 standartuli boTli bis kidev erTi sazomi erTeulia da udris 1 magnumi 0,1415 m3. davuSvaT, tvirTis SekveTaSi weria `73 tona M&M kanfeti” da Tqven darwmunebuli = 2 standartuli boTli 1 ieroboami = 4 standartuli boTli 1 reoboami = 6 standartuli boTli xarT, rom SemkveTi `tonaSi~ moculobas gu- 1 maTusala = 8 standartuli boTli lisxmobs (nacvlad wonisa an masisa, rogorc 1 salmanazari = 12 standartuli boTli es me-5 TavSia). Tu SemkveTi sinamdvileSi 1 balTazari gadaadgilebis tonas gulisxmobda, ramden zedmet amerikul buSel kanfets CatvirTav = 16 standartuli boTli = 11,356 l 1 nabuqodonosori = 20 standartuli gemSi SecdomiT, Tu SekveTaSi warmoidgen a) 73 tvirTis tonas da b) 73 registris tonas? (1m3 = 28,378 amerikuli buSeli.) 42 malaiziaSi adgilobrivi sazomiT 28,9 pikuli wonis xari iyide. 1 pikuli udris 100 jins, 1 jini – 16 tahils, 1 tahili – 10 Ciss da 1 Cisi – 10 hunas. 1 huna 0,3779 gramis tolia. Tu am xaris saxlSi wayvanas da ojaxis gancvifrebas ganizraxav, ramdeni kilogrami masa unda Cawero tvirTis deklaraciaSi? 43 drois tradiciuli sazomi erTeu lebi, magaliTad, dRis an weliwadis xan grZlivoba, dafuZnebulia astronomiul dakvirvebebze. Tumca, tibetSi SeiZleba ada mianze dafuZnebuli drois sazomi erTeulis naxvac, sadac dibugi aris CasunTqvaamosunTqvas Soris saSualo dro. gamoTvale dibugebis raodenoba erT dReSi. 44 brZaneba miiRe, rom Seni gemiT 24,5 mili gacuro aRmosavleTisken, sadac mekobreebis gemi CaiZira. Tumca, rodesac im adgilas Seni myvinTavebi gemis nasaxsac ver aRmoaCnen, mixvdebi, rom curvis manZilSi igulisxmeboda 24,5 sazRvao mili da ara Cveulebrivi mili. gamoiyene D danarTSi mocemuli sigrZeebis cxrili, raTa gamoTvalo, ramdeni kilometriT xar daSorebuli mekobreTa gemidan. 45 qorwilebis dros Rvinos Tlili boTli. 46 politikuri partiis yrilobisTvis produqtebis SeZenisas SecdomiT SeukveTe wynari okeanis saSualo zomis xamanwkebi (erT amerikul pintaSi 8-12 cali), atlantis okeanis saSualo zomis xamanwkebis nacvlad (pintaSi 26-38 cali). xamanwkiT savse WurWlis zomebia 1m × 12 sm × 20 sm, xolo amerikuli pinta 0,4732 litris tolia. ramdeni xamanwkiT naklebi SegikveTavs? 47 astronomiuli erTeuli (AU) aris dedamiwasa da mzes Soris saSualo manZili – 1,5 × 108 km. sinaTlis siCqarea 3 × 108 m/wm. gamoxate sinaTlis siCqare astronomiul erTeul/wuTebSi. 48 ra masis wyali movida wvimis dros qalaqSi me-9 amocanaSi? erTi kuburi metri wylis masaa 1×103 kg. 49 am TavSi mocemuli monacemebis gamo yenebiT gansazRvre 1 kg wyalbadis miRebisTvis saWiro wyalbadis atomebis raodenoba. wyalbadis atomis masaa 1 u. 50 Saqris natexis gverdis sigrZea 1 sm. Sen rom gqondes kubis formis yuTi, ro melSic 1 moli Saqris kubebi iqneba, rogori iqneba am kubis gverdis sigrZe? (erTi moli = 6,02 × 1023 erTeuls). 51 dietaze myofma adamianma SeiZleba kviraSi 2,3 kg daiklos. gamoxate masis da- 25 kargvis tempi miligram/wamebSi, TiTqos 56 miwis farTobis gasazomad xSirad dietaze myofi yovel wams daklebul masas gamoiyeneben heqtars, romelic 104 m2-ia. grZnobdes. qvanaxSiris Ria karieri yovelwliurad 75 52 marcvleul/Roris Sefardeba finan heqtar miwas moixmars 26 metris siRrmeze. suri terminia da Rorebis bazarze gamoiye ra moculobis miwa (kubur kilometrebSi) neba. is ukavSirdeba Roris Senaxvis xarjebs gadaiyreba am dros? am Roris gasayidad gamoyvanamde. marcv- Sefardeba erTi amerikuli buSeli marcv- 57 astronomiuli erTeuli (AU) deda miwidan mzemde saSualo manZilis – 92,9 × 106 milis tolia. parseki (pc) aris manZili, romlis toli ferdebisa da 1 sekundis toli leulis sabazro fasTan. amerikuli buSeli kuTxis mqone tolferda samkuTxedis fuZe 35,238 litris tolia. Tu marcvleul/Roris erTeulebSi 1-kilogrami Roris fasis Se- 1 AU-s tolia (sur. 1.9). sinaTlis weliwadi aris manZili, romelsac sinaTle 1 weliwadSi gaivlis 186000 mili/wm siCqariT. gamoxa fardeba 1 litri marcvleulis fasTan? te manZili dedamiwasa da mzes Soris a) par (miniSneba: ixile masaTa cxrili D danarTSi). sekebSi da b) sinaTlis weliwadebSi. leul/Roris Sefardeba ganisazRvreba, rogorc 1460 slagis masis Roris sabazro fasis Sefardeba bazarze 5,7-ia, ra iqneba metrul 53 saxlis Sida kibis erTi safexuris simaRle 19 sm-ia, xolo sigane – 23 sm. in- 1 pc Jinrebi irwmunebian, rom safexurebi ufro 1 pc usafrTxo gaxdeba, Tu maTi sigane 28 sm iqneba. Tu kibis simaRle sul 4,57 m, rogor gaizrdeba mTlianad kibis sigrZe, Tu axali siganis safexurebs gavukeTebT? 54 Zlieri wvimis dros mTis ferdobi, romlis sigane 2,5 kilometria, sigrZe – 0,8 km da sisqe – 2m, mewyris saxiT velze Camocurdeba da Tanabrad vrceldeba 0,4 km × 0,4 km zomebis velze. erTi kuburi metri talaxis masaa 1900 kg. ra masis talaxi Camova 4m2 farTobze? 55 Sen unda moamzado sadili meqsikuri saWmlis 400 moyvarulis TavyrilobisTvis. recepti moiTxovs erT adamianze 2 ialapenos wiwakas, Tumca Sen mxolod habaneros wiwakebi gaqvs. wiwakis sicxare izomeba skovilis sicxaris erTeulebSi (SHU). saSualod, erTi ialapenos wiwakis sicxarea 4000 SHU, xolo habaneros wiwakisa – 300000 SHU. sasurveli sicxaris misaRwevad, ramdeni habaneros wiwakiT unda Secvalo ialapenos wiwakebi 400 adamianisTvis sadilis mozadebis dros? 1 sekundis toli kuTxis mqone tolferda samkuTxedi 58 1 AU a) mikroskopul fizikaSi zogjer gamoiyeneba drois sazomi erTeuli – Seiki. erTi Seiki 10–8 wamis tolia. ra ufro metia – wamSi Seiki Tu weliwadSi wami? b) adamiani daaxloebiT 106 weliwadia arsebobs, samyaro ki 1010 wlisaa. Tu samyaros asaks ganvsazRvravT, rogorc `samyaros 1 dRes~, romelic Sedgeba `samyaros wamebisgan~, zustad ise, rogorc Cveulebrivi dRe Sedgeba Cveulebrivi wamebisgan, maSin ramdeni samyaros wamis win gaCnda adamiani? 59 mzis dabnelebis dros mzes mTlianad mTvare faravs. davuSvaT, mzemde manZili 400jer metia mTvaremde manZilze. a) ipove mzis diametris fardoba mTvaris diametrTan. b) ipove mzis moculobis fardoba mTvaris moculobasTan. g) daiWire patara moneta ise, rom mTvare mTlianad dafaros, da gazome Seni Tvalsawieris kuTxe, romelsac es moneta faravs. am sididesa da dedamiwasa da mTvares Soris manZilze (3,8 × 105 km) dayrdnobiT ipove mTvaris diametri. 2 wrfivi moZraoba xSirad avtrokatastrofis dros erT manqanas meore uknidan ejaxeba. aTwleulebis manZilze inJinrebi da medikosebi cdiloben axsnan, ra iwvevs aseTi Sejaxebisas wina manqanaSi mjdomis kisris dazianebas. gasuli saukunis 70-ian wlebSi daaskvnes, rom dazianebas iwvevs Tavis uecari gadavardna savarZlis zemodan, rodesac manqana mkveTri biZgiT win gadaadgildeba. am aRmoGenis Semdeg manqanebSi GaamontaJes savarZlis Tavmisadeblebi, magram aman Sejaxebebis dros kisris dazianebebis raodenoba didad ver Seamcira. ra iwvevs sinamdvileSi am dazianebebs? pasuxs am TavSi SeityobT 2.1 ra aris fizika? fizikis erT-erTi amocana sxeulis moZraobis Seswavlaa, magaliTad, imisa, Tu ra siCqariT moZraobs da ra manZils gadis is drois mocemul monakveTSi. avtoinJinrebi aqtiurad iyeneben fizikas, vinaidan misi meSveobiT adgenen manqanebis maxasiaTeblebs Sejibrebamde da Sejibris dros. geologebi fizikas iyeneben teqtonuri baqnebis moZraobis Seswavlisa da am gziT miwisZvrebis winaswari prognozirebisaTvis. medikosebs fizika sWirdebaT pacientebis sisxlis mimoqcevis suraTis Sesaqmnelad, rodesac nawilobriv daxSuli arteriis diagnozs svamen. avtomoyvarulebi fizikis saSualebiT igeben, rogor unda Seanelon avtomobili, rodesac radiolokatori gafrTxilebas iZleva. arsebobs uamravi sxva magaliTic. am TavSi SeviswavliT sxeulebis (avtomobilebis, teqtonuri baqnebis, sisxlis ujredisa an sxva sxeulebis) moZraobas wrfis gaswvriv. aseT moZraobas erTganzomilebiani anu wrfivi moZ- raoba ewodeba. 2.2 moZraoba samyaroSi yvelaferi moZraobs. erTi SexedviT uZravi sagnebic ki, magaliTad, gza, dedamiwis brunvasTan erTad moZraobs. dedamiwa mzis garSemo moZraobs, mze _ irmis naxtomis galaqtikis garSemo, da TviT es galaqtikac moZraobs sxva galaqtikebis mimarT. moZraobebis klasifikacia da Sedareba (e.w. kinematika) xSirad sakmaod rTulia. ras vzomavT da rogor vadarebT erTmaneTs? 26 27 2.3 mdebareoba da gadaadgileba vidre pasuxis gacemas SeecdebodeT, ganvixiloT moZraobis zogadi maxasiaTeblebi, romlebic sam punqtad SeiZleba Camoyalibdes: 1. moZraoba xdeba mxolod wrfis gaswvriv. wrfe SeiZleba iyos vertikaluri (magaliTad, qvis vardna), horizontaluri (magaliTad, manqanis moZraoba trasaze) an iribi, magram is aucileblad wrfe unda iyos. 2. moZraobas Zalebi iwvevs, magram am sakiTxs me-5 TavSi ganvixilavT. am TavSi ki SevexebiT mxolod moZraobasa da cvlilebebs moZraobis dros. aCqardeba Tu Seneldeba sxeuli, gaCerdeba Tu sawinaaRmdego mimarTulebiT wava? Tu moZraoba icvleba, ra dro sWirdeba am cvlilebas? 3. moZravi sxeuli an nawilakia (igulisxmeba umciresi sxeuli, magaliTad, eleqtroni) an sxeulia, romelic nawilakiviT moZraobs (anu misi yvela nawili moZraobs erTnairad da erTi mimarTulebiT). magaliTad, rodesac Rori bavSvebis saTamaSo moedanze Semorbis, is SeiZleba ganvixiloT, rogorc nawilakiviT moZravi, xolo mori, romelic mdinares moyveba – ara, vinaidan misi sxvadasxva wertilebi sxvadasxva mimarTulebiT moZraobs. 2.3 mdebareoba da gadaadgileba sagnis mdebareobis gansazRvra niSnavs dadebiTi mimarTuleba misi poziciis povnas raime aTvlis wertilis mimarT, rogoricaa, magaliTad, 2.1 suraTze gamosaxuli x RerZis saTave (anu nulovani wertili). RerZze ricxvebis (koordinatebis) zrdis mimarTulebas uaryofiTi mimarTuleba -3 -2 naaRmdego mimarTulebas _ uaryofiTi. magaliTad, sxeulis koordinati SeiZleba iyos x = 5 m, rac imas niSnavs, rom is aTv- 0 1 2 3 xm dadebiTi ewodeba (suraTze _ marjvniv), xolo sawi- -1 saTave sur. 2.1 mdebareoba ganisazRvreba RerZze, romelzec aRniSnulia sigrZis erTeulebi (am SemTxvevaSi metrebi). RerZis saxeli (aq x) yovelTvis iwereba dadebiT mxares. lis wertilidan 5 metriTaa daSorebuli dadebiTi mimarTulebiT. Tu x = –5 m, ese igi, sxeuli imave manZilzea aTvlis wertilidan, oRond uaryofiTi mimarTulebiT. RerZze –5 m koordinati naklebia, vidre –1 m koordinati, xolo orive naklebia, vidre +5 m koordinati. `pliusis” dawera saWiro ar aris, magram `minusi” yovelTvis unda iyos naCvenebi. x1-dan x2-ze gadasvlas Δx gadaadgileba ewodeba, sadac Δx = x2 – x1 (2.1) (Δ berZnuli aso “deltaa” da sididis cvlilebas aRniSnavs, rac sididis saboloo da saw yisi mniSvnelobebis sxvaobis tolia). rodesac 2.1 formulaSi x1 da x2-is nacvlad ricxvebi Caismeba, dadebiTi mimarTulebis gadaadgileba yovelTvis dadebiTi ricxvi iqneba, xolo uaryofiTi mimarTulebisa – uaryofiTi. magaliTad, _ Tu wertili gadaadgildeba x1 = 5 m- dan x2 = 12 m-sken, maSin Δx = 12 – 5 = +7. dadebiTi Sedegi aRniSnavs moZraobis dadebiT mimar- Tulebas. Tu wertili moZraobs x1 = 5 m-dan x2 = 1 m-sken, maSin Δx = 1 – 5 = – 4. uaryofiTi Sedegi niSnavs, rom moZraoba uaryofiTi mimarTulebiT xdeba. realuri ganvlili manZili gadaadgilebis sididesTan kavSirSi ar aris, vinaidan gadaadgileba moicavs mxolod sawyis da saboloo mdebareobebs. magaliTad, Tu wertili moZraobs x = 5 m-dan x2 = 200 m-sken da Semdeg ukan brundeba (x = 5 m), gadaadgileba gzis dasawyisidan dasasrulamde iqneba Δx = 5 – 5 = 0. gadaadgilebisTvis “pliusis” miniWebas mniSvneloba ara aqvs, xolo minus niSani yo- 28 wrfivi moZraoba Tavi 2 velTvis unda iyos naCvenebi. Tu gadaadgilebis niSans (da, maSasadame, mimarTulebas) ugulebelvyofT, maSin +1 dagvrCeba mxolod gadaadgilebis ricxviTi (anu ab-1 1 0 2 3 soluturi) mniSvneloba. magaliTad gadaadgilebis 4 Δ x = –4 ricxviTi mniSvnelobaa 4. gadaadgileba veqtoruli sididis magaliTia. veqtoruli ewodeba sidides, romelsac ricxviTi mniS- x (t) x(t) vnelobis garda aqvs mimarTulebac. veqtorebs ufro grafiki zRarbisTvis, dawvrilebiT me-3 TavSi SeviswavliT, amJamad ki mxo romelic uZravad dgas x = – 2 m wer- lod imis gageba gvWirdeba, rom gadaadgilebas ori tilSi. maxasiaTebeli aqvs: 1) ricxviTi mniSvneloba, romelic sur. 2.2. x-is mniSvneloba nebismieri droisTvis erTi da igivea. t aris manZili sawyis da saboloo mdebareobebs Soris, da 2) mimarTuleba sawyis da saboloo mdebareobebs Soris, romelic SeiZleba warmodgenil iqnas “plius” an “minus” niSniT, Tu moZraoba erTi wrfis gaswvriv xdeba. qvemoT xedavT sakontrolo wertils, romlis msgavsic am wignSi bevri SegxvdebaT. TiToeuli sakontrolo wertili Seicavs erT an met kiTxvas, romelze pasuxis gacemac garkveul msjelobas an gamoTvlas moiTxovs. sakontrolo wertili swrafad amowmebs, ramdenad kargad gaiazre ganxiluli sakiTxi. pasuxebi mocemulia wignis bolos. 1 . mocemulia x RerZze sawyisi da saboloo mdebareobebis sami wyvili. romeli wyvilebi gvaZlevs uaryofiT gadaadgilebas? sakontrolo wertili a) –3 m, +5 m; b) –3 m, –7 m; g) 7 m, –3 m x (m) 2.4 s aSualo siCqare da saSualo manZili x (t) mdebareobis aRweris optimaluri formaa x mde t (wm) bareobis warmodgena t drois funqciad – x(t) (Ca naweri x(t) niSnavs funqcias da ara x-is namravls t-ze). amis martivi magaliTia 2.2 suraTze naCvenebi mdebareobis x(t) funqcia uZravi zRarbisTvis (romelsac ganvixilavT, rogorc nawilaks) drois mdebareoba t=0 7-wamian monakveTSi. zRarbis mdebareoba ucvlelia, x = – 2 m. 2.3 (a) suraTi ufro sainteresoa, vinaidan masze moZraobaa naCvenebi. zRarbi t = 0 momentSi imyofeba x = –5 poziciaze, moZraobs saTavis (x = 0) mimarTulebiT, saTaves gaivlis t = 3 momentSi da x-is ufro maRali dadebiTi mniSvnelobebisken miiwevs. sur. 2.3 (b) asaxavs zRarbis wrfiv moZraobas, rac daaxloebiT isaa, rasac Cven davinaxavT. 2.3 (a) suraTze mocemuli grafiki ufro abstraqtulia da ara hgavs imas, rac realurad Cans, samagierod, informaciiT ufro mdidaria. masze zRarbis siswrafec Cans. (a) xm 0 3 4 t (wm) (b) sur. 2.3 . a) zRarbis moZraobis x(t) grafiki da masTan dakavSirebuli traeqtoria. b) x RerZis qvemoT mocemuli skala gviCvenebs im momentebs, rodesac zRarbi x-is sxvadasxva mniSvnelobebs aRwevs. 29 2.4 saSualo siCqare da saSualo manZili sityvas “siswrafe” ramdenime sidide ukavSirdeba. erT-erTi maTgania saSualo siCqare vsaS. es aris Δx gadaadgilebis Sefardeba Δt drois Sualedze, romelSic gadaadgileba moxda: νsaS = ∆x ∆t = x2 – x1 (2.2) t2 – t1 mocemuli Canaweri niSnavs, rom t1 droSi zRarbis mdebareoba aris x1, xolo t2 droSi _ x2. vsaS-s erTeulia metri wamSi (m/wm). amocanebis amoxsnisas sxva erTeulebic SeiZleba Segvxvdes, Tumca forma manZili/dro yovelTvis daculi iqneba. x-is t-sadmi damokidebulebis grafikze vsaS aris im xazis daxriloba, romelic x(t) grafikis or wertils akavSirebs. erTi wertili Seesabameba x2-s da t2-s, meore wertili _ x1-s da t1-s. gadaadgilebis msgavsad, vsaS-s gaaCnia rogorc ricxviTi mniSvneloba, aseve mimarTule- ba (e.i. veqtoruli sididea). misi ricxviTi mniSvneloba daxrilobis sididea. ν = am xazis daxriloba dadebiTi vsaS (da daxriloba) gveubneba, saS rom xazi mimarTulia zeviT da marjvniv, = uaryofiTi _ qveviT da marjvniv. saSua- ∆x ∆t lo siCqares yovelTvis igive niSani aqvs, t = (s) rac gadaadgilebas, vinaidan Δt 2.2 formulaSi yovelTvis dadebiTia. sur. 2.4 gviCvenebs, rogor vipovoT vsaS drois SualedisTvis t =1 wm-dan t =4 wm-mde. gavavlebT swor xazs, romelic akavSirebs mdebareobis grafikis wertils drois am monakveTis dasawyisSi da x = (t) mdebareobis grafikis wertils am mon- Δx = 2m – (–4 m) = 6m akveTis dasasruls. Semdeg vipoviT swori xazis daxrilobas Δx/Δt. drois moceΔt mul monakveTSi saSualo siCqare tolia νsaS = 6m 3 wm = 2 m/wm ssaS saSualo siCqare wertilis moZ- = 4s – 1s = 3s sur. 2.4 saSualo siCqaris gamoTvla t = 1 wm-sa da t = 4 wm-s Soris. saSualo siCqare im xazis daxrilobaa, romelic x(t) grafikis Sesabamis wertilebs akavSirebs. raobis siCqaris aRweris kidev erTi saSualebaa. Tu saSualo siCqare vsaS wertilis Δx gadaadgilebas ukavSirdeba, saSualo manZili ssaS ukavSirdeba mTlian ganvlil manZils (magaliTad, ganvlili metrebis raodenobas) mimarTulebis damoukideblad. anu ssaS = mTliani gavlili manZili (2.3) Δt vinaidan ssaS saSualo siCqares mimarTulebas ar gviCvenebs, mas arc algebruli niSani aqvs. zogjer ssaS da vsaS erTmaneTis tolia (niSans Tu ar CavTvliT). Tumca, rogorc amocanis amoxsnis 2.1 nimuSSi vnaxavT, rodesac sxeuli orjer gadis erTi da imave gzas, ssaS da vsaS erTmaneTisgan sakmaod gansxvavdeba. amocanis amoxsnis nimuSi Zveli satvirTo manqana swor gzaze 70 km/sT siCqariT migyavs. 8,4 km-is gavlis Semdeg manqanas benzini gauTavda da gaCerda. amis gamo daaxloebiT 30 wuTis ganmavloba- 2.1 Si 2 kilometri fexiT iare benzingasamarTi sadguris sapovnelad. a) risi tolia Seni mTliani gadaadgileba mgzavrobis dawyebidan sadguramde? 30 wrfivi moZraoba Tavi 2 amoxsna: davuSvaT, rom moZraoba xdeba x `sadguri~. rogorc viciT, saSualo siC- RerZis dadebiTi mimarTulebiT, saTavidan qare (x1=0) sadguramde, romlis mdebareoba unda iyos x2= 8,4 + 2 = 10,4 km. x RerZze Seni Δx xazis daxrilobis tolia. vsaS aris Δx-is gadaadgilebis gamosaTvlelad meore pozi- ( Δx am ori wertilis SemaerTebeli = 10,4 km) fardoba Δt-ze (Δt=0,62 sT), anu gvaZlevs vsaS = 16,8 km/sT. cias pirveli unda gamoaklo. 2.1 formulid) warmoidgine, rom benzinis Casasxme- dan gvaqvs: Δx=x2 – x1 = 10,4 – 0 = 10,4 (pasuxi) maSasadame, mTliani gadaadgileba x RerZis dadebiTi mimarTulebiT iqneba 10,4 km. b) risi tolia Δt drois Sualedi mgzav- lad, fulis gadasaxdelad da ukan fexiT dasabruneblad damatebiT 45 wuTi dagWirda. risi tolia saSualo siCqare mgzavrobis dawyebidan benziniT manqanamde dabrune- robis dawyebidan sadgurze misvlamde? bamde? amoxsna: ukve viciT, rom fexiT siarulis dro iyo Δtfex= 0,50 sT, magram ar viciT manqaniT mgzavrobis dro Δtman. samagierod, viciT, rom manqaniT mgzavrobisas Δxman gadaadgilebaa 8,4 km, xolo saSualo siCqare (vsaS.manq.) – 70 km/sT. gamoviyenoT formula 2.2. saSualo amoxsna: am SemTxvevaSic saSualo siCqa siCqare gamoiTvleba manqaniT mgzavrobisas re tolia mTliani ganvlili manZilis Se fardebisa im drosTan, rac am manZilis gavlas daWirda. mTliani manZili tolia 8,4 + 2 + 2 = 12,4 km-isa, mTliani dro ki – 0,12 + 0,5 + 0,75 = 1,37 sT-isa. maSasadame, formula 2.3 gvaZlevs gadaadgilebis SefardebiT Sesabamis drois monakveTTan: vsaS.manq. = SsaS = ∆xmanq. ∆t manq. 12,4 km 1,37 sT = 0.12 km/sT (pasuxi) aqedan gamomdinareobs, rom: ∆xmanq. 8.4 km = = 0.12 sT ∆t manq. = ν saS.manq. 70 km/sT satvirTo manqana gaCerda = 0.12 sT + 0.5 sT = 0.62 sT (pasuxi) g) risi tolia Seni saSualo siCqare (vsaS) mgzavrobis dawyebidan sadgurze misvlamde? manq an siar iT uli ∆t = ∆tmanq. + ∆tfex. mdebareoba (km) fexiT ase rom, fardobis tolia: vsaS = ∆x = 10.4 km 0.62 sT ∆t = 16.8 km/sT ≈ 17 km/sT grafikulad vsaS-is dapirvelad (pasuxi) sapovnelad Tav- avagoT x(t) funqciis grafi- ∆x(=10,4 km) ∆t(=10,4 km) ipove pasuxi ricxobrivad da grafikulad. amoxsna: aq mTavari kvlav 2.2 formulaa: mgzavrobis mTliani vsaS mTliani gadaadgilebisa (10,4) da mTliani drois Sualedis b.g. sadguri i siarul dro (sT) sur. 2.5. `manqaniT siaruliT~ da `fexiT siaru liT~ aRniSnuli xazebi, manqanis tarebis da fexiT siarulisas mdebareobis droze damo kidebulebis grafikebia. aTvlis wertilisa da benzingasamarTi sadguris damakavSirebeli xazis daxriloba mTliani mgzavrobis saSualo siCqarea. sakontrolo wertili 2 mocemul amo- canaSi davuSvaT, rom satvirTo manqanaSi benzinis Casxmis Semdeg x1 wertilSi ki, rogorc es naCvenebia 2.5 suraTze, sa- 35 km/sT siCqariT dabrundi. ras udris dac Seni saSualo siCqare mTeli mgzavrobisa- sawyisi da saboloo wertilebia, Sesabamisad, aTvlis saTave da wertili vis? 31 2.5 myisi siCqare amocanis amoxsnis taqtika taqtika 1 : gaiziare Tu ara amocana? 3: aris Tu ara Seni pasuxi safuZvliani? taqtika amosaxsneli amocana xSirad Zalian iolia da mas, ubralod, ver vigebT. saukeTeso aqvs Tu ara azri Sens pasuxs? iqneb me- testi amis gasagebad Semdegi kiTxvaa: Segi- tismetad grZeli an metismetad moklea? Zlia Tu ara amocanis pirobebis aRwera? niSani sworia? erTeulebi? magaliTad, amo- Camowere monacemebi da gamoiyene mocemuli paragrafis simboloebi (amocanis canis amoxsnis 2.1 nimuSis (g) nawilSi swori pasuxia 17 km/sT. Tu miiReb 0,00017 km/sT-s, amoxsnis 2.1 nimuSSi monacemebis saSualebiT –17 km/sT-s, 17 km/wm-s an 17000 km/sT-s, unda SegiZlia Δx ipovo gadaadgileba (a) nawilSi mixvde, rom raRaca arasworad gaakeTe. Sec- da Sesabamisi Δt drois Sualedi (b) nawilSi. domis mizezi SeiZleba iyos amoxsnis meTo- gansazRvre ucnobi da misi simbolo (amoca- di, araswori gamoTvlebi an kalkulatorze nis (g) nawilSi sapovnia vsaS saSualo siCqare). ricxvebis araswori akrefa. Semdeg ipove kavSiri ucnobsa da monacemebs Soris (kavSiris amsaxveli formulaa 2.2, ro- taqtika 4: grafikis wakiTxva 2.2, 2.3a, 2.4 da 2.5 suraTebze mocemuli melic saSualo siCqares gansazRvravs). grafikebis wakiTxva advilad unda SeZlo. yvela grafikSi horizontalur RerZze t taqtika 2: sworia Tu ara erTeulebi? darwmundi, rom ricxvebis formulebSi Casmis dros swor erTeulebs iyeneb. amocanis amoxsnis 2.1 nimuSSi aseTi erTeulebia kilometri (manZilisTvis), saaTi (drois monakveTisTvis) da kilometr/saaTi (siCqarisTvis). zogjer SeiZleba saWiro gaxdes am erTeulebis sxva erTeulebSi gadayvana. 2.5 dro gadaiTvleba, romelic marjvena mimarTulebiT izrdeba. vertikalur RerZze gadaiTvleba moZravi wertilis x mdebareoba. am RerZis dadebiTi mimarTuleba zemoTaa. yovelTvis daimaxsovre erTeulebi (wamebi Tu wuTebi, metrebi Tu kilometrebi), romelSic cvladebia mocemuli. myisi siCqare Tqven ukve ixileT sagnis moZraobis siswrafis aRweris ori gza: saSualo siCqare ssaS (speed) da saSualo veqtoruli siCqare vsaS (velocity). orive maTgani Δt drois SualedSi izomeba. Tumca sxeulis moZraobis siCqares ukeTesad aRwers ν myisi siCqare (an ubralod siCqare), romelic gviCvenebs, ra siCqariT moZraobs sxeuli drois mocemul momentSi. myisi siCqare miiReba saSualo siCqaridan, rodesac Δt drois monakveTi 0-s uaxlovdeba. vinaidan Δt mcirdeba, saSualo siCqare uaxlovdeba zRvrul mniSvnelobas, rac swored am momentSi siCqaris tolia: ν = lim ∆t→0 ∆x dx = ∆t dt (2.4) am formulaSi ν myisi siCqaris ori maxasiaTebeli Cans: 1) ν aris siCqare, romliTac wertilis x mdebareoba icvleba drois mocemul momentSi, rac imas niSnavs, rom ν aris x-is warmoebuli t-Ti; 2) ν nebismier momentSi aris sxeulis mdebareobis grafikis daxriloba im wertilSi, romelic am moments Seesabameba. siCqare veqtoruli sididea da, maSasadame, mimarTulebac aqvs. speed* aris velocity**-is ricxviTi mniSvneloba. es niSnavs, rom speed aris velocity, oRond mimarTulebisa da algebruli niSnis gareSe (yuradRebiT unda iyoT, speed da velocity erTmaneTisgan zogjer sakmaod gansxvavdeba). roca velocity aris +5 m/wm da –5 m/wm, speed orive SemTxvevaSi aris 5 m/wm. manqanis spidometri gviCveebs speed-s da ara velocity-s (mas mimarTulebis gansazRvra ar SeuZlia). * speed [spı:d] – siCqare (mxolod ricxviTi mniSvneloba) velocity [vilositi] – siCqare (rogorc ricxviTi mniSvneloba, ise mimarTuleba) ** 32 Tavi 2 wrfivi moZraoba amocanis amoxsnis nimuSi 2.6 a suraTze mocemulia liftis kabinis 2.2 RerZs zemoTaa). sur. 2.6 a gviCvenebs, rom x moZraobis x(t) grafiki. Tavdapirvelad lif- marTlac izrdeba 20 m-iT am SualedSi. asea ti uZravia, Semdeg moZraobs maRla (x RerZis Tu ise, sur. 2.6 b ar gvauwyebs x-is mniSvnelo- dadebiTi mimarTuleba) da Cerdeba. aage ν (t) bebs Sualedis dasawyisSi da boloSi. amisT- grafiki. vis gvWirdeba damatebiTi informacia, rogoricaa x-is mniSvneloba raime momentSi. amoxsna: rogorc viciT, mocemul momentSi siCqare x(t) grafikis daxrilobis tolia. x(t) grafikis daxriloba da, maSasadame, siCqare, 0 wm-dan 1 wm-mde da 9 wm-dan zeviT bulia. bc intervalSi daxriloba mudmivia da ar udris nuls, vinaidan liftis kabina mudmivi siCqariT moZraobs. aqedan gamomdinare, x(t) wrfis daxrilobas gamoviTvliT d c mdebareoba (m) nuls udris, vinaidan liftis kabina gaCere- x x=4m t = 3 wm b ∆t a Semdegnairad: 24 m - 4.0 m ∆x = + 4.0 m/wm =ν= ∆t 8.0 wm – 3.0 wm liftis kabina jer moZraobas iwyebs, Semdeg ki neldeba da Cerdeba, amitom ν icvleba, rogorc es naCvenebia SualedebSi 1 wm-dan 3 wm-mde da 8wm-dan 9wm-mde. maSasa- b c ν(t)-s daxriloba siCqare (m/wm) vinaidan d a dame, 2.6 b suraTze ufro zusti grafikia b) a davbrundeT”, raTa Sesabamisi x(t) grafiki migveRo (sur. 2.6 a). Tumca, am SemTxvevaSi ver gavigebT sxvadasxva dros x-is sxvadasxva aCqareba (m/wm2) qveTavSi). 2.6 b suraTzea mocemuli, SegviZlia “ukan t dro (wm) mocemuli (2.6 g suraTi ganxilulia 2.6 rodesac gvaqvs iseTi grafiki, rogoric a) Seneleba v dadebiTi x mimarTulebiT moZraobs. es Suulia 2.6 b suraTze. aqve unda iTqvas, rom, t dro (wm) “pliusi” niSnavs, rom liftis kabina aledebi (sadac ν=0-s da ν=4 m/wm-s) gamosax- ∆x x(t)-s daxriloba a b c d t mniSvnelobebs, vinaidan ν(t) grafiki mxolod x-is cvlilebas asaxavs. drois nebismier SualedSi x-is aseTi cvlilebis sapovnelad saWiroa gamoiTvalos ν(t) grafikiT miRebuli figuris farTobi am SualedisTvis. magaliTad, 3 wm-dan 8 wm-mde, rodesac liftis kabinas 4 m/wm siCqare hqonda, x-is cvlileba iyo Δ x = (4 m/wm) (8 wm – 3 wm) = +20 m. (farTobi dadebiTia vinaidan ν(t) grafiki t g) sur. 2.6 a) x(t) grafiki liftis kabinisTvis, romelic moZraobs maRla, x RerZis gaswvriv; b) ν(t) grafiki liftis kabinisTvis. aRsaniSnavia, rom es aris x(t) grafikis warmoebuli (ν = dx/dt); g) a(t) grafiki liftis kabinisTvis. es aris ν(t) grafikis warmoebuli (a = dν/dt). grafiks qvemoT figurebi gviCvenebs, ras igrZnobs mgzavris sxeuli aCqarebebis dros. 33 2.6 aCqareba 2.3 amocanis amoxsnis nimuSi x RerZze moZravi wertilis mdebareoba ν = 0 + 9,2 – (3)(2.1)t2 = 9,2 – 6,3t2. mocemulia formuliT: x = 7.8 + 9.2t – 2.1t 3 (2-6) Arodesac t = 3,5 wm, (2.5) ν = 9,2 – (6,3)(3,5)2 = – 68 m/wm x izomeba metrebSi, xolo t wamebSi. ras udris siCqare, rodesac t = 3,5 wm-s? siCqare (Apasuxi) rodesac t = 3,5 wm, wertili moZraobs x-is uaryofiTi mimarTulebiT 68 m/wm siCqariT. mudmivia Tu icvleba? vinaidan t sidide Sedis 2.6 formulaSi, ami- amoxsna: simartivisTvis 2.5 formulidan tom ν siCqare damokidebulia t-ze da drois erTeulebi amoRebulia, Tumca, Tu gsurT, mixedviT icvleba. SegiZliaT koeficientebis nacvlad daw- 3 qvemoT mocemu li formulebi gvaZlevs wertilis x(t) mdebareobas oTx SemTxvevaSi (yovel for mulaSi x izomeba metrebSi, t – wamebSi, sakontrolo wertili eroT 7,8 m, 9,2 m/wm da –2,1 m/wm3. unda gaviTvaliswinoT, rom siCqare aris x(t) mdebareobis funqciis pirveli xarisxis warmoebuli. aqedan gamomdinare, vwerT: t>0): (1) x = 3t – 2; (2) x = –4t2 – 2; (3) x = 2/t 2 da (4) x = –2. a) romel SemTxvevaSia wertilis ν siCqare mudmivi? b) romel SemTxvevaSia ν x-is uaryofiTi mimarTulebis? d dx 3 ν = dt = dt (7.8 + 9.2t – 2.1t ), saidanac gamomdinareobs 2.6 aCqareba rodesac wertilis siCqare icvleba, amboben, rom wertili ganicdis aCqarebas (anu Cqardeba). RerZis gaswvriv moZraobis dros asaS (saSualo aCqareba) Δt drois SualedSi tolia: asaS = ν2 - ν1 ∆ν = ∆t t2 - t1 (2-7) sadac ν1 aris wertilis siCqare t1 droSi da ν2 – t2 droSi. myisi aCqareba (an ubralod aCqareba) aris siCqaris warmoebuli droiT: a= dν dt (2-8) sxvagvarad rom vTqvaT, wertilis aCqareba nebismier momentSi aris tempi, romliTac siCqare am momentSi icvleba. grafikulad, aCqareba nebismier wertilSi aris ν(t) grafikis daxriloba am wertilSi. SegviZlia SevaerToT formulebi 2.8 da 2.4 da davweroT dν d 2x d dx (2-9) = = dt dt dt dt 2 sxvagvarad rom vTqvaT, wertilis aCqareba nebismier momentSi aris misi x(t) mdebareobis a= meore xarisxis warmoebuli droiT. aCqarebis erTeulia m/wm2. amocanebSi SeiZleba SegxvdeT sxva erTeulebi, magram forma manZili/dro2 daculi iqneba. aCqarebas gaaCnia rogorc ricxviTi mniSvneloba, aseve mimarTuleba (e.i. veqtoruli sididea). misi algebruli niSani aRniSnavs x RerZis gaswvriv mimarTulebas, rogorc es iyo gadaadgilebisa da siCqaris SemTxvevaSi. es niSnavs, rom dadebiTi mniSvnelobis aCqareba mimarTulia RerZis dadebiTi mimarTulebiT, xolo uaryofiTi mniSvnelobisa – uaryofiTi mimarTulebiT. 2.6 g suraTze mocemulia aCqarebis grafiki liftisa, romlis moZraobac ganxiluli iyo amocanis 2.2 nimuSSi. Seadare a(t) grafiki ν(t) grafiks. yoveli wertili a(t) grafikze gviCvenebs ν(t) grafikis warmoebuls (daxrilobas) Sesabamis momentSi. rodesac ν mudmivia 34 Tavi 2 wrfivi moZraoba (0 an 4 m/wm-is SemTxvevaSi), warmoebuli da, maSasadame, aCqareba nulis tolia. rodesac liftis kabina moZraobas iwyebs, ν(t) grafiks aqvs dadebiTi warmoebuli (daxriloba dadebiTia), rac niSnavs, rom a(t) dadebiTia. rodesac liftis kabina Cerdeba, ν(t) grafikis warmoebuli da daxriloba uaryofiTia da, maSasadame, a(t)-c uaryofiTia. amis Semdeg Seadare ν(t) grafikis daxrilobebi aCqarebis or periodSi. liftis SenelebasTan dakavSirebuli daxriloba ufro cicaboa, vinaidan gaCerebas orjer ufro naklebi dro dasWirda, vidre siCqaris matebas. cicabo daxriloba niSnavs, rom Senelebis ricxviTi mniSvneloba ufro maRalia, vidre aCqarebisa, rogorc es naCvenebia 2.6 g suraTze. sur. 2.7. polkovniki j.p. stapi Cqa is, rasac liftis kabinaSi igrZnob, zemoT mocemul rosnul kameraSi, rodesac is aRwevs Zalian maRal siCqares da Semdeg wrafadve muxruWdeba. suraTzea naCvenebi. rodesac kabina aCqardeba, geCveneba, TiTqos zevidan gawvebian, xolo rodesac kabina damuxruWebas iwyebs, iseTi grZnoba giCndeba, TiTqos zeviT geqaCebian. am mdgomareobebs Soris ki bevrs arafers igrZnob. sxvagvarad rom vTqvaT, Seni sxeuli reagirebs aCqarebaze da ara siCqareze. rodesac 90 km/sT siCqariT moZrav manqanaSi an 900 km/sT-iT mfrinav TviTmfrinavSi xar, Seni sxeuli moZraobas ver grZnobs. Tumca, Tu manqana an TviTmfrinavi swrafad Seicvlis siCqares, maSin araTu mkveTrad igrZnob, aramed SegeSindeba kidec. gasarTob atraqcionebze mZafri SegrZnebebi swored siCqaris swrafi cvlilebiTaa gamowveuli. ufro saintereso magaliTi mocemulia 2.7 suraTze, rodesac Cqarosnuli kamera swrafad aaCqares da Semdeg mkveTrad daamuxruWes. didi aCqarebebi zogjer gamoisaxeba g erTeulebSi: 1 g = 9,8 m/wm2 (2.10) (g aris dedamiwaze Tavisufali vardnisas sxeulis aCqareba. amas ganvixilavT 2.9 TavSi). atraqcion `rusul mTebze~ SeiZleba ganicado aCqareba daaxloebiT 29 m/wm2-mde, rac namdvilad amarTlebs am atraqcionis Rirebulebas. amocanis amoxsnis taqtika taqtika 5 : aCqarebis niSani zogadad, aCqarebis niSans aramecnieruli mniSvneloba aqvs: dadebiTi aCqareba niSnavs, rom sxeulis siswrafe imatebs, xolo uar yofiTi aCqareba niSnavs, rom siCqare klebulobs (sxeulis moZraoba neldeba). Tumca, am wignSi aCqarebis niSani mimarTulebas aRniSnavs da ara siCqaris klebas an matebas. magaliTad, Tu manqana, romlis sawyisi siCqarea ν = –25 m/wm, damuxruWda da gaCerda 5 wamSi, maSin asaS = + 5 m/wm2. aCqareba dadebiTia, Tumca manqanis siCqarem daiklo. mizezi niSnebs Soris gansxvavebaa: aCqarebis mimarTuleba siCqaris sawinaaRmdegoa. niSnebis gagebis swori gza aseTia: Tu sxeulis siCqarisa da aCqarebis niSnebi erTnairia, maSin siCqare izrdeba (matulobs ricxviTi mniSvneloba). Tu niSnebi sxvadasxvaa – siCqaris sidide mcirdeba mcirdeba. 4 vombati* moZraobs x RerZis gaswvriv. ipove aCqarebis niSani, Tu is moZraobs: a) dadebiTi mimarTulebiT da siCqare izrdeba; b) dadebiTi mimarTulebiT da siCqare mcirdeba; g) uaryofiTi mimarTulebiT da siCqare izrdeba; d) uaryofiTi mimarTulebiT da sakontrolo wertili siCqare mcirdeba. vombati – CanTosani cxoveli. gavrcelebulia av straliasa da axal zelandiaSi. * 35 2.7 TanabaraCqarebuli moZraoba: gansakuTrebuli SemTxveva amocanis amoxsnis nimuSi wertilis mdebareobis cvlileba (sur. 2.1) 2.4 qariT, anu x RerZis uaryofiTi mimarTulebiT. wertilis aCqarebaa a(0) = 0, vinaidan siC- mocemulia formuliT x = 4 – 27t + t 3 qare ar icvleba. a) ipove wertilis siCqarisa da aCqarebis fun- rodesac 0 < t < 3, wertils kvlav uaryo qciebi – ν(t) da a(t). fiTi siCqare aqvs da, maSasadame, uaryofiTi amoxsna: ν(t) siCqaris funqciis misaRebad vpoulobT x(t) funqciis warmoebuls droiT. mimarTulebiT moZraobs. Tumca, aCqareba Cvens SemTxvevaSi ν = –27 + 3t2 (pasuxi) a(t) aCqarebis funqciis misaRebad vpoulobT ν(t) siCqaris funqciis warmoebuls droiT, rac gvaZlevs: a = +6 t (pasuxi) b) arsebobs Tu ara dro, rodesac ν = 0-s? amoxsna: ν(t) = 0-is Casma mogvcems 0 = –27 + 3t2, aqedan gamomdinare, t = ± 3 wm. g) aRwere wertilis moZraoba t ≥ 0-sTvis. amoxsna: SeviswavloT x(t), ν(t) da a(t) funqciebi. rodesac t = 0, wertili imyofeba x(0) = + 4 m wertilSi da moZraobs ν(0) = –27 m/wm siC- 2.7 aRar aris 0-is toli, vinaidan izrdeba da dadebiTia. siCqarisa da aCqarebis niSnebi urTierTsawinaaRmdegoa, maSasadame, wertilis moZraoba neldeba. marTlac, rogorc ukve viciT, rodesac t = 3 wm, wertili Cerdeba. rogorc vxedavT 2.1 suraTze, am dros wertili maqsimalurad aris daSorebuli aTvlis wertilidan marcxniv. Tu x(t) gamosaxulebaSi CavsvamT t = 3 wm, miviRebT, rom wertilis pozicia aris x = – 50 m. misi aCqareba kvlav dadebiTia. rodesac t > 3, wertili marjvniv moZra obs. misi aCqareba dadebiTia da izrdeba. siCqare axla ukve dadebiTia da misi ricxviTi mniSvnelobac izrdeba. TanabaraCqarebuli moZraoba: gansakuTrebuli SemTxveva ,m mis Tanabaria. magaliTad, SegiZlia manqanis aCqareba daaxloebiT Tanabari tempiT, rodesac SuqniSanze wiTels mwvane cvlis. Seni moZraobis, siCqarisa da aCqarebis grafikebi 2.8 suraTze mocemuli grafikebis msgavsi iqneba (gaiTval- mdebareoba mravali saxis moZraobis dros aCqareba sruliad an TiTq- daxriloba icvleba iswine, rom 2.8 g suraTze a(t) Tanabaria, risTvisac, 2.8 b su- a) raTze ν(t)-s unda hqondes Tanabari daxriloba). mogvianebiT, biT Tanabari iyos. msgavsi SemTxvevebi imdenad xSiria, rom gamoyvanil iqna formulebis specialuri jgufi. am formulebis miRebis siCqare Tu manqanas daamuxruWeb, aCqareba aseve SeiZleba daaxloe- daxriloba = a erT-erTi meTodi naCvenebia am TavSi, xolo meore meTodi mocemulia Semdeg TavSi. orive Tavis wakiTxvisas da saSinao b) sur. 2.8 a) TanabaraCqarebulad moZravi wertilis mdebareoba x(t); b) wertilis siCqare ν(t) – mocemuli x(t) grafikis TiToeul wertilSi daxrilobis Sesabamisad; g) wertilis aCqareba (Tanabari), tolia ν(t) grafikis (mudmivi) daxrilobisa. aCqareba davalebebis Sesrulebisas gaiTvaliswine, rom es formule- daxriloba = 0 g) , wm 36 Tavi 2 wrfivi moZraoba bi gamosadegia mxolod Tanabari aCqarebis dros (an situaciebSi, rodesac aCqarebis mniSvneloba miaxloebulia TanabarTan). rodesac aCqareba Tanabaria, saSualo aCqareba da myisi aCqareba erTmaneTis tolia. amis gamo formula 2.7 SegiZlia Semdegnairad SevcvaloT: ν - ν0 a = asaS = t - 0 aq ν0 aris siCqare, rodesac t = 0, xolo ν – siCqare nebismieri momdevno t-sTvis. Cven SegviZlia es formula gadavweroT, rogorc ν = ν0 +at. (2.11) Sesamowmeblad gaiTvaliswineT, rom rodesac t = 0, ν = ν0. Semdgomi SemowmebisTvis ga- vawarmooT 2.11 toloba. miviRebT: dν/dt = a, rac a-s gansazRvrebaa. 2.8 b suraTze naCvenebia formula 2.11-is ν(t) funqciis grafiki. funqcia wrfivia da, maSasadame, grafikic wrfea. aseTive xerxiT SegviZlia gadavweroT formula 2.2 (aRniSvnebSi mcireodeni cvlile bebiT): saidanac viRebT: vsaS = x – x0 t–0 x = x0 + vsaS t , (2.12) sadac x0 wertilis mdebareobaa t = 0 momentSi, xolo vsaS – saSualo siCqare drois romelime monakveTSi (vTqvaT, t = 0-sa da romelime momdevno t-s Soris). formula 2.11-Si mocemuli siCqaris wrfivi funqciisTvis, saSualo siCqare drois nebismier monakveTSi (vTqvaT, t = 0-sa da romelime momdevno t-s Soris) aris sawyis siCqaresa (ν0) da drois monakveTis bolos wertilis siCqaris (ν) saSualo mniSvneloba. t = 0-dan t-mde saSualo siCqare iqneba vsaS = 1 (ν + ν). 2 0 (2.13) formula 2.11-is marjvena nawilis gardaqmnis Sedegad viRebT: vsaS = ν0 + formula 2.14-is 2.12-Si CasmiT miviRebT: 1 2 at x – x0 = ν0t + 1 at 2 2 (2.14) (2.15) Sesamowmeblad gaiTvaliswineT, rom t = 0 momentSi x = x0. amasTan, formula 2.15-is gawarmoeba gvaZlevs 2.11-s, rac asec unda iyos. 2.8a suraTze mocemulia formula 2.15-is grafiki. funqcia kvadratulia da amitom es grafiki mrudia. formulebi 2.11 da 2.15 TanabaraCqarebuli moZraobis ZiriTadi formulebia. maTi gamoyeneba SesaZlebelia Tanabar aCqarebasTan dakavSirebuli nebismieri amocanis amosaxsnelad. amasTanave, SegviZlia gamoviyvanoT sxva formulebi, romlebic gamosadegi iqneba garkveul specifikur situaciebSi. unda aRiniSnos, rom Tanabar aCqarebasTan dakavSirebul amocanebSi SeiZleba Sediodes 5-mde sidide – x – x0 , ν, t, a da ν0 . Cveulebriv, erT-erTi sidide aseT amocanaSi ar monawileobs, arc cnobili da arc ucnobi sididis saxiT. amitom, gvaqvs sami mocemuli sidide da unda movZebnoT meoTxe. formulebSi 2.11 da 2.15 oTxi aseTi sididea, Tumca es oTxeuli erTnairi rodia. formula 2.11-Si ar gvaqvs gadaadgileba x – x0, formula 2.15-Si – siCqare ν. es formulebi SegviZlia SevaerToT sami gziT, raTa gamoviyvanoT sami damatebiTi formula, romelTagan TiToeuli Seicavs sxvadasxva àrarsebul cvlads~. pirvel rigSi gamovricxoT t: ν2 = ν 20 + 2a(x – x0) (2.16) es formula gamosadegia, roca ar viciT t da mis povnas arc gvTxoven. meore rigSi gamovricxoT a aCqareba, raTa miviRoT formula, sadac a ara gvaqvs: x – x0 = 1 (ν0 + ν)t 2 dasasruls, gamovricxoT ν0: (2.17) 37 2.7 TanabaraCqarebuli moZraoba: gansakuTrebuli SemTxveva x – x0 = νt – 1 at 2 2 (2.18) yuradReba miaqcieT am formulasa da formula 2.15-s Soris arsebul mcire gansxvavebas. erTi Seicavs sawyis siCqares ν0-s, xolo meore – ν siCqares t droSi. cxrili 2.1 2.1 cxrilSi CamoTvlilia Ta TanabaraCqarebuli moZraobis formulebi nabari * aCqarebiT moZraobis ZiriTadi formulebi (2.11 da formulis Nnomeri formula ararsebuli cvladi 2-11 ν = ν0 + at x - x0 2.15) da maTgan miRebuli sxva formulebi. TanabaraCqarebul moZraobasTan dakavSirebuli 2-15 x – x 0 = ν 0t + 1 2 2 at ν 2-16 ν = ν0 + 2a(x – x0) t 2-17 x – x0 = 12 (v + v0)t a Tan gaqvs). airCie 2-18 x – x0 = vt – 1 2 2 at ν0 romelSic ucnobi sidide amo- 2 2 martivi amocanis amosaxsnelad SegiZlia gamoiyeno am siaSi mocemuli formulebi (Tu sia canaSi * sanam cxrilSi mocemul formulebs gamoiyeneb, darwmundi, rom aCqareba marTlac Tanabaria ufro formula, sapovneli martivia formulebi 2.11 cvladia. daimaxsovro da 2.15 da gamoiyeno isini, rodesac dagWirdeba. magaliTi mocemulia amocanis amoxsnis 2.5 nimuSSi. sakontrolo wertili 5 mocemuli formulebi gansazRvravs wertilis x(t) mdebareobas oTx SemTxvevaSi: 1) x = 3t – 4; 2) x = –5t3 + 4t2 + 6; 3) x = 2/t2 – 4/t; 4) x = 5t2 – 3. romel SemTxvevas Seesabameba 2.1 cxrilis formulebi? amocanis amoxsnis nimuSi policiis danaxvisas sakuTar porSes amuxruWeb 100 km/sT siCqaridan 80 km/sT-mde TanabaraCqarebulad da am dros 88 m-s ga- 2.5 formulisTvis. t ucnobis gamosaricxad gamoviyenoT formula 2.11 da davweroT: ν–ν t= a 0 dixar. a) risi tolia aCqareba? amoxsna: warmovidginoT, rom moZraoba xdeba x RerZis dadebiTi mimarTulebiT. si- Tu am gamosaxulebas CavsvamT 2.15 formulaSi, miviRebT: x - x0 = ν0 ν - ν0 + 1 a ν - ν0 2 a a martivisTvis, damuxruWebis dawyebis mdebareobad miviCnioT x0, xolo drod t = 0. unda gavixsenoT, rom Tanabari aCqarebis dros manqanis aCqareba SeiZleba davukavSiroT siCqares da gadaadgilebas ZiriTadi formulebis saSualebiT (2.11 da 2.15). sawyisi siCqare ν0 = 100 km/sT = 27,78 m/wm, gadaadgileba x–x0=88 m-s, xolo siCqare Senelebis Semdeg ν = 80,0km/sT = 22,22 m/wm. ar viciT a aCqareba da t dro, romelic saWiroa orive (2.19) 2 CavsvaT cnobili sidieebi da gamoviTvaloT a: a= ν2 - ν 20 2(x – x0) = (22,22 m/wm)2 - (27,78 m/wm)2 2(88 m) = - 1.58 m/wm2 (pasuxi) gaiTvaliswine, rom a-s gamosaTvlelad SegveZlo formula 2.16-is gamoyeneba, vinaidan t ucnobi am formulaSi ara gvaqvs. 38 Tavi 2 wrfivi moZraoba b) ra dro dasWirda siCqaris mocemuli Tu siCqares gadaaWarbe da Semdeg misi sididiT Semcirebas? Semcireba scade, gaxsovdes, rom policiels amoxsna: ukve viciT a da SegviZlia gamovi yenoT formula 2.19 t-s mosaZebnad: ν–ν 22,22 m/wm - 27,78 m/wm t= a 0 = - 1,58 m/wm2 = 3,519 wm ≈ 3,52 wm (pasuxi) uamravi dro rCeba Seni siCqaris gasazomad. amocanis amoxsnis taqtika taqtika 6: Seamowme erTeulebi siCqare aris manZili Sefardebuli dro- Teulebis Sesamowmeblad gaiTvaliswine, roca formulaSi eWvi gepareba, erTeulebi rom x-isa da x0-is ganzomileba aris manZili, ν0t-s – (manZili/dro) · t (dro), rac Sekvecis 1 Semdeg aseve manZilia. 2 at2-s ganzomilebaa (manZili/dro2) (dro2), rac aseve manZilia. Seamowme. maSasadame formula sworia. sTan, xolo aCqareba – manZili Sefardebuli drois kvadratTan. nebismier formulaSi yvela erTeuli erTnairi unda iyos. formula 2.15-Si (x – x0= ν0t + 2.8 1 2 2 at ) er- kidev erTxel TanabaraCqarebuli moZraobis Sesaxeb 2.1 cxrilSi pirveli ori formula ZiriTadia da maTgan gamomdinareobs danarCeni. es ori formula SeiZleba miviRoT aCqarebis integraliT, im pirobiT, rom a mudmivia. formula 2.11-is misaRebad gadavweroT aCqarebis gansazRvreba (2.8), rogorc dν = adt Semdeg davweroT ganusazRvreli integrali orive mxares: dν = a dt vinaidan a aCqareba mudmivia (Tanabaria), misi gamotana integralidan SeiZleba. Semdeg miviRebT an dν = a dt ν = at + C (2.20) integrirebis C mudmivas gamosaTvlelad, davuSvaT, rom t = 0 da am momentSi ν = ν0. CavsvaT es mniSvnelobebi formula 2.20-Si, miviRebT ν0 = (a)(0) + C = C CavsvaT es formula 2.20-Si da miviRebT formula 2.11-s. formula 2.15-is misaRebad gadavweroT siCqaris gansazRvreba (formula 2.4), rogorc dx=νdt orive mxares davweroT ganusazRvreli integrali: dx = ν dt zogadad, ν ar aris mudmivi, ase rom, misi gamotana integralidan ar SeiZleba. Tumca, ν-s Secvla SegviZlia formula 2.11-iT: dx = (ν0 + at) dt vinaidan ν0 da a aCqareba mudmivia, formula SeiZleba gadaiweros aseTi saxiT: 2.9 Tavisufali vardnis aCqareba dx = ν0 dt + a t dt integralis aReba mogvcems x = ν0t + 1 at 2 + C´, 2 (2.21) sadac C´ integrebis kidev erTi mudmivaa. t = 0 momentSi x = x0. CavsvaT es mniSvnelobebi formula 2.21 da miviRebT x0 = C´. formula 2.21-Si C’ SevcvaloT x0-iT, miviRebT formula 2.15-s. 2.9 Tavisufali vardnis aCqareba Tu odesme raime sagani maRla agigdia an zevidan qveviT Camogigdia, SeniSnavdi, rom vardnisas sxeulis aCqareba Tanabaria. am aCqarebas Tavisufali vardnis aCqareba ewodeba da misi ricxviTi mniSvneloba warmodgenilia g simboloTi. aCqareba ar aris damokidebuli sxeulis iseT maxasiaTeblebze, rogoricaa masa, simkvrive an forma. is Tanabaria nebismier i sxeulisTvis. Tavisufali vardnis aCqarebis ori magaliTi mocemulia 2.9 suraTze. aq gamosaxulia vaSlisa da bumbulis stroboskopuri fotoebi. vinaidan es sxeulebi miwaze ecema, amitom maTi aCqareba orive SemTxvevaSi g-s tolia. maSasadame, maTi siCqare erTi da imave tempiT imatebs da miwaze orive erTdroulad ecema. g-s mniSvneloba mcirediT icvleba sxvadasxva ganedsa da simaRleze. dedamiwis Sua ganedebze g zRvis doneze udris 9,8 m/wm2-s. am wignSi mocemuli amocanebis amoxsnisas swored es ricxvi unda gamoviyenoT, Tu raime specialuri miniSneba ar arsebobs. 2.1 cxrilSi mocemuli TanabaraCqarebuli moZraobis formulebi gamosadegia dedamiwis zedapirze Tavisufali vardnis SemTxvevaSi. es niSnavs, rom isini miesadageba vertikalurad moZrav (maRla an dabla) sagans, rodesac haeris zemoqmedeba ugulebelyofilia. gaiTvaliswine, sur. 2.9 bumbuli da vaSli vakuumSi Tavisuflad vardeba Tanabari g aCqarebiT. aCqareba zrdis manZils, yovel momdevno gamosaxulebebs Soris. haeris ararsebobis pirobebSi bumbuli da vaSli Zirs erTdroulad daecema. rom Tavisufali vardnisTvis: 1) moZraobis mimarTulebebi y vertikaluri RerZis gaswvrivaa x RerZis nacvlad (es mniSvnelovania momdevno TavebSi, rodesac horizontaluri da vertikaluri moZraoba erToblivad ganixileba); 2) Tavisufali vardnis aCqareba uaryofiTia da mimarTulia qveviT dedamiwis centrisaken. maSasadame, formulebSi gveqneba – g. Tavisufali vardnis aCqareba dedamiwis zedapiris maxloblad aris a = g = – 9,8 m/wm2, xolo misi ricxviTi mniSvneloba aris g = 9,8 m/wm2. warmoidgine, rom pomidori maRla aisrole sawyisi (dadebiTi) siCqariT ν0 da Semdeg isev daiWire. Tavisufali vardnis dros (xelis gaSvebidan daWeramde) pomidoris moZraobas miesadageba 2.1 cxrilis formulebi. aCqareba yovelTvis udris –9,8 m/wm2-s, uaryofiTia da qveviT mimarTuli. samagierod siCqare icvleba, rogorc es asaxulia formulebSi 2.11 da 2.16. 39 40 Tavi 2 wrfivi moZraoba maRla moZraobis dros pomidvris dadebiTi siCqaris ricxviTi mniSvneloba mcirdeba, vidre 0-s ar miaRwevs. vinaidan pomidori am dros Cerdeba, amitom es misi maqsimaluri simaRlea. daSvebis dros siCqaris (axla ukve uaryofiTi) ricxviTi mniSvneloba izrdeba. amocanis amoxsnis nimuSi 1993 wlis 26 seqtembers deiv mandei ventilaciiT aRWurvili rkinis sferoTi ni- 2.6 Zils gadioda TiToeul wamSi. gansazRvre misi mdebareoba yoveli wamis Semdeg. agaras CanCqerSi daeSva da 48 m simaRlidan gadmoxta. davuSvaT, rom misi sawyisi siCqare 0-is toli iyo da ugulebelvyoT burTze haeris zemoqmedeba dacemis dros. a) ramden xans gagrZelda mandeis vardna wylis zedapiris miRwevamde? amoxsna: unda gaviTvaliswinoT, rom mandeis daSveba Tavisufali vardna iyo, maSasadame, aq 2.1 cxrilis formulebi gamogvadgeba. gavavloT y RerZi mandeis vardnis traeqtoriis gaswvriv, y = 0 iyos sawyisi wertili, xolo dadebiTi mimarTuleba iyos zeviT (sur. 2.10). am RerZis gaswvriv aCqareba aris a = – g, xolo wylis zedapiria simaRle y = – 48 m (y uaryofiTia, vinaidan (y = 0)-saT- avis qvemoTaa). davuSvaT, rom vardna daiwyo t = 0 mometSi sawyisi ν0 = 0 siCqariT. 2.1 cxrilidan avirCevT formula 2.15s (oRond x-is nacvlad y cvladiT), vinaidan sur. 2.10 Tavisuflad vardnili sxeulis mdebareoba, siCqare da aCqareba. am SemTxvevaSi sxeuli rkinis sferoa, romelsac deiv mandei marTavs nia garas CanCqerze. is saWiro t dros Seicavs da masSi Semavali cvladebis mniSvnelobebic viciT. miviRebT: amoxsna: kvlav viyenebT formula 2.15-s, y - y0 = ν0t - 1 gt 2, 2 1 (9.8 m/wm2)t 2, - 48 m - 0 = 0t 2 Tumca am SemTxvevaSi rig-rigobiT CavsvamT da t = 1, 2, 3 wm-s da vipoviT mandeis y mdebareobebs. Sedegebi naCvenebia 2.10 suraTze. t 2 = 48/4,9, t = 3,1 wm mniSvnelobebs g) ra siCqariT moZraobda mandei, rode(pasuxi) sac wylis zedapirs miaRwia? aRsaniSnavia, rom mandeis gadaadgileba amoxsna: sawyisi monacemebiT siCqaris y – y0 uaryofiTi sididea, vinaidan mandei qveviT moZraobs anu miemarTeba y RerZis sapovnelad gadavweroT formula 2.16 y cv- uaryofiTi mimarTulebiT. agreTve SevniS- ladebiT da CavsvaT CvenTvis cnobili si- noT, rom 48/4,9-s aqvs ori kvadratuli fes- dideebi: vi: 3,1 da -3,1. Cven avirCieT dadebiTi fesvi, vinaidan mandei wylis zedapirs namdvilad t = 0 momentSi vardnis dawyebis Semdeg aRw- ν2 = ν 02 - 2g (y - y0 ) = 0 - (2) (9.8 m/wm2) (- 48 m), ase rom, ν = - 30.67 m/wm ≈ - 31 m/wm = - 110 km/sT (pasuxi) evs. b) mandeim SeZlo daeTvala Tavisufali vardnis sami wami, magram ver dainaxa ra man- uaryofiTi fesvi im mizeziT avirCieT, rom siCqares uaryofiTi mimarTuleba aqvs. 2.9 Tavisufali vardnis aCqareba d) ra siCqariT moZraobda mandei yovel = 0 - (9.8 m/wm2 (1.0 wm) = - 9.8 m/wm. wams? grZnobda Tu ara is siCqaris zrdas? amoxsna: sawyisi monacemebiT siCqareebis sapovnelad, amocanis b) nawilSi miRebuli mdebareobebis mniSvnelobaTa gamoyenebis gareSe, davuSvaT, rom formula 2.11-Si a = –g, da rig-rigobiT CavsvaT t-s mniSvnelobebi t = 1, 2 da 3 wm. magaliTad: ν = ν0 - gt amocanis amoxsnis nimuSi (pasuxi) sxva Sedegebi naCvenebia 2.10 suraTze. mandei Tavisufali vardnis procesSi iyo da siCqaris zrdas ver grZnobda, vinaidan aCqareba yovelTvis iyo -9,8 m/wm2. Tumca mandeim mkveTrad igrZno wyalze dacema, radgan aCqareba uecrad Seicvala (mandei gadarCa, magram didi jarimis gadaxda mouxda daudevari qmedebisTvis). 2.7 2.11 suraTze beisbolis momwodebeli Tu droebiT ugulebelvyofT erTeu- burTs isvris 12 m/wm siCqariT y RerZis gas- lebs (im pirobiT, rom isini sworia), Segvi- wvriv. Zlia davweroT: 4,9t2 – 12t + 5 = 0 a) ra dro dasWirdeba burTs maqsimaluri simaRlis misaRwevad? amoxsna: burTs asrolidan ukan (xelSi) dabrunebamde aqvs Tavisufali vardnis aCqareba a = – g. vinaidan aCqareba Tanabaria, amitom am moZraobas 2.1 cxrili Seesabameba. isic unda gaviTvaliswinoT, rom ν siCqare maqsimalur simaRleze nulis toli unda iyos. ase rom, viciT ν, a da sawyisi siCqare kvadratuli gantolebis amoxsna gvaZlevs t = 0,53 wm da t = 1,9 wm (pasuxi) aseTi momenti oria! es namdvilad ar aris gasakviri, vinaidan burTi orjer gaivlis y = 5 m wertils, erTxel zeviT, meored ki qveviT moZraobisas. ν0 = 12 m/wm, da veZebT t dros, risTvisac viy- burTi enebT formula 2.11-s, romelic oTx cvlads Seicavs. t= 0 - 12 m/wm ν - ν0 = 1.2 wm (pasuxi) = a - 9.8 m/wm 2 b) risi tolia burTis frenis maqsimal- umaRles wertilSi uri simaRle? amoxsna: burTis gaSvebis wertilad aviRoT y0 = 0. davweroT formula 2.16 y cvladebiT, CavsvaT y – y0 = y da ν = 0 (maqsimalur simaRleze) da vipovoT y. miviRebT: 2 y= ν2 - ν0 0 - (12 m/wm) = = 7.3 m 2a 2(- 9.8 m/wm2) (pasuxi) vardnis dros zeviT moZraobis dros a = – g, speed mcirdeba, xolo velocity naklebad dadebiTi xdeba. a = – g, speed izrdeba. xolo velocity metad uaryofiTi xdeba. g) ra dro dasWirdeba burTs gaSvebis adgilidan 5m-is misaRwevad? amoxsna: viciT ν0 , a = –g da gadaadgileba y – y0 = 5 m. vinaidan unda vipovoT t, amitom avirCioT formula 2.15. gadavweroT y-sTvis da CavsvaT y0 = 0, miviRebT: an y = ν0t - 1 gt 2, 2 5.0 m = (12 m/wm)t - (1 )(9.8 m/wm2)t 2 2 sur. 2.11. beisbolis momwodebeli burTs haerSi isvris. Tavisufali vardnis formulebi gamosadegia rogorc zemoT, aseve qvemoT moZravi sagnebisTvis, xolo haeris zemoqmedeba SeiZleba ugulvebelvyoT. 41 42 Tavi 2 wrfivi moZraoba sakontrolo wertili 6 a) mocemul amocanaSi, ra iqneba burTis gadaadgilebis niSani zemoT asvlis dros, gaSvebidan umaRles wertilamde? b) ra iqneba es niSani vardnis dros, umaRlesi wertilidan gaSvebis wertilamde? g) ra iqneba burTis aCqareba umaRles wertilSi? amocanis amoxsnis taqtika taqtika es yovelTvis asea da ar aris damokidebuli 7 : “minus” niSnebis mniSvneloba sxeulis 2.6 da 2.7amocanebSi SemoviReT vertikal- zeviT, ise qveviT moZraobis dros. y = 0 wertili aviReT. 2.6 amocanaSi aTvlis wertili vardnis dawyebis wertili, xolo 2.7 taqtika amocanaSi – beisbolis burTis momwodeblis pasuxebs, rogorc moxda, magaliTad, 2.7 g rom sxeuli imyofeba aTvlis wertils qvemoT. amocanaSi. Tu miiRe ufro meti pasuxi, vidre uaryofiTi siCqare niSnavs, rom sxeuli moZ- elodi, nu uaryof im pasuxebs, romelic, raobs y RerZis uaryofiTi mimarTulebiT anu erTi qveviT. es yovelTvis asea, ganurCevlad imisa, sxeulis Seusabamoa. yuradRebiT fizikuri mniSvnelobe- mniSvnelobac ki SeiZleba raimes niSnavdes. rodesac amocana sxeulebis vardnas Seexeba. erTad maTi bi. Tu Seni cvladi droa, misi uaryofiTi aCqarebas uaryofiTs virCevT (–9,8m/wm2), msvlelobasTan SexedviT, gamoikvlie sad mdebareobs sxeuli. rom 8: moulodneli pasuxebi maTematika xSirad iZleva moulodnel xeli. y-is uaryofiTi mniSvneloba niSnavs, niSnavs, da uaryofiTia (qveviTaa mimarTuli) rogorc Tad avirCieT. Semdeg y RerZis aTvlis saTaved aCqareba siswrafesa mimarTulebaze. 2.7 amocanaSi burTis aCqareba uri RerZi y da misi zeda mimarTuleba dadebi- uaryofiTi mdebareobaze, uaryofiTi dro, ubralod, aris dro t = 0 mo- drois mentamde, rodesac drois aTvla daiwyo. velocity naklebad dadebiTi an metad uaryofiTi xdeba. 2.10 grafikuli integrali moZraobis SeswavlaSi rodesac mocemuli gvaqvs sxeulis aCqarebis droze damokidebulebis grafiki, misi integrirebiT SegviZlia vipovoT sxeulis siCqare drois nebismier momentSi. vinaidan a aCqareba ganisazRvreba, rogorc dv , amitom dt ν1 - ν0 = t1 t0 a dt (2.22) gamosaxulebis marjvena mxare gansazRvruli integralia (gvaZlevs ricxviT Sedegs da ara funqcias), ν0 aris siCqare t0 momentSi, xolo ν1 - siCqare t1 momentSi. gansazRvruli integrali SeiZleba gamoviTvaloT a(t) grafikidan, rogorc es 2.12 a suraTzea. kerZod: t1 t0 a dt = aCqarebis mrudiTa da t0-dan t1-mde drois RerZiT SemosazRruli figuris farTobi (2.23) Tu aCqarebis erTeulia 1 m/wm2, xolo droisa – 1 wm, maSin farTobis sidide grafikze iqneba (1 m/wm2) (1 wm) = 1 m/wm, rac, arsebiTad, siCqaris erTeulia. rodesac aCqarebis grafiki drois RerZs zemoTaa, farTobi dadebiTia, xolo rodesac aCqarebis grafiki drois RerZs qvemoTaa – uaryofiTi. 43 2.10 grafikuli integrali moZraobis SeswavlaSi analogiurad, vinaidan ν siCqare ganisazRvreba, ν = dx/dt, amitom x1 - x0 = t1 t0 a farTobi (2.24) v dt, sadac x0 aris mdebareoba t0 momentSi, xolo x1 – t1 mo- t0 mentSi. formula 2.24-is marjvena mxares gansazRvruli integrali SeiZleba gamoviTvaloT ν(t) grafikidan, ro- farTobi siCqaris mrudsa da v dt = drois RerZs Soris t -dan t -mde 0 1 t0 a) ν gorc es naCvenebia 2.12 b suraTze. kerZod: t1 t t1 farTobi (2.25) t0 Tu siCqaris erTeulia 1 m/wm, xolo droisa - 1 wm, farTobis sidide grafikze iqneba t1 b) t sur. 2.12. farTobi grafiksa da hor- t0a aCqarebis t (1 m/wm) (1 wm) = 1 m, izontalur drois RerZs Soris dan t1-mde naCvenebia: a) droze damokidebulebis grafikisT- rac, arsebiTad, manZilisa da gadaadgilebis erTeulia. farTobi dadebiTia Tu uaryofiTi, ganisazRvreba iseve, rogorc 2.12 a suraTze mocemuli a(t) grafikisT- vis; b) vis aRvwereT. amocanis amoxsnis nimuSi ν siCqaris t droze damokidebulis grafikisTvis. 2.8 anebis Seswavlis mizniT moxalise qamriT miabes manqanaSi, romelsac Sejaxebis simulaciisTvis uknidan daejaxa 10,5 km/sT-iT moZravi meore manqana. 2.13 a suraTze moce- aCqareba (m/wm2) zurgidan Sejaxebis dros kisris dazi Tavi mkerdi mulia moxalisis mkerdisa da Tavis aCqareba Sejaxebis dros, romelic t = 0 momentSi daiwyo. mkerdis aCqareba savarZlis zurgma 40 mwm-iT Seayovna, xolo Tavis aCqareba kidev 70 mwm-s Semdeg daiwyo. rogori iyo mkerdis siCqare, rodesac Tavma aCqareba daiwyo? a A amoxsna: SegviZlia gamovTvaloT sxeulis siCqare nebismier momentSi, Tu vipoviT farTobs mkerdis aCqarebis a(t) grafikze. viciT, rom t0 = 0 momentSi mkerdis sawyisi siCqa- re ν0 = 0. unda gavigoT sxeulis ν1 siCqare t1 = 110 mwm dros anu im momentSi, roca Tavi aCqarebas iwyebs. SevaerToT formulebi 2.22 da 2.23 da davweroT ν1 – ν0 = ( farTobi aCqarebis grafiksa da drois RerZs Soris t0-dan t1-mde ) dro, mwm (a) B C t (b) a(t) funqciis grafiki uknidan Sejaxebis simulaciis dros; b) grafiksa da drois RerZs Soris figuris farTobis gamoTvla. sur. 2.13. a) moxalisisa Tavis da mkerdis simartivisTvis grafiksa da RerZs Soris miRebuli figura sam nawilad davyoT (sur. 2.13 b). pirvel nawils (0-dan 40 mwm-mde) farTobi ar gaaCnia: (2.26) farTobi A = 0 meore nawils, 40-dan 100 mwm-mde, samkuTxedis forma aqvs, romlis farTobia 44 wrfivi moZraoba Tavi 2 1 ν1 = 2 m/wm = 7,2 km/sT anu farTobi B = 2 (0,060 wm)(50 m/wm2) = 1,5 m/wm (pasuxi) Tavis moZraobis dawyebisas mkerds ukve mesame nawils, 100-dan 110 mwm-mde, marTkuTxedis forma aqvs, romlis farTobia aqvs 7,2 km/sT siCqare. mecnierebi amtkiceben, rom uknidan Sejaxebis dros siCqareeb- farTobi C = (0,010 wm)(50 m/wm2) = 0,5 m/wm Si swored es gansxvaveba iwvevs kisris da- am mniSvnelobebisa da ν0-is 2.26 formu- zianebas. Tavis ukan gadavardna mogvianebiT laSi CasmiT miviRebT: xdeba da, rom ara savarZlis Tavmisadeblebi, dazianeba ufro mZime iqneboda. ν1 – 0 = 0 + 1,5 m/wm + 0,5 m/wm, mimoxilva da Sejameba mdebareoba. wertilis x mdebareoba aTvlis saTavis mimarT am wertilis ganla- damokidebulia am monakveTSi wertilis mier ganvlil manZilze: gebaze metyvelebs. mdebareoba dadebiTi an uaryofiTia imis mixedviT, Tu saTavis romel mxares imyofeba wertili. mdebareoba 0-is tolia, Tu wertili aTvlis saTavezea. ssaS = si siCqare (an, ubralod, siCqare) aris ν = lim dadebiTi ricxvebis zrdadi mimarTuleba, mdego mimarTuleba. gadaadgileba. wertilis Δx gadaadgileba aris mdebareobis cvlileba Δx = x2 _ x1 (2.1) gadaadgileba veqtoruli sididea. is dadebiTia, Tu wertili x RerZis dadebiTi mimarTulebiT gadaadgilda, da uaryofiTia, Tu wertili uaryofiTi mimarTulebiT gadaadgilda. saSualo siCqare. (velocity) rodesac wertilis mdebareoba icvleba x1-dan x2-mde drois Δt=t2–t1 monakveTSi, misi saSualo siCqare aris: νsaS = ∆x ∆t = x2 – x1 t2 – t1 (2-2) νsaS-s niSani asaxavs moZraobis mimarTu lebas (νsaS veqtoruli sididea). saSualo (2.3) ∆t myisi siCqare. moZravi wertilis ν myi- RerZis dadebiTi mimarTulebaa saTavidan xolo uaryofiTi mimarTuleba _ sawinaaR- mTliani manZili ∆t→0 dx ∆x = ∆t dt (2.4) sadac Δx da Δt gansazRvrulia formuliT 2.2. myisi siCqare (konkretul momentSi) aris x-is t-ze damokidebulebis grafikis daxriloba. speed aris myisi siCqaris ricxviTi mniSvneloba. saSualo aCqareba. saSualo aCqareba tolia siCqaris Δν cvlilebis Sefardebisa Δt drois monakveTTan: ∆ν asaS = ∆t (2.7) algebruli niSani asaS-s mimarTulebas gviCvenebs. myisi aCqareba. a myisi aCqareba aris ν(t) siCqaris pirveli xarisxis warmoebuli da x(t) mdebareobis meore xarisxis warmoe buli d 2x a = dν = 2 dt dt (2.8, 2.9) siCqare ar aris damokidebuli wertilis ν-s t-ze damokidebulebis grafikze a aCqareba nebismier t momentSi aris grafikis mier ganvlil manZilze, is damokidebulia daxriloba am momentis Sesabamis wertilSi. sawyis da saboloo mdebareobebze. x-is t-ze damokidebulebis grafikze sa Sualo siCqare Δt drois monakveTSi aris daxrili swori xazi, romelic am Sualedis sawyisi da saboloo momentebis Sesabamis wertilebs erTmaneTTan akvSirebs. saSualosiCqare (speed). wertilis ssaS saSualo siCqare (speed) drois Δt monakveTSi Tanabari aCqareba. 2.1 cxrilSi mocemuli xuTi formula aRwers wertilis TanabaraCqarebul moZraobas: ν = ν0 + at, x – x0 = ν0t + 12 at2, ν2 = ν02 + 2a(x – x0), x - x0 = 12 (ν0 + ν)t, x – x0 = νt - 12 at2 . (2.11) (2.15) (2.16) (2.17) (2.18) 45 es formulebi gamousadegaria, rodesac aC- aRniSvnebSi Segvaqvs ori cvlileba: 1) moZraobas mivmarTavT vertikaluri y RerZis qareba Tanabari ar aris. aCqareba. gaswvriv; 2) a-s CavanacvlebT – g -iT, sadac wrfivi TanabaraCqarebuli moZraobis maga- g aris Tavisufali vardnis aCqarebis ricx- liTia sagnis Tavisufali vardna dedamiwis viTi mniSvneloba. dedamiwis zedapirTan ax- zedapirisken. TanabaraCqarebuli moZrao- los g = 9,8 m/wm2. Tavisufali vardnis bis formulebi am moZraobas aRwers, magram SekiTxvebi 1. 2.14 suraTze naCvenebia sxeulis moZraobis oTxi traeqtoria sawyisidan saboloo wertilamde drois erTsa da imave monakveTSi. es traeqtoriebi erTmaneTisgan Tanabrad dacilebul paralelul xazebs kveTs. daalage traeqtoriebi: a) vsaS mixed- viT; b) ssaS -s mixedviT dididan mcireze. 2. 2.15 suraTze mocemulia wertilis mdebareobis t droze damokidebulebis grafiki. sur. 2.14 kiTxva 1. a) ra iqneba wertilis mdebareobis niSani, rodesac t = 0? b) wertilis siCqare dadebi- x Tia, uaryofiTia Tu nulis tolia, rodesac t = 1 wm? g) rodesac t = 2 wm? d) rodesac t = 3 wm? e) ramdenjer iqneba wertili x = 0 t (s) mdebareobaSi? 3. 2.16 suraTze naCvenebia RerZis gaswvriv moZravi wertilis siCqare. wertili 1 grafikze yvelaze maRlaa, xolo wertili 4 _ yvelaze dabla, wertilebi 2 da 6 erTsa da imave simaRlezea. ra mimarTulebiT xde- sur. 2.15 kiTxva 2. ba moZraoba t = 0-is dros? b) wertil 4-Si? g) eqvsi danomrili wertilidan romelSi v icvleba moZraobis mimarTuleba? d) daalage eqvsi wertili aCqarebis ricxviTi mniSvnelobis klebadobis mixedviT. 4. mocemuli formulebi gvaZlevs wertilis ν (t) siCqares oTx SemTxvevaSi: a) ν = 3; b) ν = 4t2 + 2t – 6; g) ν = 3t – 4; d) ν = 5t2 – 3. t sur. 2.16 kiTxva 3. romel SemTxvevas Seesabameba 2.1 cxrilis formulebi? 5. 2.17 suraTze mandarini maRla aagdes d) mandarinis Δν siCqaris cvlilebis mixed- sami Tanabari zomis fanjris gavliT. daal- viT. age fanjrebi: a) maTSi mandarinis saSualo 6. rodesac t = 0, x RerZis gaswvriv ravi wertilis mdebareobaa x = –20 m. tilis ν0 sawyisi siCqaris (t0 momentSi) Tanabari aCqarebis niSnebia: 1) +, +; 2) siCqaris (ssaS) mixedviT; b) maTi gavlis drois mixedviT; g) maTi gavlis dros mandarinis aCqarebis ricxviTi mniSvnelobis mixedviT; moZwerda a +, –; 46 Tavi 2 wrfivi moZraoba 3) –, +; 4) –, – . CamoTvlilTagan cxisTvis (gra romel variantSi: a) gaCerdeba fikebi A da B, wertili myisierad? b) gadakveTs aTvlis saTaves? g) ar ga- C, D da E, da aseve F da G pa dakveTs aTvlis saTaves? ralelurebia). 7. lurji manqanis (80 km/sT) mZRoli ucbad xedavs, rom wi- 9. rodesac t = 0 da x = 0, Tel manqanas (60 km/sT) ejaxeba. uZravi lurji Sejaxebis manqana Tavidan asacile- v A B t 0 C iwyebs blad, ra maqsimaluri siCqare TanabaraCqare unda hqondes lurj manqanas, bul moZraobas rodesac wiTel manqanas miuaxlovdeba? 2 m/wm aCqarebiT x RerZis 8. warmoidgine, rom xidis moa- dadebiTi mi- 2 D G F E sur. 2.18 kiTxva 8. jirze gadayudebuls kvercxi sur. 2.17 marTulebiT. rodesac t = 2 wm-s, imave mi- gagivarda (sawyisi siCqare nu- kiTxva 5. marTulebiT moZravi wiTeli manqana Cauv- lis tolia), xolo meore kver- lis x = 0 wertils 8 m/wm siCqariT da 3 m/wm2 cxi ganzrax isrole. 2.18 suraTis romel Tanabari aCqarebiT. formulebis romeli grafikebzea gamosaxuli ν(t) siCqare: a) ga- wyvili unda amoixsnas imis gasagebad, rodis vardnili kvercxisTvis; b) nasroli kver- gaaswrebs wiTeli manqana lurjs? amocanebi SSM – amoxsnebi mocemulia Student Solutions Manual-Si. www – amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday ILW – Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday. • – ••• – wertilebis raodenoba gviCvenebs amocanis sirTules ramdenime amocanaSi davalebuli gaqvs mde- (`vazrovneb, ese igi swrafad mivdivar!~). bareobis, siCqarisa da aCqarebis droze 2001 wels, sem uitingemma haberis rekordi damokidebulebis grafikis ageba. rogorc daamxo (mis siCqares 19km/sT-iT gadaaWarba). wesi, ubralo monaxazebi sakmarisia, Tu naT- ra droSi gaiara uitingemma 200 m manZili? lad Cans wrfivi funqciebi da mrudebi. Tu • 3 avtomobili 40 km-s gadis 30 km/sT kompiuteri da grafikuli kalkulatori siCqariT. Semdeg imave mimarTulebiT kidev gaqvs, SegiZlia grafikebis asagebad gamoiy- 40km-s gadis 60 km/sT siCqariT. a) ras udris eno. avtomobilis saSualo siCqare 80 km-is gav- Tavi 2-4 saSualo velocity (vsaS) da saSualo speed (ssaS). • 1 Zlieri ceminebisas Tvali 0,5 wm-iT ixuWeba. Tu ceminebis dros manqanas 90km/sT siCqariT marTav, ra manZils gaivli am droSi? lis Semdeg? (dauSvi, rom is dadebiTi x mimarTulebiT moZraobs); b) ras udris saSualo siCqare? g) aage x-is t-ze damokidebulebis grafiki da aCvene, rogor vipovoT grafikze saSualo siCqare. SSM WWW rekordi ekuTvnoda kris habers. misi dro • 4 manqana adis borcvze 40km/sT siCqariT da Semdeg ukan brundeba 60km/sT siC- 200 m-ze iyo 6,509 wami, razec haberma aseTi qariT. gamoTvale mTliani mgzavrobis saSu- komentari gaakeTa: `Cogito ergo zoom!” alo siCqare (ssaS). • 2 1992 wels velosipedis siCqaris 47 • 5 x RerZis gaswvriv moZravi sxeulis mdebareoba mocemulia formuliT x = 3t – 4t2+ t3, sadac x izomeba metrebSi, t – wamebSi. ipove sxeulis mdebareoba t-s mocemuli mniSvnelobebisTvis: a) 1wm; b) 2wm; g) 3wm; d) 4wm; e) ra iqneba sxeulis gadaadgileba t = 0-sa da t = 4 wm-s Soris? v) ra iqneba sxeulis saSualo siCqare t = 2 wm-dan t = 4 wm-mde? z) aage x-is tze damokidebulis grafiki, rodesac 0 ≤ t ≤ 4, meti iyo pirveli bilikis L1 sigrZeze. ramdenad didi unda iyos L2 – L1, raTa davaskvnaT, rom pirveli morbenali ufro swrafia? ILW. • • 10 ori matarebeli 30km/sT siCqariT moZraobs erTmaneTisken. rodesac matareblebs Soris manZili 60km-ia, 60km/sT-iT moZravi Citi erTi matareblis win frindeba, meorisken mifrinavs da, roca meores miaRwevs, isev ukan brundeba (warmodgena ara da aCvene, rogor SeiZleba am grafikiT vipo- gvaqvs, ratom unda moiqces Citi ase). ra man- voT v)-s pasuxi. SSM Zils gaivlis Citi, sanam matareblebi erTma- • 6 gamoTvale Seni saSualo siCqare mocemul or SemTxvevaSi: a) miseirnob 73,2 m-is manZilze 1,22 m/wm siCqariT da Semdeg garbixar 73,2 m-s 3,05m/wm siCqariT; b) 1 wT-is ganmavlobaSi miseirnob 1,22m/wm siCqariT da Semdeg 1wT-is ganmavlobaSi garbixar 3,05m/wm siCqariT; g) aage x-is t-ze damokidebulebis neTs daejaxebian? 55km/sT siCqariT, xolo meore naxevars – 90km/sT siCqariT. ukan dabrunebisas naxevar manZils gadixar 55km/sT siCqariT, xolo meore naxevars – 90km/sT siCqariT. risi tolia grafiki da aCvene, rogor unda iqnas napovni Seni saSualo siCqaris ricxviTi mniSvneloba: saSualo siCqare am grafikze. a) san-antoniodan hiustonamde? b) hiustoni- • • 11 san-antoniodan hiustonisken manqaniT miemgzavrebi. drois naxevars moZraob • 7 x RerZis gaswvriv moZravi werti- dan san-antoniomde? g) mTeli mgzavrobis lis mdebareobis x(t) funqciaa x = 4 – 6t2, manZilze? d) ras udris Seni saSualo veq- sadac x izomeba metrebSi, xolo t – wamebSi. toruli siCqare (velocity) mTeli mgzavrobis a) ra dros da b) sad gaCerdeba wertili? ra manZilze? e) aage x-is t-ze damokidebulebis g) uaryofiT momentSi da d) dadebiT momentSi grafiki a)-sTvis, imis gaTvaliswinebiT, rom gaivlis wertili aTvlis saTaves? e) aage x-is moZraoba x-is dadebiTi mimarTulebiT xdeba. t-ze damokidebulebis grafiki –5 wm-dan +5 wm-mde; v) x(t) grafikis marjvniv dasaZravad unda SevitanoT +20 t Tu –20 t dro? z) Setana gazrdis Tu Seamcirebs x-is im mniSvnelobas, grafikze aCvene, rogor unda vipovoT saSua- rodesac wertili Cerdeba? • • 8 intervius asaRebad unda gaemgzav ro sxva qalaqSi 300km-ian Cqarosnul tra saze. interviu daniSnulia 11:15-ze. Sen gegmav 100km/sT siCqariT moZraobas, amitom saxlidan gadixar 8:00-ze. pirveli 100km-is manZilze midixar 100km/sT siCqariT, magram samSeneblo samuSaoebis gamo iZulebuli xar, Semdegi 40km gaiaro 40km/sT siCqariT. ra minimaluri siCqariT moZraoba dagWirdeba dar- lo veqtoruli siCqare (velocity). Tavi 2-5 myisi siCqare • 12 a) Tu wertilis mdebareoba mocemulia formuliT x = 4 – 12t + 3t2 (sadac t izo meba wamebSi, xolo x – metrebSi), risi tolia siCqare, rodesac t = 1 wm? b) x-is dadebiTi Tu uaryofiTi mimarTulebiT moZraobs wertili? g) risi tolia misi siCqare am dros?; d) siCqare izrdeba Tu mcirdeba? (Semdegi ori kiTxvis pasuxis gacema scade gamoTvlebis gareSe) e) arsebobs Tu ara momenti, rodesac siCqare nulis tolia? Tu arsebobs, daasax- Cenil manZilze, interviuze rom ar dagagvi- ele; v) arsebobs Tu ara momenti t = 3 wm-is andes? Semdeg, rodesac wertili x-is uaryofiTi • • 9 1 km manZilze Sejibrebisas pirvel bilikze morbenali sportsmeni ufro swra fad garbis (2wuTi da 27,95 wami), vidre meore bilikze morbenali (2 wuTi da 28,15 wami). Tumca, meore bilikis L2 sigrZe SeiZleba odnav mimarTulebiT moZraobs? Tu arsebobs, daasaxele. ••13 x RerZis gaswvriv moZravi werti lis mdebareoba mocemulia formuliT x = 9,75 + 1,5t 3 , sadac t mocemulia wameb- 48 Tavi 2 wrfivi moZraoba Si, xolo x _ santimetrebSi. gamoTvale tolia g) νsaS da d) asaS t = 3 wT-dan t = 9 wT- a) saSualo siCqare t = 2 wm-dan t = 3 wm- mde? e) aage x-is t-ze da ν-s t-ze damokideb- mde; b) myisi siCqare, rodesac t = 2 wm; ulebis grafikebi, gansazRvre, rogor unda g) myisi siCqare, rodesac t = 3 wm; d) myisi miviRoT a)-s pasuxi d)-dan grafikis saSu- siCqare, rodesac t = 2,5 wm; e) myisi siCqare, alebiT. rodesac wertili imyofeba sawyisi da sab- • • 19 x RerZis gaswvriv moZravi wertilis mdebareoba, x = ct2_ bt3 formulis Tanaxmad, damokidebulia droze, sadac x metrebSia, xolo t wamebSi. ra iqneba erTeulebi: a) mudmiva c-sTvis da b) mudmiva b-sTvis? davuSvaT oloo mdebareobebs Soris, t = 2wm da t = 3 wm; v) aage x-is t-ze damokidebulebis grafiki da aCvene pasuxebi grafikulad. maTi ricxobrivi mniSvnelobebia, Sesabam- Tavi 2-6 aCqareba • 14 a) Tu wertilis mdebareoba mocemulia formuliT x = 20 t – 5t 3 (sadac x metrebSia, xolo t wamebSi), aris Tu ara wertilis 4 wm-mde monakveTSi d) ra manZils gaivlis siCqare nulis toli? Tu aris, rodis? b) ro- wertili da e) risi toli iqneba misi gadaadg- dis aris wertilis aCqareba nulis toli? ileba? ipove wertilis siCqare Semdeg mo- g) drois romel SualedSi (dadebiTSi Tu mentebSi: v) 1 wm; z) 2 wm; T) 3 wm da i) 4 wm. ipove uaryofiTSi) aris a aCqareba nulis toli? wertilis aCqareba Semdeg momentebSi: k) 1 wm; d) dadebiTi? e) aage grafikebi x(t), ν(t) da a(t). l) 2 wm; m) 3 wm da n) 4 wm. • 15 garkveul momentSi wertilis siCqare iyo 18 m/wm x-is dadebiTi mimarTulebiT, Tavi 2-7 Tanabari aCqareba: gansakuT 2,4 wm-is Semdeg ki misi siCqare gaxda 30 m/wm isad, 3 da 2. g) ra dros miaRwevs wertili maqsimalur dadebiT x mdebareobas? 0-dan SSM WWW. sawinaaRmdego mimarTulebiT. ras udris rebuli SemTxveva. • 20 miu-mezoni (elementaruli nawilaki) wertilis saSualo aCqareba drois am mon- moZraobs 5×106 m/wm siCqariT da Semdeg nel- akveTSi? SSM deba 1,25×1014 m/wm2 aCqarebiT. a) ra manZils • 16 x RerZis gaswvriv moZravi eleqtronis mdebareoba mocemulia formuliT x = 16te–t , sadac t wamebSi izomeba. ra manZiliT iqneba eleqtroni daSorebuli aTvlis wertilidan gaCerebis momentSi? • 17 x RerZis gaswvriv moZravi wertilis mdebareoba mocemulia formuliT x = 12 t – 2t , sadac x metrebSia, xolo t wameb2 3 Si. gansazRvre wertilis: a) mdebareoba, b) siCqare da g) aCqareba, rodesac t = 3 wm; gaivlis miu-mezoni gaCerebamde? b) aage x-is t-ze da ν-s t-ze damokidebulebis grafikebi miu-mezonisTvis. • 21 eleqtroni moZraobs TanabarCqarebulad +3,2 m/wm2 aCqarebiT. garkveul momentSi misi siCqarea +9,6 m/wm. ras udris siCqare: a) 2,5 wm-iT adre da b) 2,5 wm-iT gvian? • 22 mSral gzaze karg manqanas SeuZlia daamuxruWos 4,92 m/wm2 Tanabari aCqarebiT. a) ra dro dasWirdeba 24,6 m/wm siCqariT moZ- d) ra maqsimalur dadebiT koordinats mi- rav manqanas gasaCereblad? b) ra manZils aRwevs wertili da e) ra dros? v) ra maqsim- gaivlis am dros manqana? g) aage x-is t-ze da ν-s t-ze damokidebulebis grafikebi. • 23 davuSvaT, kosmosuri xomaldi moZ- alur dadebiT siCqares miaRwevs wertili da z) ra dros? T) ras udris wertilis aCqareba, rodesac wertili ar moZraobs (t gansxvavebulia nulisgan); i) gansazRvre wertilis saSualo siCqare t = 0-dan da t = 3 wm-mde. • 18 t = 0-dan t = 5 wT-mde kaci dgas, t = 5 wT-dan t = 10 wT-mde ki moZraobs wrfivad mudmivi siCqariT 2,2 m/wm. risi tolia: a) misi νsaS saSualo siCqare da b) asaS sa- Sualo aCqareba t = 2 wT-dan t = 8 wT-mde? risi raobs Tanabari aCqarebiT 9,8 m/wm2, rac frenis dros normaluri gravitaciis iluzias qmnis. a) Tu xomaldi moZraobas daiwyebs uZravi mdgomareobidan, ra dro dasWirdeba mas sinaTlis siCqaris (3 × 108 m/wm) erTi meaTedis misaRwevad? b) ra manZils gaivlis am drois ganmavlobaSi? SSM.? • 24 eleqtroavtomobili uZravi mdgo- 49 mareobidan 2m/wm2 Tanabari aCqarebiT moZ- lazeruli Tofi mainc gaigebs). b) aage x-is raobas iwyebs, aRwevs 20 m/wm siCqares, Semdeg t-ze da ν-s t-ze damokidebulebis grafikebi ki TanabarSenelebulad Cerdeba 1 m/wm2-iT. a) SenelebisTvis. ra dro gavida moZraobis dawyebidan gaCere- • 29 liftis mTliani samoZrao manZilia bamde? b) ra manZils gaivlis avtomobili 190 m, maqsimaluri siCqare _ 305 m/wT, xolo dawyebidan gaCerebamde? aCqareba _ 1,22 m/wm2. a) ra manZils gaivlis • 25 ν0=1,5 × 105 m/wm sawyisi siCqaris mqone lifti maqsimaluri siCqaris miRwevamde? eleqtroni Sedis L = 1 sm-is sigrZis kamera- b) ra dro dasWirdeba 190 m-is SeuCereblad Si, sadac eleqtrulad aCqardeba ν = 5,7 × 10 gavlas? 6 m/wm-mde (sur. 2.19). ras udris misi aCqareba, ILW. • • 30 manqana moZraobs x RerZis gaswvr- (msgavsi iv 900 m-is manZilze. moZraobas iwyebs x = 0 procesi mimdinareobs Zvel televizoreb- wertilSi da amTavrebs x = 900 wertilSi. Si). distanciis pirvel meoTxedSi manqanis aC- romelic Tanabradaa miCneuli? SSM. Tanabari moZraobis monakveTi aCqarebis monakveTi qarebaa +2,25 m/wm2, darCenil manZilze ki –0,75m/wm2. risi tolia: a) 900 m-is gavlis dro? b) maqsimaluri siCqare? g) aage x mdebareobis, ν siCqarisa da a aCqarebis t-ze damokidebulebis grafikebi. eleqtronis moZraobis wiri • • 31 2.20 suraTze naCvenebia x Re- x (m) rZis gaswvriv TanabaraCqarebulad maRali Zabvis wyaro moZravi wertilis moZraoba. ras udris t (s) wertilis aCqarebis sur. 2.19 amocana 25 • 26 siCqaris msoflio rekordi ekuTvnis polkovnik jon p. staps, romelmac 1954 a) ricxviTi mniSvneloba da b) mimarTuleba? • • 32 warmoid- sur. 2.20 amocana 31 wlis martSi reaqtiuli Zravis mqone manqan- gine, rom manqana migyavs da mobiluri tele- iT 1020km/sT siCqares miaRwia. manqana 1,4wm- foniT emociurad saubrob, am dros ki Sens Si gaCerda (ix. sur. 2.7). ramdeni g-s toli aC- win 25 m-is daSorebiT policiis manqana mid- qareba ganicada pilotma gaCerebisas? is. Senc da policiac 110km/sT siCqariT moZ • 27 56 km/sT siCqariT moZravi manqana 24 raobT. telefoniT laparakis gamo polici- m-iT iyo daSorebuli barieridan, rodesac is manqana 2 wm-is ganmavlobaSi ver SeamCnie. mZRolma daamuxruWa. manqana bariers 2 wm-is am 2 wamis dawyebisas policielma uecrad Semdeg Seaskda. a) rogoria Sejaxebamde man- daamuxruWa 5 m/wm2 aCqarebiT. a) ra manZili qanis Tanabari aCqarebis ricxviTi mniSvn- iqneba or manqanas Soris, roca policias eloba? b) ra siCqare aqvs manqanas Sejaxebis SeamCnev? warmoidgine, rom kidev 0,4 wm dag- dros? Wirda xifaTis gasaazreblad da amis Semdeg SSM ILW • 28 Seni manqanis muxruWebs SeuZlia daamuxruWe. b) Tu Senc 5 m/wm2 aCqarebiT Tanabrad Seanelos moZraoba 5,2 m/wm2-iT. daamuxruWe, rogori iqneba Seni siCqare, a) Tu midixar 137 km/sT siCqariT da uecrad rodesac policiis manqanas daejaxebi? patruli dainaxe, ra minimaluri dro dagWirdeba siCqaris 90 km/sT limitamde Sesam- • • 33 manqanebi A da B erTi da imave mimarTulebiT moZraobs. A manqanis x mdebareoba cireblad? (pasuxSi Cans, rom damuxruWebas 0-dan 7 wm-mde monakveTSi mocemulia 2.21 azri ara aqvs, vinaidan siCqares radari an suraTze. rodesac t = 0, B manqana imyofeba wrfivi moZraoba Tavi 2 x = 0 wertilSi, moZraobs 12 m/wm siCqariTa da da uaryofiTi Tanabari aCqarebiT aBB. a) ras unda udrides aB, raTa manqanebi erTmaneTis gverdiT aRmoCndnen, rodesac t = 4 wm? b) aBs am mniSvnelobisas ramdenjer aRmoCndeba manqanebi erTmaneTis gverdiT? g) aage x mde- v (m/wm) 50 bareobis t droze damokidebulebis grafiki B manqanisTvis. ramdenjer aRmoCndeba manqanebi gverdigverd, Tu aBB aCqarebis ricxviTi mniSvneloba d) metia da e) naklebia a) kiTxvis pasuxze? t (s) sur. 2.22 amocana 35 • • 36 2.23 suraTze, wiTeli da mwvane manqana erTmaneTis mimarTulebiT moZraobs x RerZis paralelurad. rodesac t = 0, wiTeli manqana xwT = 0 wertilSia, xolo mwvane - xmw = 200 m wertilSi. Tu wiTeli manqana moZraobs 20 km/sT mudmivi siCqariT, manqax (m) nebi erTmaneTs Cauvlian x = 44,5 m wertilSi, xolo Tu 40km/sT mudmivi siCqariT, maSin x = 76,6 m wertilSi. risi tolia mwvane manqanis: a) sawyisi siCqare da b) aCqareba? t (wm) sur. 2.21 amocana 33 sur. 2.23 amocana 36 • • 34 a) Tu metros matareblis mgzavrebisTvis maqsimaluri dasaSvebi aCqarebaa • • • 37 Cqari samgzavro matarebeli moZ- 1,34m/wm , xolo metros sadgurebi ganlageb- raobs 161 km/sT siCqariT. uecrad memanqanem ulia 806 m-is daSorebiT, ras udris matare- SeniSna, rom mis liandagze SecdomiT gver- blis maqsimaluri siCqare sadgurebs Soris? didan amave mimarTulebiT moZravi lokomo- b) ras udris sadgurebs Soris mogzaurobis tivi Semovida da misgan D = 676 m manZilzea. dro? g) Tu matarebeli yovel sadgurze lokomotivi moZraobs 29 km/sT siCqariT. Cerdeba 20 wm-is ganmavlobaSi, ras udris swrafi matareblis memanqanem saswrafod matareblis maqsimaluri saSualo siCqare daamuxruWa. a) ras unda udrides Tanabari erTi sadguridan meoremde? d) aage x, ν da a- Senelebis ricxviTi mniSvneloba Sejaxebis s t-ze damokidebulebis grafikebi gaCerebi- Tavidan asacileblad? b) warmoidgine, rom 2 dan gaCerebamde. • • 35 ori matareblis memanqaneebma ue- memanqane imyofeba x = 0 wertilze da t = 0, rodesac pirvelad dainaxavs lokomotivs. crad SeniSnes, rom maTi matareblebi erT- aage x(t) grafikebi lokomotivisa da swrafi maneTis Semxvedrad moZraobs. 2.22 suraTze matareblisTvis, rodesac Sejaxeba Tavidan mocemulia matareblebis ν siCqareebi, ro- iqneba da ar iqneba acilebuli. gorc drois funqciebi Senelebis dros. memanqaneebma matareblebis Seneleba daiwyes, rodesac maT Soris manZili 200 m iyo. rogori iqneba matareblebs Soris manZili gaCerebis Semdeg? swrafi matarebeli lokomotivi sur. 2.24 amocana 37 51 uravidan qvas isvris 12 m/wm sawyisi siCqa- y Tavi 2-9 Tavisufali vardnis aCqareba • 38 xuligani 30m simaRlis Senobis saxriT. a) ra dro dasWirdeba qvas Zirs dacemisTvis? b) ra siCqare eqneba qvas dacemisas? • 39 samSeneblo moedanze qanCsaWeri gad- t (wm) movarda da miwas 24 m/wm siCqariT daeca. a) ra sur. 2.25 amocana 45 simaRlidan Camovarda qanCsaWeri? b) ramden xans ecemoda? g) aage qanCsaWeris y-is, ν-s • • 46 2.26 suraTze naCvenebia y RerZis da a-s t droze damokidebulebis grafikebi. gaswvriv maRla asrolili burTis ν siCqa- SSM ris y simaRleze damokidebulebis grafiki. • 40 wvimis wveTebi miwaze 1700 m simaRleze myofi Rrublidan ecema. a) rom ara yAA simaRleze siCqarea νA, xolo yBB simaRleze – 1/3 νA. ras udris νA siCqare? haeris winaaRmdegoba, ra siswrafe eqneboda v wveTebs miwaze dacemisas? b) usafrTxo iqneboda Tu ara wvimis dros gareT gasvla? • 41 a) ra siCqariT unda iqnas burTi dedamiwis zedapiridan asrolili, 50 m simaRles rom miaRwios? b) ramden xans iqneba 1 ν 3 A burTi haerSi? g) aage y-is, ν-s da a-s t droze damokidebulebis grafikebi. pirvel grafikze aCvene dro, rodesac burTi 50 m-s sur. 2.26 amocana 46 miaRwevs. SSM WWW • 42 SeSinebuli maCvi haerSi axta da yB yA or pirvel 0,2 wm-Si 0,544 m simaRles miaRwia. • • 47 gasaRebi gadmovarda 45 m simaRlis a) ras udris maCvis sawyisi siCqare? b) ras xididan da mudmivi siCqariT moZrav navze udris siCqare 0,544 m simaRleze? g) ra si- daeca. gasaRebis gadmovardnis momentSi maRleze axteba maCvi? navi dacemis adgils 12 m-iT iyo daSorebu- • 43 cxeli haeriT savse burTi 12 m/wm li. risi tolia navis siCqare? SSM ILW siCqariT zeviT adis da dedamiwidan 80 m si- • • 48 uzarmazari sahaero burTi maRla maRleze iqidan yuTi vardeba. a) ra dro das- adis 19,6 m/wm siCqariT. am dros masze mi- Wirdeba yuTs Zirs dasavardnad? b) ra siC- magrebuli instrumentebis yuTi mowydeba qariT daecema is miwaze? da miwaze vardeba. 2.27 suraTze mocemulia SSM • 44 50 m simaRlis xididan WanWiki Cam- yuTis vertikaluri siCqaris damokideb- ovarda. a) ra droSi gaivlis WanWiki dacemis uleba droze vardnis dasawyisidan miwaze ukanasknel 20%-s? rogori iqneba misi siC- dacemamde. a) burTidan mowyvetis wertili- qare, b) rodesac iwyebs dacemis ukanasknel dan ra maqsimalur simaRles miaRwevs yuTi? 20%-s, da g) rodesac miwas miaRwevs? b) dedamiwidan ra manZilzea daSorebuli • 45 t = 0 momentSi xididan trasaze pir veli vaSli Camovarda, mogvianebiT ki imave mowyvetis wertili? v simaRlidan meore vaSli isroles. 2.25 suraTze mocemulia vardnis dros vaSlebis t (wm) y mdebareobebis droze damokidebulebis grafikebi. daaxloebiT ra siCqariT isroles meore vaSli? sur. 2.27 amocana 48 Tavi 2 wrfivi moZraoba •• 49 sveli Tixis burTi miwaze 15 m simaR- ra dro dasWirdeba moTamaSes: a) naxtomis lidan ecema. miwasTan Sexeba burTis srul zeda 15 sm-is gasavlelad da b) qveda 15 sm-is gaCerebamde 20 mwm grZeldeba. a) ras udris gasavlelad? burTis saSualo aCqarebis ricxviTi mniSvn- ••• 57 saxuravidan Camovardnil rkinis burTs eloba im periodSi, rodesac is miwas exeba? 0,125 wm dasWirda fanjaramde mosaRwevad (burTi wertili); da drois am monakveTSi 1,2 m gaiara. Semdeg b) saiT aris mimarTuli saSualo aCqareba burTi trotuarze daeca da kvlav fanjris – zeviT Tu qveviT? SSM win axta, rasac 0,125 wm moandoma. davuSvaT, •• 50 qva miwidan vertikalurad aisroles t = 0 momentSi. 1,5 wm-is Semdeg qvam maRali rom axtoma dacemis zusti analogia. dro, koSkis wvers Cauara da kidev erTi wamis Sem- 2 wm-is tolia. ra simaRlisaa Senoba? warmoidgine, rogorc romelic burTma fanjris qvemoT gaatara, isaa koSki? •• 51 CogburTis burTi xarisxis Sesamowme- planetis zedapir- blad 4 m simaRlidan iatakze daagdes. burTi idan burTi ver (m) •• 58 Soreuli mzis sistemis erT-erTi 2 m simaRleze axta. burTi iataks 12 mwm-is tikalurad aisro y deg umaRles wertils miaRwia. ra simaRl- ganmavlobaSi exeboda. a) risi tolia Sexebis les. 2.28 suraTze periodSi saSualo aCqarebis ricxviTi mniS- naCvenebia y-is t-ze vneloba? b) zeviTaa saSualo aCqareba mimar- damokidebulebis Tuli Tu qveviT? SSM. grafiki burTisT- •• 52 43,9 m simaRlis xididan mdinareSi qva vis, sadac y bur- Caagdes. 1 wamis Semdeg kidev erTi qva is- Tis avardnis simaRlea, xolo burTis gaS- roles vertikalurad qveviT. orive qva vebis momentSi t = 0. ras udris a) planetaze mdinareSi erTdroulad Cavarda. a) risi Tavisufali vardnis aCqareba da b) burTis tolia meore qvis sawyisi siCqare? b) aage sawyisi siCqare? t (wm) sur. 2.28 amocana 58 siCqaris droze damokidebulebis grafiki TiToeuli qvisTvis. nulovani dro miiCnie Tavi 2-10 grafikuli integrireba pirveli qvis vardnis dawyebis drod. •• 53 onkanidan wyali 200 sm-iT daSorebul moZraobis Seswavlisas • 59 amocanis 2.8 nimuSSi Tavis maqsimal- iatakze wveTavs. wveTebi drois Tanabar Su- uri aCqarebis dros, risi tolia: a) Tavis da aledebSi ecema da, amasTan, pirveli wveTi b) mkerdis siCqare? iataks maSin exeba, rodesac meoTxe wveTi •• 60 karatistis muSti dartymisas moZra vardnas iwyebs. pirveli wveTis dacemisas obas iwyebs welidan da swrafad win var- ra manZiliT aris daSorebuli a) meore da b) deba mxris srulad gaSlamde. 2.29 suraTze mesame wveTi onkanidan? naCvenebia muStis ν(t) siCqare. ra manZilze •• 54 sagani h simaRlidan ecema. manZilis gadaadgilda muSti: a) rodesac t = 50 mwm da naxevars is 1 wm-Si gadis. ipove: a) vardnis b) rodesac muStma maqsimalur siCqares mi- dro da b) simaRle. g) axseni, ratomaa dauS- aRwia? vebeli kvadratuli gamosaxulebidan miRebuli t-s mniSvneloba. •• 55 mTvlemare kata uecrad yvavilis qoTans xedavs, romelic Ria fanjris miRma jer zeviT adis, Semdeg ki qveviT Camodis. qo- v (m/wm) 52 Tani 0,5 wm-is ganmavlobaSi moCans, fanjris simaRlea 2 m. ra manZiliT ascdeba fanjris zeda mxares yvavilis qoTani? ••• 56 kalaTburTeli 76 sm simaRleze axta. t (mwm) sur. 2.29 amocana 60 53 65 t = 0 momentSi kldeze mcocavs uecrad soli Camouvarda. garkveuli xnis Semdeg v (m/wm) kldeze 10 m-iT maRla myofma misma partniorma soli Tavad Camoagdo. 2.32 suraTze mocemulia solebis y mdebareobebis damokidebuleba t droze. ra siCqariTaa nas- t (wm) roli meore soli? sur. 2.30 amocana 61 • • 61 ra manZils gaivlis 16 wamSi morbenali, grafiki mocemulia 2.30 suraTze? y romlis siCqaris droze damokidebulebis • • 62 rodesac fexburTeli Camowodebul burTs Tavs urtyams, Tavis aCqareba SeiZleba sakmaod mniSvnelovani iyos. 2.31 suraTze naCvenebia fexburTelis Tavis a(t) aCqareba, SiSveli da qudiani TavisTvis. ra gansxvavebaa SiSveli da qudiani Tavebis siCqareebs Soris, rodesac t = 7 mwm? sur. 2.32 amocana 65 66 videoTamaSSi wertili daprogramebulia ekranze gadaadgilebisTvis x = 9t – 0,75t3 formulis Sesabamisad, sadac x aris ekranis marcxena kididan daSoreba santimetrebSi, qudiani Tavi a (m/wm) 2 SiSveli Tavi t (wm) xolo t mocemulia wamebSi. rodesac wertili miaRwevs ekranis kides (x = 0 an x =15 sm), t t (mwm) sur. 2.31 amocana 62 damatebiTi amocanebi nulis toli xdeba da wertili kvlav iwyebs x(t)-s Sesabamisad moZraobas. a) dawyebidan ra droSi gaCerdeba wertili? b) x-is ra mniSvnelobisTvis moxdeba es? g) ras udris am dros wertilis aCqareba (rogoria misi 63 SuqniSanze mwvane Suqis anTebisas, avto- niSani)? d) marcxniv Tu marjvniv moZraobs mobili iwyebs TanabaraCqarebul moZraobas wertili gaCerebamde? e) gaCerebis Semdeg? 2,2 m/wm aCqarebiT. am dros mas 9,5 m/wm mud- v) ra dros miaRwevs wertili pirvelad ekra- mivi siCqariT moZravi satvirTo manqana ew- nis kides (t > 0)? eva da uswrebs. a) SuqniSnidan ra manZilze 67 Senobis saxuravidan qva vertikalurad daeweva avtomobili satviTo manqanas? b) ra aisroles. qvam maqsimalur simaRles 1,6 wm- siCqare eqneba am dros avtomobils? is Semdeg miaRwia. vardnis dros qva Senobas 64 72 km/sT siCqariT moZravi wiTeli matare- ascda da 6 wm-Si miwaze daeca. SI sistemaSi: a) beli da 144km/sT siCqariT moZravi mwvane ra siCqariT aisroles qva? b) Senobis saxur- matarebeli erTmaneTis Semxvedri mimar- avidan ra maqsimalur simaRles miaRwia qvam? 2 TulebiT midian. rodesac matareblebs So- g) ra simaRlisaa Senoba? ris manZili 950 m-ia, TiToeuli memanqane 68 TanabaraCqarebulad moZravma manqanam meore matarebels SeniSnavs da amuxruWebs. or wertils Soris 60 m manZili 6 wamSi da- matareblebi muxruWdebian 1 m/wm2 aCqarebiT. fara. meore wertilis gavlisas manqanis moxdeba Tu ara Sejaxeba? Tu moxdeba, ipove siCqare iyo 15 m/wm. a) ras udrida manqanis wiTeli da mwvane matareblebis siCqare Se- siCqare pirvel wertilSi? b) risi tolia aC- jaxebis dros. Tu ara, ipove matareblebs qarebis ricxviTi mniSvneloba? g) pirveli Soris manZili gaCerebis dros. wertilidan ra manZilis daSorebiT iyo man- 54 wrfivi moZraoba Tavi 2 qana gaCerebuli? d) aage x-is t-ze da ν-s t-ze damokidebulebis grafikebi. 73 200 km/sT siCqariT moZravi manqanis gaCereba SesaZlebelia 170 m-Si. davuSvaT, 69 yinulmWreli mudmivi siCqariT aR- aCqareba Tanabaria. ipove misi ricxviTi mniS- mosavleTisken miemarTeboda, rodesac ue- vneloba a) SI sistemaSi da b) g erTeulebSi. carma qarma mas Tanabari aCqareba mianiWa g) ra Tmux droa saWiro dasamuxruWeblad? 3 wm-is ganmavlobaSi. 2.33 suraTze naCvenebia Seni reaqciis droa 400 mwm, romelic gWird- x-is t-ze damokidebulebis grafiki, sadac t = 0 aris momenti, rodesac qarma daubera, xolo x RerZi mimarTulia aRmosavleTisken. eba, raTa situacia aRiqva, fexi muxruWis a) risi tolia yinulmWrelis aCqareba 3 wm-is yvelaze swrafad moZraobs, jei-alaia. bur- ganmavlobaSi? b) risi tolia yinulmWrelis Tis siCqare 303 km/sT-s aRwevs. Tu jei-al- siCqare 3-wamian monakveTSi? g) Tu aCqareba ais moTamaSe, romelsac burTi uaxlovdeba, Tanabari rCeba kidev 3 wm-is ganmavlobaSi, Zalauneburad Tvals daxuWavs 100 mwm-is ra manZils gaivlis yinulmWreli am meore 3- ganmavlobaSi, ra manZils gaivlis am droSi wamian monakveTSi? burTi? pedals daado da daamuxruWo. 74 sportis saxeoba, romelSic burTi 75 x RerZis gaswvriv moZravi wertilis aCqarebaa a = 5t, sadac t mocemulia wamebSi, xolo a - m/wm2-ebSi. rodesac t = 2 wm, wer- x (m) tilis siCqarea +17 m/wm. ras udris siCqare, rodesac t = 4 wm? 76 plekoptera frTebs Zlierad ver iqnevs da amitom am mwers frena ar SeuZlia. t (wm) magram, rodesac wylis zedapirze imyofeba, sur. 2.33 amocana 69 is frTebs haerSi swevs da gemiviT micuravs. warmoidgine, rom plekoptera mudmivi siC- 70 qva Camovarda 100 m simaRlis kldidan. qariT moZraobs garkveuli sigrZis swor xa- ra dro dasWirdeba qvas a) pirveli 50 m-is da zze. ialqaniviT gaSlili frTebiT am manZi- b) bolo 50 m-is gasavlelad? lis gavlas 7,1 wm sWirdeba, xolo dakecili 71 SuqniSanze yviTeli Suqi ainTo. Seni frTebiT _ 25 wami. a) risi tolia fardoba manqanis siCqare zustad siCqaris daSvebuli νgaSlili/νdakecili? b) ra droSi gaivlis zRvris, ν0= 55 km/sT-s tolia. Seni manqanis mweri pirvel 2 m-s gaSlili da dakecili Senelebisas AaCqarebis ricxviTi mniSvnelo- frTebiT? ba 5,18 m/wm2-is tolia. Seni reaqciis dro 77 aSS-s nacionaluri satransporto damuxruWebis dawyebamde T = 0,75 wm-s udris. usafrTxoebis sabWos mier sahaero katas SuqniSanze wiTeli sinaTlis anTebis dros trofis kvlevisas yvelaze mniSvnelovania xazis asacileblad, TviTmfrinavis frenis Sesaxeb informaciis unda ganagrZo 55 km/sT-iT moZraoba Tu Camwerze ganTavsebuli informacia. am Cam- unda daamuxruWo, rodesac xazamde manZili wers `Savi yuTi~ ewodeba, miuxedavad imisa, da yviTeli Suqis anTebis xangrZlivoba aris rom forToxlisferia. Camwerma unda gauZ- gadakveTis Tavidan los katastrofas 3400 g aCqarebis dros 6,5 a) 40 m da 2,8 wm; b) 32m da 1,8wm? 72 matareblis ori sadguri erTmane- mwm-is ganmavlobaSi. Tu aseTi katastrofis Tisgan 1100 m manZilzea. Tu matarebeli dros Camwersa da TviTmfrinavs nulovani distanciis aCqardeba siCqare aqvs drois monakveTis dasasruls, +1,2 m/wm -iT da meore naxevarze Seneldeba rogori iqneba maTi siCqare Sualedis dawye- –1,2 m/wm -iT, risi toli iqneba a) mgzavrobis bisas? dro da b) maqsimaluri siCqare? aage x-is, ν-s 78 pirvel naxevarze 2 2 da a-s t-ze damokidebulebis grafikebi. 2.34 suraTze mocemulia wrfivad moZravi wertilis x-is t-ze damokidebulebis 55 tilis myisi siCqare t = 4,5 wm momentSi? g) 82 2.35 suraTze mocemulia x RerZis gaswvriv moZravi wertilis a aCqarebis t droze damokidebulebis grafiki. rodesac t = 0, risi tolia wertilis saSualo aCqareba 0- wertilis koordinatia 4 m, xolo siCqare – grafiki. a) risi tolia wertilis saSualo siCqare 0-dan 4,5 wm-mde? b) risi tolia wer- dan 4,5 wm-mde? d) risi tolia wertilis myisi 2 m/wm. a) ra siCqare eqneba wertils, rodesac aCqareba t = 4,5 wm momentSi? t = 2 wm? b) dawere gamosaxuleba ν(t)-sTvis (m/wm-ebSi), romelic gamosadegi iqneba Su- x a (m/wm 2) (m) aledisTvis 2 ≤ t ≤ 4. t (wm) sur. 2.34 amocana 78 79 sxeulis mdebareoba mocemulia formuliT x = 2 t 3 , sadac x izomeba metrebSi, xolo t – wamebSi. ipove a) saSualo siCqare da b) saSualo aCqareba t = 1-dan t = 2 wm-mde. Semdeg ipove g) ν myisi siCqare da d) a myisi aCqareba, rodesac t = 1 wm. amis Semdeg ipove e) ν da v) a, rodesac t=2 wm. z) Seadare myisi da saSualo mniSvnelobebi da TiToeul SemTx- t (wm) sur. 2.35 amocana 82 83 matarebelma moZraoba daiwyo TanabaraCqarebulad. garkveuli drois ganmavlobaSi matarebeli midioda 30 m/wm siCqariT, xolo Semdeg 160 m 50 m/wm siCqariT gaiara. gamoTvale a) aCqareba, b) 160 m-is gavlis dro, g) 30 m/wm siCqaris miRwevis dro da d) manZili, romelic matarebelma gaiara uZraobis mdgomareobidan 30 m/wm siCqaris miR- vevaSi axseni, ratom aris erTi meoreze meti. wevamde. e) aage x-is t-ze da ν-s t-ze damokide- T) aage x-is t-ze da ν-s t-ze damokidebulebis bulebis grafikebi. grafikebi da aCvene maTze a)-s pasuxi. 80 tyviis burTi 5,2 m simaRlis tram- 84 120 m simaRlis Senobis bolo sarTulze gaCerebuli carieli liftis bagiri gaw- plinidan tbaSi Cavarda. is wyals garkveuli yda. a) ra siCqariT daecema lifti miwaze? siCqariT daeca da fskerzec imave Tanabari b) ramden xans gagrZeldeba vardna? g) rogo- siCqariT CaiZira. burTma fskers Camovard- ri iqneba liftis siCqare, rodesac naxevar nidan 4,8 wm-Si miaRwia. a) ra siRrmisaa tba? gzas gaivlis? d) ra dro dasWirdeba gzis gamoTvale burTis saSualo siCqaris b) ricx- pirveli naxevris gavlas? viTi mniSvneloba da g) mimarTuleba (maRla 85 reaqtiulZraviani manqana, romelic Tu dabla). davuSvaT, tbaSi wyali daaSres. swor gzaze moZraobs, gamoiyeneba adamian- tramplinidan Camovardnili burTi fsker- ebze didi aCqarebis zemoqmedebis Seswavlis ze kvlav 4,8 wm-Si ecema. gamoTvale burTis mizniT. aseTi manqana 1,8 wm-Si 1600 km/sT siC- sawyisi siCqaris d) ricxviTi mniSvneloba da qares aRwevs. gamoTvale a) aCqareba g-s er- e) mimarTuleba. TeulebSi da b) ganvlili manZili. 81 1977 wlis 26 ianvridan 1983 wlis 18 se- 86 sprinteris maqsimaluri siCqarea 11 qtembramde britanelma jorj miganma fexiT m/wm. Tu sprinteri TanabaraCqarebul moZ- gaiara 30600 kilometri uSuaiadan (samxre- raobas daiwyebs, maqsimalur siCqares 12 m-is Ti amerikis ukiduresi samxreTi wertili) gavlis Semdeg miaRwevs da am siCqares 100 m- pradhous yuremde (aliaska). ramden m/wm-s iani trasis bolomde inarCunebs. a) ra droSi udris misi saSualo siCqare? gaivlis sprinteri 100 m-s? b) drois gasaum- 56 Tavi 2 wrfivi moZraoba jobeseblad sprinteri cdilobs maqsimal- ze. Semdeg megobari muyaos xels gauSvebs, uri siCqaris miRwevis manZili Seamciros. xolo Seni amocanaa _ is rac SeiZleba male risi toli unda iyos es manZili, rom man daiWiro. niSani im adgilze, romliTac muy- trasa 10 wm-Si dafaros? aos daiWer, reaqciis dros gaCvenebs. a) mar- 87 sportuli avtomobili 60 km/sT siCqa- cxena wertilidan ra manZilze moaTavsebdi res 5,4 wm-Si aRwevs. a) ra aCqarebiT moZrao- 50 m/wm niSans? ra manZilze ganalagebdi niS- bs avtomobili am periodSi? b) Tu aCqareba nebs b) 100 g) 150, d) 200 da e) 250 mwm? (mag- Tanabaria, ra manZils gaivlis 5,4 wm-Si? g) ra aliTad, iqneboda Tu ara 100 mwm niSani Savi dro dasWirdeba avtomobils 0,25 km-is gas- wertilidan 2-jer ufro Sors, vidre 50 mwm avlelad, Tu aCqareba a)-Si miRebuli mniSvn- niSani?). elobis tolia. 88 reaqciis dro (mwm) wertili moZraobas iwyebs aTvlis saTavidan t = 0 momentSi da gadaadgildeba x RerZis dadebiTi mimarTulebiT. siCqaris droze damokidebulebis grafiki mocemusur. 2.37 amocana 90 lia 2.36 suraTze. a) ra aris wertilis koordinati, rodesac t = 5 wm? b) ra siCqare aqvs wertils, rodesac t = 5 wm? g) ra aCqarebiT moZraobs wertili, rodesac t = 5 wm? d) ra saSualo siCqare aqvs wertils t = 1-dan t = 5 wm-mde? e) ra saSualo aCqareba aqvs wertils drois am monakveTSi? 91 60 m simaRlis Senobis saxuravidan qva Camovarda. miwidan ra simaRleze imyofeba qva davardnamde 1,2 wm-iT adre? 92 36,6 m simaRlis Senobis saxuravidan burTi vertikalurad Camoagdes. burTma miwidan 12,2 m simaRleze myof fanjaras Camogdebidan 2 wm-is Semdeg Cauara. ra siCqare aqvs burTs, rodesac fanjaras Cauvlis? v (m/wm ) 93 Jongliori burTebs H simaRleze isvris. ra simaRleze unda aisrolos Jongliorma burTebi, rom isini haerSi orjer ufro didxans iyvnen? 94 ori margaliti erTi da imave simaRt (wm) sur. 2.36 amocana 88 lidan Tavisuflad ecema 1 wm-is SualediT. pirveli margalitis vardnis dawyebidan ramden xanSi iqneba margalitebs Soris 10 m manZili? 89 qva vertikalurad aisroles. zeviT frenis dros qvam A wertils ν siCqariT Cauara, xolo B wertils, romelic A-ze 3 m-iT maRlaa _ ν 2 siCqariT. gamoTvale a) ν siCqare da b) qvis mier miRweuli maqsimaluri simaRle. 90 2.37 suraTze naCvenebia adamianis reaqciis drois gasazomi martivi xelsawyo. xelsawyo ubralo muyaoa, romelzec gamosaxulia skala da ori Savi wertili. megobars muyao vertikalurad uWiravs ise, rom misi cera da saCvenebeli TiTebi marjvena wertils exeba, xolo Seni cera da saCvenebeli TiTebi damiznebulia marcxena wertil- 95 manqanis dasamuxruWeblad jer si tuaciis gasaazrebeli dro gWirdeba, ris Semdegac manqana Tanabrad Senelebulad Cerdeba. warmoidgine, rom pirveli fazis ganmavlobaSi (damuxruWebis dawyebamde) manqanam 56,7 m gaiara sawyisi 80,5 km/sT siCqariT, xolo meore fazis ganmavlobaSi – 24,2 m sawyisi 48,3 km/sT siCqariT. gamoTvale a) Seni reaqciis dro da b) aCqarebis ricxviTi mniSvneloba. 96 Tu beisbolis momwodebeli burTs 160 km/sT siCqariT isvris horizontalurad, ra dro dasWirdeba burTs 18,4 m-iT daSorebul bazamde misaRwevad? 57 97 x RerZis gaswvriv protonis moZraobis formulaa x = 50t+10t2, sadac x metrebSi izomeba, xolo t – wamebSi. gamoTvale a) pro- mwm-is manZilze grZeldeba. ra manZils gaiv- tonis saSualo siCqare moZraobis pirveli lis daxamxamebaSi, Tu misi saSualo siCqarea 3 wm-is ganmavlobaSi; b) protonis myisi siC- 3400 km/sT? 102 Tvalis daxamxameba daaxloebiT 100 lis gamanadgurebli “mig-25” pilotis Tva- qare, rodesac t = 3 wm, da g) protonis myisi 103 ra dro moiges niu-iorkis gasasv- aCqareba, rodesac t = 3 wm. d) aage x-is t-ze lelidan bufalos Sesasvlelamde 700 km-ian damokidebulebis grafiki da aCvene, rogor monakveTze avtomobilistebma, roca niu- SeiZleba am grafikze a)-s pasuxis povna. iorkis avtostradaze siCqaris zRvari 55 e) grafikze aCvene b)-s pasuxi; v) aage ν-s t-ze ml/sT-dan 65 ml/sT-mde gazardes? damokidebulebis grafiki da masze aCvene 104 motocikleti moZraobda 30 m/wm siCqariT, rodesac mZRolma daamuxruWa. g)-s pasuxi. 98 1,2 m sigrZis lulis mqone Tofidan damuxruWebis dawyebidan 3 wamSi siCqare 15 gasrolili tyviis siCqare 640 m/wm-ia. dauS- m/wm-mde Semcirda. ra manZils gaivlis mo- vi, rom aCqareba Tanabaria, da gamoTvale tocikleti damuxruWebis dawyebidan ga- dro, romelsac tyvia gaatarebs lulaSi gas- Cerebamde? 105 paraSutisti TviTmfrinavidan xte- rolis Semdeg. 99 2.38 suraTze mocemulia x RerZis ba da 50 metrs Tavisuflad vardeba. Semdeg gaswvriv moZravi wertilis a aCqarebis t-ze paraSuti ixsneba da Senelebisas aCqareba damokidebulebis grafiki. rodesac t = –2 wm, 2 m/wm2-is tolia. paraSutisti miwaze 3 m/wm wertilis siCqarea 7 m/wm. gamoTvale werti- siCqariT ecema. a) ramden xans imyofeba para- lis siCqare, rodesac t = 6 wm. Sutisti haerSi? b) ra simaRleze daiwyo vardna? a (m/wm2) 106 piloti horizontalurad mifrinavs 1300 km/sT siCqariT dedamiwidan 35 mis simaRleze (h = 35). magram t = 0 momentSi dedamiwis zedapiris daxrilobaa θ = 4,3°. Tu piloti TviTmfrinavis mimarTulebas ar Secvlis, ra t momentSi Seejaxeba dedamiwas? t (wm) θ h sur. 2.38 amocana 99 sur. 2.39 amocana 106 100 maRaros urika gorakze 20 km/sT siCqariT avida da iqidan 35 km/sT siCqariT 107 rodesac mZRoli manqanas mkveTrad Camovida (urikis gorakis Tavze yofnis dro amuxruWebs, gaCerebis manZili gamoiTvle- ugulebelyofilia). gamoTvale urikis saSu- ba `reaqciis manZilisa~ da `damuxruWebis alo siCqare mTliani mgzavrobis manZilze. manZilis~ SejamebiT. `reaqciis manZili~ 101 1889 wels iubulporSi (indoeTi) ba- aris sawyisi siCqarisa da mZRolis reaqciis giris gadaqaCvis Sejibri 2 saaTsa da 41 wuTs drois namravli. cxrilSi mocemulia tipi- gagrZelda. gamarjvebulma gundma bagiris uri ricxviTi mniSvnelobebi. a) ra reaqciis centri 3,7 m-iT gadaaadgila. gamoTvale Se- dro aqvs mZRols? b) ra manZilis Semdeg gaC- jibris dros bagiris centris saSualo siC- erdeba manqana, Tu misi sawyisi siCqarea 25 qare sm/wT-ebSi. m/wm? 58 Tavi 2 wrfivi moZraoba tempSi da b) ra iqneba saSualo siCqaris sawyisi siCqare (m/wm) reaqciis manZili (m) 10 7,5 20 30 damuxruWebis manZili (m) gaCerebis manZili (m) ricxviTi mniSvneloba m/wm-ebSi? Tu gundi 5 12,5 saSualo siCqaris ricxviTi mniSvneloba 15 20 35 22,5 45 67,5 TiToeul tempSi gadis 8 m manZils, g) ra dro dasWirdeba TiToeul temps da d) ra iqneba m/wm-ebSi? 114 motociklisti Cqardeba horizontalur swor trasaze. trasis 6 Tanabar, 10 m-iT 108 Saflbordis (samagido TamaSi) diski kiis dartymis Sedegad Tanabrad aCqardeba da 1,8 m-is manZilze 6 m/wm siCqares aRwevs. am dros diski iwyebs Tanabarad Senelebas 2,5 m/wm2 aCqarebiT da Cerdeba. a) ra dro gava diskis aCqarebidan gaCerebamde? b) ra manZils gaivlis diski? 109 aRmosavleTiT mimarTuli x RerZis gaswvriv moZrav motociklets aqvs aCqareba a = (6,1–12t) m/wm2, drois 0 ≤ t ≤ 6 SualedSi. rodesac t = 0, motociklis siCqare da mdebareoba, Sesabamisad, aris 2,7 m/wm da 7,3 m. a) ra maqsimalur siCqares miaRwevs motocikleti? b) ra manZils gaivlis motocikle- daSorebul monakveTebze damontaJebulia boZebi, romlebzec motociklistis moZraobis dro izomeba. cxrilSi mocemulia erTi cdis Sedegebi. a) ipove, ras udris TiToeuli d manZili t-s mimarT da motociklis a aCqareba. b) cxrilis monacemebis gamoyenebiT aage d -s t2-ze damokidebulebis grafiki. g) ipove motociklistis aCqarebis ricxviTi mniSvneloba. boZis nomeri 1 2 3 4 5 6 ganvlili manZili (m) 0 10 20 30 40 50 dro (wm) 0 1,63 2,33 2,83 3,31 3,79 ti t = 0-dan t = 6 wm-mde? 110 msxverplze Tavdasxmisas Cxriala gvelis Tavis aCqarebaa 50 m/wm2. Tu manqanis aCqarebac igivea, ra droSi miaRwevs manqana 100 km/sT siCqares? 111 giganturma avialainerma afrenisTvis 360km/sT siCqares unda miaRwios. ra minimaluri Tanabari aCqarebaa saWiro 1,8 km-is sigrZis asafreni zolidan afrenisTvis? 112 avtomobilis mZRolma 0,5 wamSi siCqares Tanabarad moumata 25 km/sT-dan 115 inglisis nacionalur fizikur la boratoriaSi Tavisufali vardnis g aCqarebis gazomvisTvis vakuumur kameraSi minis burTi maRla aisroles. 2.40 suraTze ΔTdab aris burTis mdebareobis or wertils Soris droiTi intervali konkretul dabal simaRleze, xolo ΔT maR – droiTi intervali konkretul maRal doneze, H aris manZili am or simaRles Soris. daamtkice, rom 55 km/sT-mde. motociklistma ki _ 0,5 wamSi g = 0-dan 30 km/sT-mde. gamoTvale a) manqanis aC- 8H 2 ∆Tdab – ∆T2maR . qareba da b) motociklis aCqareba. dros maspinZelTa gundi mimdevrobiT oTx tempSi moZraobs: a) swrafad (120 nabiji wuTSi, 30 duimi nabijSi), simaRle 113 merilendSi saomari wvrTnebisas gamarTuli Sejibris ∆T maR 2) swrafad da naxevari nabijiT (120 nabiji ∆T dab wuTSi, 15 duimi nabijSi), 3) ormagi tempiT dro (180 nabiji wuTSi, 36 duimi nabijSi), 4) or- H sur. 2.40 amocana 115 magi tempiT da naxevari nabijiT (180 nabiji wuTSi, 18 duimi nabijSi). Tu gundi TiToeul tempSi moZraobs 5 wm-is ganmavlobaSi, a) ra manZils (metrebSi) gaivlis gundi TiToeul 116 NASA-s uwonadobis kvlevis centrSi 145 m simaRlis koSki dgas. es aris carie- 59 li vertikaluri Senoba, romelSic SeiZleba 117 h simaRlidan ν0 sawyisi siCqariT 1 m diametris sferos Camogdeba. a) ra drois burTi vertikalurad dabla isroles. a) ra manZilze iqneba sfero Tavisufali vardnis siCqare eqneba burTs miwaze dacemis dros? mdgomareobaSi? b) rogori iqneba sferos b) ra droSi miaRwevs burTi miwas? ra iqneba siCqare, rodesac is koSkis fskerze arsebul pasuxebi g) a)-sTvis da d) b)-sTvis, Tu burTs damWer mowyobilobas miaRwevs? g) daWeris imave simaRlidan da imave siCqariT maRla Semdeg, sfero neldeba 25 g uaryofiTi aCqa- aisvrian? sanam formulebs dawer, moifiqre, rebiT da bolos Cerdeba. ra manZils gaivlis g)-s da d)-s pasuxebi meti iqneba a)-sa da b)-s sfero Senelebis dros? pasuxebze, naklebi Tu toli. 3 veqtorebi udabnos WianWvela saharis udabnoSi cxovrobs. budidan sakvebis mosaZeb nad gamosuli WianWvela SemTxveviTi traeqtoriiT moZraobs, rogorc es suraTzea naGvenebi. aseTi traeqtoriiT WianWvela udabnos qviSaze zog jer 500 m-ze met manZilsac gadis, magram rogorc ki ukan dabrunebas ganizraxavs, yovelgvari orientiris gareSe pirdapir saxlisken mirbis. rogor agnebs udabnoSi budisken gzas WianWvela niSnebis gareSe? pasuxs am TavSi SeityobT 3.1 ra aris fizika? fizikaSi gvxvdeba uamravi sidide, romelTac aqvs ricxviTi mniSvneloba da mimarTuleba. maSasadame, am sidideebis asaxvisTvis saWiroa specialuri maTematikuri ena – veqtorebis ena. es ena gamoiyeneba inJineriaSi, mecnierebasa da yoveldRiur saubarSic _ ki. Tu odesme vinmesTvis giTqvams: `xuT korpuss rom gascdebi, marcxniv Seuxvie~, maSin swored veqtorebis ena gamogiyenebia. zRvaosnoba mTlianad veqtorebzea dafuZnebuli. fizikosebi da inJinrebi veqtorebs iyeneben brunvisa da magnituri Zalebis aRwerisTvis. am TavSi yuradRebas gavamaxvilebT mxolod veqtorebis enaze. 3.2 veqtorebi da skalarebi swori xazze moZrav wertils SeiZleba hqondes ori mimarTuleba. SegviZlia erTi mimarTulebiT moZraoba dadebiTad miviCnioT, meore mimarTulebiT moZraoba ki – uaryofiTad. magram, samganzomilebian sivrceSi moZravi wertilis mimarTulebis aRsaniSnavad `plius~ an `minus~ niSani sakmarisi aRar aris. saWiroa veqtoris gamoyeneba. veqtors aqvs ricxviTi mniSvneloba da mimarTuleba. veqtori emorCileba masze moqmedebis garkveul wesebs, romlebsac am TavSi ganvixilavT. veqtorul sidides aqvs rogorc ricxviTi mniSvneloba, aseve mimarTuleba da, maSasadame, SeiZleba gamovsaxoT veqtoris saxiT. veqtoruli fizikuri sidideebia gadaadgileba, siCqare da aCqareba. am wignSi kidev bevr veqtorul sidides gavecnobiT, amitom maTTan dakavSirebuli moqmedebebis wesebis Seswavla momdevno TavebSi Zalian dagexmarebaT. 60 61 3.3 veqtorebis geometriuli Sekreba zogierT fizikur sidides, magaliTad, temperatu- B´´ ras, wnevas, energias, masas da dros, mimarTuleba ar gaaCnia. aseT sidideebs skalaruls uwodeben da maT Cveulebrivi algebris kanonebs miusadageben. skalaruB´ li sidide gamoisaxeba ricxviTi mniSvnelobiT da niSniT (magaliTad, temperatura - 400 F). A´´ B (a) umartivesi veqtoruli sididea gadaadgileba anu mdebareobis cvlileba. veqtors, romelic gadaadgile- A A´ bas gamosaxavs, gadaadgilebis veqtors uwodeben B (arsebobs agreTve siCqaris da aCqarebis veqtorebi). roca wertilis mdebareoba icvleba A-dan B-ze (sur. 3.1 a), gadaadgilebas gamovsaxavT am wertilebis SemaerTebeli mimarTuli isriT. isari veqtoris grafikuli gamosaxulebaa. veqtoris simbolos, am wignSi mocemuli sxva tipis isrebisgan gamosarCevad, aqvs samkuTxedis formis isari. 3.1 a suraTze isrebs A-dan B-mde, A´-dan B´-mde da A´´dan B´´-mde erTnairi sidide da mimarTuleba aqvs. maSasa- (b) A sur. 3.1 a) samive isars Tanabari sigrZe da mimarTuleba aqvs, maSasadame, samive erTsa da imave gadaadgilebas asaxavs. b) ori wertilis SemaerTebeli samive xazi gadaadgilebis erTsa da imave veqtors Seesabameba. dame, es gadaadgilebis erTnairi veqtorebia da aRwers wertilis mdebareobis erTnair cvlilebas. veqtoris gadaadgilebisas misi ricxviTi mniSvneloba ar icvleba, Tu misi sigrZe da mimarTuleba ar Seicvala. gadaadgilebis veqtori wertilis moZraobis realur traeqtorias ar gviCvenebs. magaliTad, 3.1 b suraTze A da B wertilebis SemaerTebeli gadaadgile-bis samive wiri imave veqtors Seesabameba, romelsac 3.1 asuraTze vxedavT. gadaadgilebis veqtori mxolod moZraobis saboloo Sedegs asaxavs da ara Tavad moZraobas. 3.3 veqtorebis geometriuli Sekreba davuSvaT, rogorc 3.2a suraTze naCvenebi, wertili moZraobs A-dan B-sken da Semdeg B-dan C-sken. sruli gadaadgilebis asaxva (traeqtorias mniSvneloba ara aqvs) SeiZleba AB da BC gadaadgilebis veqtorebis saSualebiT. am ori gadaadgilebis jami aris gadaadgileba A-dan C-mde. AC-s vuwodebT AB da BC veqtorebis jams (an Sedegs). es Cveulebrivi algebruli jami ar aris. 3.2b suraTze 3.2a suraTidan gadavitaneT veqtorebi da aRvniSneT Tavze patara isriani simboloTi, rogoricaa, magaliTad, namdvili traeqtoria B . Tu gvsurs veqtoris mxolod ricx- viTi mniSvnelobis Cveneba (niSnisa da mimarTulebis gareSe), viyenebT ubralod simboloebs, magaliTad a, b, s da a.S. Tavze isriani simbolo asaxavs veqtoris orive A (a) C sruli gadaadgileba veqtorebis jamia Tvisebas – sididesa da mimarTulebas. 3.2b suraTze naCvenebi sami veqtoris urTierTdamokidebuleba SeiZleba gamoisaxos veqtoruli for- (b) muliT = rac imas niSnavs, rom + , (3.1) veqtori da veqtorebis sur. 3.2 a) AC veqtori AB da BC veqtorebis jamia. b) igive veqtorebi sxva saxelebiT. 62 veqtorebi Tavi 3 jamia. 3.1 formulaSi `+~ niSans da sityvebs `jami~ veqtorebis jami + dasawyisi da `mimateba~ gansxvavebuli mniSvnelobebi aqvs, vinaidan gulisxmobs ricxviT mniSvnelobasac da dasasruli + mimarTulebasac. 3.2 suraTze naCvenebia organzomilebiani da veqtorebis geometriuli Sekrebis procedura. 1) daxaze SenTvis saWiro masStabis sur. 3.3. da veqtorebis Sekreba SeiZleba orive TanmimdevrobiT. ix. formula 3.2. 2) daxaze aseTive masStabis misi saTave veqtoris veqtori. veqtori ise, rom bolos emTxveodes. 3) veqtoruli jami aris veqtori, romelic gaivleba -s saTavidan -s bolomde. veqtorTa Sekrebas ori Tviseba aqvs: 1) Sekrebis Tanmimdevrobas mniSvnelobas ar aqvs. + + + + -s + + + ( ( ( + + + ) ) ) a , b da c veqtorebis Sekrebisas maTi nebismieri mimdevrobiT dalageba SeiZleba. ixile formula 3.3. sur. 3.4 da -s Sekreba gvaZlevs imave Sedegs, rasac + = + -s Sekreba (sur. 3.3). maSasadame, -s da (gadanacvlebadobis kanoni) (3.2) 2) rodesac mocemuli gvaqvs orze meti veqtori, maTi Sekreba nebismieri TanmimdevrobiT SeiZleba. maSasadame, Tu gvinda , da da maT jams mivumatoT , an jer da veqtorebis Sekreba, SegviZlia jer SevkriboT da mere mivumatoT da SevkriboT . Sedegi orive SemTx- vevaSi erTi da igive iqneba, rogorc es 3.4 suraTzea naCvenebi. veqtor – -s aqvs ( + )+ = +( + ) (uRlebadobis kanoni) (3.3) -s Tanabari sidide, magram sawinaaRmdego mimarTuleba (sur. 3.5). 3.5 suraTze naCvenebi ori veqtoris Sekreba mogvcems + (– ) = 0 maSasadame, – -s damateba igivea, rac veqtorebs Soris sxvaobis dasadgenad: Tu = = – = anu veqtorebis veqtorisa da – sxvaoba – -s gamokleba. veqtorTa am Tvisebas viyenebT +(– ) , maSin (veqtorebis gamokleba) (3.4) veqtoris SekrebiT vipoveT. 3.6 suraTze igive moqmedeba naCvenebia geometriulad. iseve, rogorc algebraSi, veqtoris simbolos gadatana gamosaxulebis erTi mxaridan meoreze daSvebu- – da – Tanabari sidide da urTierTsawinaaRmdego mimarTuleba aqvs. sur. 3.5 veqtorebs lia, magram saWiroa niSnis Secvla. magaliTad, Tu gvaqvs formula 3.4 da gvWirdeba + = an = + -s povna, SeiZleba daiweros . miuxedavad imisa, rom mxolod gadaadgilebis veq- 63 3.3 veqtorebis geometriuli Sekreba – – = (a) sur. 3.6 a) veqtorebi , da – . b) veqtors – gamosaklebad -sTvis – veqtoris (b) veqtors vumatebT. torebi ganvixileT, Sekrebisa da gamoklebis wesebi saerToa yvela tipis veqtorisTvis, iqneba es siCqare, aCqareba Tu raime sxva veqtoruli sidide. Tumca, dasaSvebia mxolod erTi tipis veqtorebis Sekreba. magaliTad, SegviZlia SevkriboT ori gadadgileba, ori siCqare, magram gadaadgilebisa da siCqaris Sekreba yovelgvar azrsaa moklebuli. es igivea, rom SevkriboT 21 wami da 12 metri. . sakontrolo wertili 1 = + . da da gadaadgilebebis sidideebia 3 m da 4 m, xolo -s mimarTulebebis gaTvaliswinebiT, rogori iqneba: a) -s maqsimaluri SesaZlo sidide da b) minimaluri SesaZlo sidide? 3.1 amocanis amoxsnis nimuSi sportuli orientirebis gakve Tilze Seni mizania banakidan rac – SeiZleba Sors wasvla, risTvisac – mxolod sami wrfivi gadaadgilebis 30° = ufleba gaqvs. SegiZlia nebismieri – gadaadgilebebi: – , 2 km aRmosavleTiT; , 1 km dasavleTiT. -Ti, ki – SegiZlia Secvalo -Ti. rogoria is maqsimaluri manZili, romliTac SegiZlia banaks daSorde mesame gadaadgilebis dasasruls? amoxsna: gavavloT veqtorebi , , , – da – , rogorc naCvenebia 3.7a suraTze. amis Semdeg furcelze gadavaadgiloT sami veqtori da ise davalagoT, rom erTis saTave emTxveodes meoris bolos. vipovoT am veqtorebis masStabi km-Si (a) b) , 2 km, 30° Crdilo-aRmosavleTiT; g) – (b) TanmimdevrobiT gamoiyeno Semdegi a) + jami. pirveli veqtoris saTave emTxveva banakis mdebareobas, xolo mesame veqtoris bolo – gaCerebis wertils. jamuri veqtori gaivleba pirveli veqtoris saTavidan mesame veqtoris bolomde. misi d si- 0 1 2 sur. 3.7. a) gadaadgilebis veqtorebi. maTgan mxolod sami unda gamoviyenoT. b) banakidan Seni daSoreba udidesi iqneba, Tu nebismieri TanmimdevrobiT SeaerTeb , da – veqtorebs. dide banakidan daSorebis manZilia. irkveva, rom d manZili udidesi iqneba , da – veqtorebisTvis. veqtorebis dalageba nebismieri TanmimdevrobiT SegviZlia, vinaidan jami ar Seicvleba. 3.7b suraTze naCvenebi TanmimdevrobisTvis veqtorebis jami iqneba = + + (– ) 3.7 a suraTze naCvenebi masStabis gamoyenebiT vzomavT veqtorebis jamis d sigrZes: d = 4,8 k (pasuxi) 64 veqtorebi Tavi 3 y y y θ θ x θ y (g) x x x (b) (a) veqtoris ax da ay komponentebi. b) veqtoris gadaadgilebisas komponentebi ar icvleba, Tu veqtoris sidide da mimarTuleba ucvlelia. g) komponentebi Seadgens marTkuTxa samkuTxedis kaTetebs, romlis hipotenuza veqtoris sididis tolia. sur. 3.8. a) 3.4 veqtorebis komponentebi veqtorebis geometriuli Sekreba sakmaod damRlelia. arsebobs amis ufro martivi da zusti meTodi, romelic veqtorebis marTkuTxa koordinatTa sistemaSi ganlagebas moiTxovs. x da y RerZebi, rogorc wesi, sibrtyeze ixazeba, ise, rogorc es 3.8a suraTzea naCvenebi. z RerZi gamodis koordinatTa saTavidan, Tumca amjerad mas ugulebelvyofT da mxolod organzomilebian veqtorebs ganvixilavT. veqtoris komponenti aris veqtoris gegmili RerZze. magaliTad, 3.8 a suraTze ax aris veqtoris komponenti x RerZze, xolo ay – veqtoris komponenti y RerZze. RerZze veqtoris gegmilis sapovnelad am veqtoris boloebidan RerZze perpendikularul xazebs davuSvebT. x RerZze veqtoris gegmili aris x komponenti, xolo y RerZze –y komponenti. komponentebis povnis process veqtoris dagegmileba ewodeba. veqtoris komponents igive mimarTuleba aqvs (RerZis gaswvriv), rac Tavad veqtors. 3.8 suraTze ax da ay dadebiTia, vinaidan TulebiT. Tu veqtori mimarTulia orive RerZis dadebiTi mimar- veqtors SemovabrunebT, maSin orive komponenti uaryofiTi gaxdeba. 3.9 su- raTze b veqtoris dagegmileba mogvcems dadebiT bx komponents da uaryofiT by komponents. sazogadod, veqtors aqvs sami komponenti, Tumca 3.8 a suraTis SemTxvevaSi, z RerZze misi komponenti nulis tolia. 3.8 a da b suraTebze Cans, rom Tu veqtors mimarTulebis Secvlis gareSe gadavaadgilebT, misi komponentebi ar Seicvleba. 3.8a suraTze veqtoris komponentebis povna SeiZleba trigonometris daxmarebiT: sadac θ aris kuTxe θ ax = a cos θ da ay= a sin θ , (3.5) veqtorsa da x RerZis dadebiT mimarTulebas Soris, xolo a aris bx = 7 m veqtoris sidide. 3.8 g suraTze Cans, rom veqtori da misi komponentebi x da y RerZebze marTkuTxa samkuTxeds qmnis. suraTze aseve Cans, rom komponentebis by = –5 m saSualebiT SeiZleba veqtoris ageba, Tu komponentebs ise davalagebT, rom erTis saTave meores bolos daemTxves, Semdeg ki avagebT marTkuTxa samkuTxeds, romlis hipotenuza erTi komponentis saTavidan meoris bolosken mimarTuli veqtori iqneba. koordinatTa RerZebze veqtoris dagegmilebis veqtoris komponenti x RerZze dadebiTia, xolo y RerZze _ uaryofiTi. sur. 3.9. Semdeg SeiZleba misi komponentebis gamoyeneba Tavad veqtoris nacvlad. magaliTad, 3.8 a suraTze naCvenebi 65 3.4 veqtorebis komponentebi veqtori moicema (srulad ganisazRvreba) a-Ti da θ-Ti. imave veqtoris asaxva ax da ay komponentebis saSualebiTac SeiZleba. monacemebis orive wyvili erTsa da imave informacias Seicavs. Tu veqtori mocemuli gvaqvs komponenturi saxiT (ax da ay) da gvsurs misi aRniSvna sidide-kuTxuri saxiT (a da θ ), viyenebT Semdeg formulebs: ay a = √ ax2 + ay2 da tg θ = a x (3.6) ufro zogad, samganzomilebian SemTxvevaSi, veqtoris mosacemad gvWirdeba sidide da ori kuTxe (vTqvaT, a, θ da φ) an sami komponenti (ax, ay da az). sakontrolo wertili 2. suraTze naCvenebi meTodebidan, romelia swori x da y RerZebis komponentebidan veqtoris asagebad? y y ax y ax x ax x ay x ay ay (a) (b) (g) y y y x ay ax ax x ay x ay ax (d) (e) amocanis amoxsnis nimuSi (v) 3.2 y patara TviTmfrinavi Rrublian dRes aeroportidan afrinda. amis Semdeg TviT mimarTulebiT samxreTiT iyo TviTmfrinavi aeroportidan am momentSi? 22° m triT CrdiloeTiT da ramdeni kilometriT 5k kilome- 100 21 Crdilo-aRmosavleTis mifrinavda 22° kuTxiT. ramdeni manZili (km) is P 200 mfrinavi SeniSnes aqedan 215 km-Si, rodesac amoxsna: amocanaSi mocemulia veqtoris si- θ dide (215 km) da kuTxe (22° Crdilo-aRmosavleTiT). unda vipovoT veqtoris komponentebi. avagoT xy koordinatTa sistema, sadac 0 0 manZili (km) 100 x da x RerZis dadebiTi mimarTuleba aRmosavleTi iqneba, xolo y RerZisa – CrdiloeTi (sur. 3.10). simartivisTvis, aTvlis saTave sur. 3.10. TviTmfrinavi aeroportidan frindeba da garkveuli drois Semdeg P wertils aRwevs. 66 veqtorebi Tavi 3 aeroportis mdebareobas davamTxvioT. TviT mfrinavis gadaadgileba _ veqtori aer Tebs aTvlis saTavesa da im wertils, sadac TviTmfrinavi dainaxes. veqtoris komponentebis sapovnelad, gamoviyenoT formula 3.5, dx = d cos θ = (215 km)(cos 68°) = 81 km (pasuxi) amocanis amoxsnis nimuSi dy = d sin θ = (215 km) (sin 68°) = 199 km ≈ 200 km, (pasuxi) sadac θ = 68° (= 90° – 22°): maSasadame, TviTmfrinavi aRmosavleTiT 81 km-iT, xolo CrdiloeTiT 200 km-iT aris daSorebuli aeroportidan. 3.3 ori aTwleulis manZilze mRvimeebis mkv- zeviT levarTa jgufebi eZebdnen gzas kentukiSi Cr. mdebare flint rijis gamoqvabulTa sistemasa da mamontis gamoqvabuls Soris. rode- 2,6 km das. sac, sabolood, es gza aRmoaCines, gamoqvab- dasasruli ulebis erTiani sistema msoflios yvelaze 25 m aRm. dasawyisi qveviT grZel gamoqvabulad aRiares (200 km-ze meti samx. sigrZis). gzis maZieblebma uricxv gadasasv- (a) lelebze xoxviTa da mokuntuli siaruliT da 25 m zeviT. risi tolia maTi gadaadgileba dasawyisidan dasasrulamde? zeviT dasasruli gaiares 2,6km dasavleTiT, 3,9 km samxreTiT 0,025 km d dh amoxsna: mocemuli gvaqvs samganzomilebiani θ veqtoris komponentebi da gvWirdeba veqtoris ricxviTi mniSvneloba da ori kuTxe, qveviT (b) raTa veqtoris mimarTuleba ganvsazRvroT. Tavdapirvelad komponentebi gamovsaxoT, rogorc 3.11a suraTzea naCvenebi. horizontaluri komponentebi (2,6 km dasavleTiT da 3,9km samxreTiT) marTkuTxa samkuTxedis ka- sur. 3.11 a) mRvimeebis mkvlevarTa jgufis mTliani gadaadgilebis komponentebi da maTi horizontaluri gadaadgileba dh. b) gverdiTi xedi, romelic aCvenebs dh-s da jgufis mTliani gadaadgilebis veqtors. Tetebs qmnis. mkvlevrebis horizontaluri gadaadgileba samkuTxedis hipotenuzaa da misi dh sigrZe gamoiTvleba piTagoras Teoremis saSualebiT: dh = √ (2.6 km)2 + (3.9 km)2 = 4.69 km. gamosayeneblad, 3.11a suraTs SevxedoT Crdilo-dasavleTidan. miviRebT 3.11b suraTs, romelzec vertikaluri komponenti da dh horizontaluri gadaadgileba kidev 3.11a suraTze naCveneb samkuTxedze vxedavT, erT marTkuTxa samkuTxeds qmnis. amjerad, rom horizontaluri gadaadgileba mimar- mkvlevrebis mTliani gadaadgileba samkuT Tulia samxreT-dasavleTisken θh kuTxiT, xedis hipotenuzas emTxveva, romlis d sig- romelic gamoiTvleba Semdegnairad: rZe gamoiTvleba formuliT: 3,9 km tg θh = 2,6 km cxrilidan, θh = 56° d = √(4.69 km)2 + (0.025 km)2 = 4.69 km ≈ 4.7 km. (pasuxi) mTliani gadaadgilebis mimarTulebis sa povnelad saWiro ori kuTxidan erTi vipo gadaadgileba (pasuxi) mimarTulia horizon taluri gadaadgilebidan zeviT, θ kuTxiT. cxrilidan θ = 0.3°. (pasuxi) veT. maSasadame, mkvlevrebis gadaadgilebis veq- vertikaluri komponentis (25 m = 0,025 km) toris sididea 4,7km, xolo mimarTuleba 3.4 veqtorebis komponentebi – 56° kuTxiT samxreT-dasavleTiT da 0,3° ba gamoqvabulidan gasasvlelis sapovnelad. kuTxiT zeviT. saerTo vertikaluri moZ- maT mier ganvlili gzis traeqtoria mniS- raoba horizontalur moZraobasTan Sedare- vnelovnad gansxvavdeba gadaadgilebis veq- biT umniSvneloa. Tumca, es ar iyo didi nug- torisgan, romelic ubralod swori xazia eSi mkvlevrebisTvis, romelTac mravaljer sawyis da saboloo mdebareobebs Soris. mouwiaT maRla acoceba da dabla Camococe- amocanis amoxsnis taqtika taqtika 1: kuTxeebi gradusebsa da radianebSi kuTxe x RerZis dadebiTi mimarTulebis mimarT dadebiTia, Tu saaTis isris moZ raobis sawinaaRmdego mimarTulebiT izo meba da uaryofiTia, Tu izomeba saaTis isris moZraobis mimarTulebiT. magaliTad, 210° da –150° erTi da igive kuTxea. kuTxeebi izomeba radianebSic. am ori Tad, 3.13 suraTis SemTxvevaSi am codniT SegiZliaT daadginoT, aris Tu ara kalkulatorze miRebuli Sedegi swori. sxvadasxva meoTxedebSi funqciebis niSnebis codnac sakmaod gamosadegia. taqtika 3: Sebrunebuli trigonometriuli funqciebi Sebrunebuli funqciebis (sin–1, cos–1 da erTeulis dasakavSireblad gaixsene, rom tg–1) SemTxvevaSi, kalkulatorze miRebuli Sekruli wre 360° da 2π radiania. magaliTad, pasuxis garda arsebobs sxva SesaZlebeli 40°-is radianebSi gadasayvanad, daiwereba 2π rad 40° = 0,7 rad 360° taqtika 2: trigonometriuli funqciebi aucileblad dagWirdebaT ZiriTadi tri gonometriuli funqciebis – sinusis, kosi- pasuxic, romelsac kalkulatori ar iTvlis. 3.13 suraTze mocemulia Sebrunebuli trigonometriuli funqciebis Sedegebis diapazoni kalkulatorisTvis. magaliTad, sin–10,5 asocirdeba kuTxeebTan 30° (romelic kalkulatorze moCans, vinaidan is Sedis mis diapazonSi) da 150°. orive mniSvnelobis gasagebad 3.13 a suraTze gaavle horizonta- nusisa da tangensis gansazRvrebebis codna, luri xazi 0,5-is gaswvriv da daakvirdi, sad vinaidan isini xSirad gamoiyeneba mecniere- kveTs is sinusis funqcias. baSi. 3.12 suraTze naCvenebia am funqciebis gansazRvrebebi, sadac samkuTxedis gverdebi aRniSvnebis gareSea. aucilebelia, agreTve, imis codna, Tu rogor ganvasxvavoT swori pasuxi? swori pasuxi isaa, romelic yovel konkretul SemTxvevaSi ufro gonivruli gveCveneba. magaliTad, ganvixiloT θh kuTxis gamoT- rogor icvleba trigonometriuli funqci vlis procesi amocanis 3.3 nimuSSi, sadac ebi kuTxis cvlilebasTan erTad. magali- tg θh = 3,9/2,6 = 1,5. kalkulatorze tg–1 1,5-is sin θ = cos θ = tg θ = θ kuTxis mopirdapire kaTeti hipotenuza θ kuTxis mimdebare kaTeti hipotenuza θ kuTxis mopirdapire kaTeti θ kuTxis hipotenuza θ θ kuTxis mopirdapire kaTeti θ kuTxis mimdebare kaTeti mimdebare kaTeti sur. 3.12 trigonometriuli funqciebis gansazRvrisTvis gamoyenebuli samkuTxedi. ixile, agreTve, danarTi E. 67 68 veqtorebi Tavi 3 +1 - 900 sin 0 900 1800 2700 3600 +1 -1 - 900 (a) cos 0 900 1800 2700 3600 -1 (b) tg +2 +1 (g) - 900 0 900 1800 2700 3600 -1 sur. 3.13 sami saWiro funqcia. kalkulatoris gamoTvlis diapazoni Sebrunebuli trigonometriuli funqciebisTvis naCvenebia grafi-kebis gamuqebuli nawiliT. -2 gamoTvla gvaZlevs Sedegs θh = 56°, magram θh reTve formula 3.6 tg θ-sTvis gamosadegia = 236°(=180°+56°) kuTxis tangensi aseve 1,5- mxolod im SemTxvevaSi, Tu kuTxe izomeba ia. romeli pasuxia swori? fizikuri situ- x RerZis dadebiTi mimarTulebis mimarT. aciidan gamomdinare (sur. 3.11a) 56° sworia, raime sxva mimarTulebisadmi kuTxis gazom- xolo 236° – aSkarad araswori. taqtika 4: veqtorebis kuTxeebis gazomva triuli funqciebi unda Seicvalos, xolo formula 3.6-Si wiladi unda Sebrundes. bevrad iolia kuTxis gazomva x RerZis dadebiTi 3.5 formulebi cos θ-sa da sin θ-sTvis, ag 3.5 vis SemTxvevaSi 3.5 formulaSi trigonome- mimarTulebis mimarT. erTEulovani veqtorebi erTeulovani veqtoris sidide 1-is tolia da romelime konkretul mimarTulebas mi- aniSnebs. erTeulovan veqtors ar gaaCnia ganzomileba da erTeuli. misi erTaderTi daniSnuleba mimarTulebis Cvenebaa. x, y da z RerZebis dadebiTi mimarTulebis mqone erTeulovani veqtorebi aRiniSneba , da simboloebiT, sadac ` ^ ~ niSani gamoiyeneba isris nacvlad (sur. 3.14). 3.14 suraTze RerZebis ganlagebas marjvena koordinatTa sistemas uwodeben. es sistema aseTived rCeba RerZebis Tanabrad Semo- y brunebis SemTxvevaSi. am wignSi mxolod mxolod marjvena koordinatTa sistemebs viyenebT. erTeulovani veqtorebi Zalian sasargebloa sxva veqtorebis asaxvisTvis. magaliTad, 3.8 da 3.9 suraTebze veqtorebi da da = ax + ay (3 .7 ) = bx + by (3.8) es ori formula ilustrirebulia 3.15 suraTze. sidideebs ax x SeiZleba gamoisaxos, rogorc da ay uwodeben veqtoris veqtorul z sur. 3.14. , da erTeulovani veqtorebi asaxavs mimarTulebebs marjvena koordinatTa sistemaze. 69 3.6 veqtorebis Sekreba komponentebis saSualebiT komponentebs. ax da ay skalaruli sidideebia da veqtoris skalaruli komponentebi (an ubralod komponentebi) ewodeba. magaliTad, amocanis 3.3 nimuSSi erTeulovan veqtorebSi gamovsaxoT mRvimeebis mkvlevagadaadgileba. Tavdapirvelad 3.11a suraTis mixedviT avagoT 3.14 suraTze naCvenebi rebis koordinatTa sistema. Semdeg , da mivmarToT, Sesabamisad, aRmosavleTisken, zeviT da samxreTisken. maSasadame, mogzaurobis dasawyisidan dasasrulamde gadaadgileba erTeu- lovani veqtorebis saSualebiT gamoisaxeba Semdegnairad: = –(2,6 km) aq –(2,6km) + (0,025 km) x RerZze veqtoris dx +(3,9km) (3.9) komponentia, xolo – (2,6 km) – dx komponenti. 3.6 veqtorebis Sekreba komponentebis saSualebiT veqtorebis Sekreba SesaZlebelia geometriulad, naxazis saSualebiT. veqtorebi aseve ikribeba zogierTi tipis kalkulatorzec. Sekrebis mesame gza gulisxmobs maTi komponentebis Sekrebas RerZebis mixedviT, rasac axla SeviswavliT. Tavdapirvelad, ganvixiloT toloba = romelic ambobs, rom + + veqtori igivea, rac (3.10) , veqtori. maSasadame, veqtoris TiToeu- + veqtoris Sesabamisi komponentis toli: rx = ax + bx (3.11) ry = ay + by (3.12) rz = az + bz. (3.13) li komponenti unda iyos sxvagvarad rom vTqvaT, ori veqtori erTmaneTis tolia, Tu maTi Sesabamisi komponentebi tolia. formulebi 3.10-3.13 gvauwyebs, rom veqtorebis Sesakrebad: 1) veqtorebi unda da davSaloT skalarul komponentebad; 2) SevkriboT es komponentebi, raTa jamuri komponentebi miviRoT, da 3) gavaerTianoT veqtoris komponentebi Tavad veqtoris veqtoris mis- aRebad. mesame etapze SegviZlia gadavwyvitoT, erTeulovan-veqtorul CanawerSi (rogorc formula 3.9-Si) gamovsaxoT veqtori Tu sidide-kuTxur CanawerSi (amocanis 3.3 nimuSis pasuxi). veqtorebis komponentebiT Sekrebis procedura gamosadegia veqtorebis gamoklebis SemTx- = vevaSic. gavixsenoT, rom gamokleba + (– ). gamoklebis mizniT, = da – – SeiZleba gadaiweros, rogorc mimateba SevkriboT komponentebis saSualebiT, miviRebT: dx= ax – bx , dy= ay – by da dz= az – bz , sadac = dx + dy + dz . y d ay bx 0 x θ 0 (a) y θ b x ax by (b) sur. 3.15 a) veqtoris komponentebi; b) veqtoris komponentebi. 70 veqtorebi Tavi 3 sakontrolo wertili 3. -is da 1 -is komponentebis niSani x RerZze? b) ra iqneba 2 -is da 1 -is komponentebis niSani y RerZze? g) ra iqneba 2 + 1 y a) mocemul suraTze ra iqneba d2 d1 -is komponentebis niSnebi x da y RerZebze? 2 x 0 amocanis amoxsnis nimuSi 3.4 y 3.16 a suraTze naCvenebia sami veqtori: da = (4,2 m) – (1,5 m) 3 = (–1,6 m) + (2,9 m) 2 = (–3,7 m) 1 ra iqneba veqtorTa jami , romelic asevea -3 -2 1 suraTze naCvenebi? -1 amoxsna: sami veqtoris Sekreba SesaZlebelia -2 RerZebze maTi komponentebis saSualebiT. x RerZisTvis SevkriboT , da 2 3 4 -3 veqtorebis x komponentebi, raTa miviRoT jamuri toris x komponenti: x -1 veq(a) rx= ax+ bx+ cx= 4,2 m –1,6 m + 0 =2,6 m. aseve moviqceT y RerZisTvis: y 2.6 ry = ay+ by+ cy = –1,5 m + 2,9 m – 3,7 m = – 2,3 m. gavaerTianoT es komponentebi -3 -2 veq -1 1 2 3 x 4 -1 toris Casawerad erTeulovani veqtorebis -2.3 -2 saSualebiT: -3 = (2,6 m) – (2,3 m) (pasuxi) (b) sadac (2,6 m) veqtoris komponenti x aris RerZze, xolo -(2,3 m) - y RerZze. 3.16 b suraTze naCvenebia veqtoris komponentebis dalagebis erTi meTodi veqtoris misaRebad. Cvens kiTxvaze pasuxis gacema aseve SesaZlebelia, Tu gavigebT veqtoris ricxviT sur. 3.16. xolo kuTxe (x RerZis dadebiTi mimarTule bisadmi) -2,3 m θ = tg–1 (2,6 m) = – 41°, mniSvnelobas da kuTxes. 3.6 formulidan gamomdinare, ricxviTi mniSvnelobaa r = √ (2,6 2 m) + (–2,3 m) 2 veqtori aris danarCeni sami veqtoris jami. (pasuxi) `minus~ niSani gvauwyebs, rom kuTxe saaTis is(pasuxi) ris moZraobis mimarTulebiT izomeba. ≈ 3,5 m amocanis amoxsnis nimuSi eqsperimentebiT dadginda, rom am Tavis 3.5 raxavs, RerZebis gaswvriv gadaadgilebebs dasawyisSi suraTze gamosaxuli udabnos Seajamebs da gansazRvravs veqtors, ro- WianWvela moZraobis traeqtorias Tavis melic pirdapir budisken aris mimarTuli. gonebaSi arsebul koordinatTa sistemaSi magaliTisTvis miviCnioT, rom WianWvela avlebs. rodesac budeSi dabrunebas ganiz- xy koordinatTa sistemaSi 3.17a suraTze 3.6 veqtorebis Sekreba komponentebis saSualebiT y naCvenebi mimarTulebebiT xuTjer garbis 6 sm-s. mexuTe garbenis bolos, ra iqneba WianWvelas mTliani gadaadgilebis veqtoris 71 dasasruli 3 j sidide da kuTxe? ra iqneba WianWvelas sabo loo mdebareobis da saxlis damakavSirebeli veqtoris saxli 5 120° 30° sidide da kuTxe? j 1 gadaadgi daadgilebis veqtori: 1 + 2 + (a) y dasasr. 3 + 4 + . j 5 Semdeg gamovTvaloT x komponentebis jami: 3.8 sm 8.2 sm dj, x = d1x + d2x+ d3x + d4x + d5x , (3-14) da y komponentebis jami: y (3-15) saxli mTliani veqtori x da y bolos avagoT x saxli (b) dasasr. dj, y = d1y + d2y + d3y + d4y + d5y . 30° 4 lebis gasagebad unda SevajamoT xuTi ga j= 2 saxli amoxsna: amocanis amoxsnis gza sami nawilisgan Sedgeba. Tavdapirvelad, 150° x saxli komponentebisgan. (g) formula 3.14-is gamosaTvlelad, viy- sur. 3.17. a) xuTi garbenis saZiebeli traeqtoria; b) j veqtori da misi x da y komponentebi. g) saxli saxlTan damakavSirebeli veqtori. enebT 3.5 formulas: d 1x = (6 sm) cos 0° = + 6 sm d 2x = (6 sm) cos 150° = – 5,2 sm d 3x = (6 sm) cos 180ο = –6 sm d 4x = (6 sm) cos (–120ο) = –3 sm d 5x = (6 sm) cos 90ο = 0. amis Semdeg 3.14 formula mogvcems: dj, x = + 6 sm + (–5,2 sm) + (–6 sm) analogiurad vafasebT xuTi veqtoris y komponentebs 35. formulis daxmarebiT. Sedegebi 3.1 cxrilSia mocemuli. am Sedegebis 3.15 formulaSi CasmiT viRebT: dj, y = + 3,8 sm SualebiT j veqtoris sididisa da kuTxis j 2 2 = √d j,x + d j,y = √ (-8.2 sm)2 + (3.8 sm)2 = 9,0 sm. kuTxis gasagebad (x RerZis dadebiTi mimarTulebis mimarT) aviRoT Sebrunebuli tangensi: θ = tg–1 = tg–1 dy veqtori da gasagebad gamoviyenoT formula 3.6. veqto- 3.1 dx (sm) j ris sidide tolia: + (–3 sm) + 0 = –8,2 sm cxrili 3.17b suraTze mocemulia misi x da y komponentebi. komponentebis sa- (sm) 1 2 3 4 5 + 6.0 - 5.2 - 6.0 - 3.0 0 0 + 3.0 0 - 5.2 + 6.0 mTliani - 8.2 + 3.8 dj, y dj, x 3,8 sm –8.2 sm = – 24.86° SeniSvna: gaixsene amocanebis amoxsnis me-3 taqtika, sadac vTqviT, rom Sebrunebuli tangensis gamoTvlisas kalkulatorma SeiZleba swori pasuxi ar mogvces. –24,86° niSnavs, rom mTliani veqtori x y koordi- x 72 veqtorebi Tavi 3 natTa sistemis meoTxe meoTxedSia. magram, komponentebidan veqtoris agebis Semdeg (sur. 3.17b) davinaxavT, rom j veqtori meore meoTxedSia. maSasadame, kalkulatoris pasuxi unda gavasworoT 180°-is damatebiT: θ = –24,86° + 180° = 155,14° ≈ 155° WianWvelis j j-is mimarTuleba (sur. 3.17g). saxli sawinaaRm- (– 24,86° ≈ – 25°). maSasadame j-is sidide da kuTxe tolia: gadaadgilebis sidide da saxli = 9 sm – 25° kuTxiT (pasuxi) udabnos WianWvela, romelic 500 m-ze met 155° kuTxiT veqtors, romelic (pasuxi) mimarTu- lia WianWvelas mdebareobidan budisken, amocanis amoxsnis nimuSi am amocanaSi mocemuli veqtorebis Sek reba kalkulatoris saSualebiT SeuZlebelia. sami veqtori akmayofilebs pirobas: j-is dego mimarTulebisTvis kuTxe ukve gvaqvs kuTxe tolia: dj = 9 sm toli sidide aqvs da sawinaaRmdego B=A+C (3.16) A -s ricxviTi mniSvnelobaa 22 erTeuli, xolo kuTxe x RerZis dadebiTi mimarTulebis mimarT _ – 47° (saaTis isris moZraobis mimarTulebiT). C -s ricxviTi mniSvnelobaa 17 erTeuli, xolo kuTxe x RerZis dadebiTi mimarTulebis mimarT _ φ (saaTis isris moZraobis sawinaaRmdego mimarTulebiT). B mimarTulia x RerZis dadebiTi mimarTulebiT. ras udris B -s sidide? amoxsna: am SekiTxvaze pasuxs kalkulatorze gamoTvliT ver gavcemT, vinaidan ar viciT C -s φ kuTxe. samagierod, 3.16 formula SegviZlia CavweroT x da y RerZebze veqtoris komponentebis saSualebiT. vinaidan B mimarTulia x RerZis gaswvriv, virCevT am manZils gadis, uamrav gansxvavebul gadaadgilebas Seasrulebs, magram bolos mainc uSecdomod gamoiTvlis j-is (am Tavis wakiTxvis gareSec). 3.6 B cos 0° = 22 cos (–47°) + 17 cos φ (3.17) B-s povna ar SegviZlia, vinaidan φ-jer kidev ucnobia. axla formula 3.16 gamovsaxoT y komponentebis saSualebiT By = Ay + Cy gamovTvaloT y komponentebi 3.5 for mulis saSualebiT da CavsvaT cnobili monacemebi, miviRebT: B sin 0° = 22 sin (– 47°) + 17 sin φ , saidanac gamomdinareobs: 0 = 22 sin (– 47°) + 17 sin φ φ-s amoxsna gvaZlevs: φ = – sin–1 22 sin (– 47°) = 71,17° 17 CavsvaT es Sedegi 3.17 formulaSi da miviRebT: B = 20,5 erTeuls (pasuxi) daakvirdiT meTods, romelic am amocanis amoxsnisas gamoviyeneT: rodesac x kom- RerZs da vwerT Bx = Ax + Cx Semdeg TiToeuli x komponenti gamovsa xoT 3.5 formulis saSualebiT da CavsvaT CvenTvis cnobili monacemebi, miviRebT: ponentebze vatarebdiT moqmedebebs, Segvxvda ori ucnobi – CvenTvis saintereso B da CvenTvis uintereso φ. Semdeg moqmedebebi CavatareT y komponentebze da gamovTvaleT φ, Semdeg isev x RerZs davubrundiT da gamovTvaleT B. 73 3.7 veqtorebi da fizikis kanonebi 3.7 veqtorebi da fizikis kanonebi am wignis nebismier suraTze mocemul koordinatTa sistemaSi x da y RerZebi wignis kideebis paraleluria. maSasadame veqtoris ax da ay komponentebic kideebis paraleluria (sur. 3.18 a). RerZebis mimarTulebis gansazRvris erTaderTi mizezi (a) simartivea da aq sxva raime dafaruli azri ar arsebobs. SegviZlia 3.18b suraTze mocemuli RerZebis (da ara veqtoris) Semobruneba φ kuTxiT, ris Semdegac komponentebs axali mniSvnelobebi eqnebaT. vuwodoT am mniSvnelobebs ´ da x ´. vinaidan y φ-s sidide usasrulo ricxvTa simravlea, aseve usasruloa φ veqtoris komponentebis wyvilTa mniSvnelobebi. romelia komponentebis `swori~ wyvili? nebismieri, vinaidan TiToeuli wyvili erTsa da imave veqtors gansxvavebulad aR- wers. yvela wyvili erTnairi sididisa da mimarTulebis mqone veqtors qmnis. 3.18 suraTze gvaqvs: (3.18) da (b) sur. 3.18 a) veqtori da misi komponentebi. b) igive veqtori koordinatTa sistemis RerZebis φ kuTxiT Semobrunebis Semdeg. θ = θ´ + φ. (3.19) es imas niSnavs, rom koordinatTa sistemis arCevanSi sruli Tavisufleba gvaqvs, vinaidan veqtorebs Soris kavSiri (magaliTad, veqtorebis Sekreba 3.1 formuliT) ar aris damokidebuli aTvlis saTavis mdebareobaze an koordinatTa sistemis RerZebis mimarTulebaze. igive Seexeba sxva fizikur sidideebs Soris kavSirebs: arc erTi maTgani ar aris damokidebuli koordinatTa sistemis arCevaze. am yvelafers daumateT veqtorebis enis ubraloeba da simdidre da mixvdebiT, ratom gamoisaxeba fizikis kanonebi TiTqmis yovelTvis am enaze. magaliTad, mxolod 3.10 formulas SeuZlia warmoadginos sami (an meti) damokidebuleba, magaliTad, formulebi 3.11, 3.12 da 3.13. 3.8 veqtorebis gamravleba* veqtorebis gamravlebis sami meTodi arsebobs, magram arc erTi ar aris Cveulebrivi algebruli gamravlebis analogiuri. am Tavis wakiTxvisas gaiTvaliswine, rom kalkulatori veqtorebis gamravlebaSi mxolod im SemTxvevaSi dagexmareba, Tu gamravlebis ZiriTad wesebs gaigeb. veqtoris gamravleba skalarze veqtoris s skalarze gamravleba gvaZlevs axal veqtors. am veqtoris sidide veqtoris sididisa da s-is absoluturi mniSvnelobis namravlia. misi mimarTuleba emTxveva mimarTulebas, Tu s dadebiTia, da sawinaaRmdegoa, Tu s uaryofiTia. -s -s s-ze gayofisTvis -s vamravlebT 1/s-ze. * am masalas mxolod mogvianebiT (me-7 da me-11 TavebSi) gamoviyenebT, amitom Tqven maswavlebels SeuZlia am qveTavis Seswavla im droisTvis gadados. 74 veqtorebi Tavi 3 veqtoris veqtorze gamravleba veqtoris veqtorze gamravlebis ori gza arsebobs. erTi gza gvaZlevs skalars (skalarul namravls), xolo meore _ axal veqtors (veqtorul namravls). studentebi gamravlebis am or meTods xSirad erTmaneTSi ureven, amitom yuradRebiT gaiazre gansxvaveba maT Soris. skalaruli namravli da veqtorebis skalaruli namravli (sur. 3.19) iwereba rogorc · da ganisazRvreba Sem- φ degnairad: · sadac a aris b– = ab cos φ, (3.20) (a) veqtoris ricxviTi mniSvneloba, -s komponenti -s mimarTulebis gaswvriv udris a cos φ-s. veqtoris ricxviTi mniSvneloba, xolo φ – kuTxe da veqtorebs Soris (ufro zustad, maT mimarTulebebs Soris). arsebobs ori aseTi φ kuTxe: φ da 360° – φ. 3.20 formulaSi orives gamoyeneba SeiZleba, vinaidan maTi kosinusebi erT- -s komponenti -s mimarTulebis gaswvriv udris b cos φ-s. maneTis tolia. aRsaniSnavia, rom 3.20 formulis marjvena (b) mxares mxolod skalarebia (maT Soris cos φ). ma- · Sasadame, formulis marcxnivac da , maT Soris φ kuTxiT. b) TiToeul veqtors aqvs komponenti meore veqtoris mimarTulebis gaswvriv. sur. 3.19. a) veqtorebi skalaruli sididea. skalaruli namravli ori sididis namravlia. erTi sididea erT-erTi veqtoris ricxviTi mniSvneloba, xolo meore – meore veqtoris pirveli veqtoris mimarTulebis gaswvriv skalaruli komponenti. magaliTad, 3.19b suraTze -s skalaruli komponenti dan -s mimarTulebis gaswvriv aris a cos φ. am komponents -ze daSvebuli perpendikulari gansazRvravs. aseve, gaswvriv aris -s komponenti -s bolo- -s mimarTulebis cos φ. roca or veqtors Soris φ kuTxe aris 0°, erTi veqtoris komponenti meoris gaswvrivaa da, aqedan gamomdinare, veqtorebis skalaruli namravli maqsimaluria. Tu φ = 90°, maSin erTi veqtoris komponenti meoris gaswvriv nulia da skalaruli namravli nulis tolia. formula 3.20 SeiZleba davweroT Semdegnairad: · = (a cos φ (b) = (a) (b cos φ) (3.21) gadanacvlebadobis kanons skalaruli namravlisTvis aseTi saxe aqvs: · = · . rodesac ori veqtori warmodgenilia erTeulovan-veqtoruli saxiT, maTi skalaruli namravli daiwereba Semdegnairad: · = (ax + ay + az ) · (bx + by +bz ) (3.22) SesaZlebelia am formulis gavrcobac jufTebadobis kanonis mixedviT: Tu pirveli veqtoris TiToeul komponents gavamravlebT meore veqtoris TiToeul komponentze, miviRebT: · = ax bx + ay by + azbz (3.23) 3.8 veqtorebis gamravleba* C da D veqtorebis sidideebia, Sesabamisad, 3 erTeuli da 4 sakontrolo wertili 4. erTeuli. ras udris C -s da D -s mimarTulebebs Soris kuTxe, Tu C · D tolia a) nulis, b) 12 erTeulis da g) –12 erTeulis? amocanis amoxsnis nimuSi risi tolia φ kuTxe = 3 – 4 -sa da 3.7 veqtoruli saxiT warmovadgenT da gamoviyenebT jufTebadobis kanons: = –2 + 3 -s Soris? amoxsna: unda iTqvas, rom aq mocemuli bevri moqmedebis Sesruleba kalkulatoriTac SeiZleba, magram Tu gsurT ukeTesad Seiswav- · = (3 – 4 )(–2 + 3 ) = = 3 (–2 ) + 3 (+3 ) + (– 4 )(–2 )+(– 4 )(+3 ). Semdeg formula 3.20 gamoviyenoT bolo loT skalaruli namravli, kalkulators nu gamosaxulebis TiToeuli wevrisTvis. pirve- gamoiyenebT. ori veqtoris mimarTulebas Soris kuTxe li wevrisTvis erTeulovan veqtorebs Soris skalaruli namravlis gansazRvrebaSi Sedis kuTxea 0°, sxva wevrebisTvis ki aris 90°. mi- (formula 3.20): viRebT: · = ab cos φ am formulaSi, a aris (3.24) veqtoris sidide, anu a = √32 + (–4)2 = 5, xolo b aris (3.25) veqtoris sidide b = √(–2) + 3 = 3,61. 2 2 · = (–6)(1) + (9)(0) + (8)(0) – (12)(0) = – 6. CavsvaT es ricxvi da formulebis 3.25 da 3.26 Sedegebi 3.24 formulaSi, miviRebT: – 6 = (5)(3,61) cos φ. ase rom, (3.26) SegviZlia calke gamovTvaloT formula 3.24- φ = cos–1 –6 = 109° ≈ 110°. (5)(3,61) (pasuxi) is marcxena mxare, Tu veqtorebs erTeulovan- veqtoruli namravli da × veqtorebis veqtoruli namravli iwereba, rogorc da aris veqtori, romlis sidide: sadac φ aris da c = ab sin φ , (3.27) -s Soris umciresi kuTxe (mcire kuTxes imitom viyenebT, rom sin φ da sin(360° – φ) algebruli niSniT gansx- φ vavdeba). (a) Tu da paraleluria an antiparaleluria, maSin × =0. × -s absoluturi sidide, romelic asec SeiZleba daiweros × , maqsimaluria, Tu da erTmaneTis φ perpendikularulia. sur. 3.20. marjvena xelis wesis ilustracia veqtorebis namravlisTvis. (a) marjvena xelis TiTebiT veqtori veqtorSi gadaitaneT. gawvdili cera TiTi aCvenebs = × veqtoris mimarTulebas. b) naCvenebia, rom × da × erTmaneTis sapirispiroa. (b) 75 76 Tavi 3 veqtorebi veqtoris mimarTuleba -s Semcveli sibrtyis perpendikularulia. 3.20a sura- -s da Tze naCvenebia, rogor ganvsazRvroT da = × -s mimarTuleba e.w. marjvena xelis wesiT. veqtorebi mimarTulebis Seucvlelad ganalage ise, rom saerTo saTavidan gamodiodnen. Semdeg warmoidgine xazi, romelic veqtorebis sibrtyis perpendikularulia. warmoidgine, rom es xazi marjvena xeliT giWiravs ise, rom TiTebi veqtoridan veqtorisken iyos mimarTuli am veqtorebs Soris arsebuli umciresi kuTxiT. gaSlili ceri veqtoris mimar- Tulebas aCvenebs. veqtoruli gamravlebis dros Tanmimdevroba mniSvnelovania. 3.20 b suraTze = × -s mimarTulebas marjvena xelis wesiT vsazRvravT. gaSlili ceri mimarTulia wina SemTxvevis sawinaaRmdego mimarTulebiT, maSasadame ´ = – × . es niSnavs, rom = – ( × ) (3.28) sxvagvarad rom vTqvaT, gadanacvlebadobis kanoni veqtoruli namravlisTvis ar moqmedebs. erTeulovan-veqtoruli Canaweris SemTxvevaSi gvaqvs × = (ax + ay + az ) × (bx + by + bz ) (3.29) frCxilebis gaxsna SegviZlia jufTebadobis kanonis mixedviT. es imas niSnavs, rom pirveli veqtoris TiToeuli komponenti veqtorulad unda gamravldes meore veqtoris TiToeul komponentze. erTeuli veqtorebis veqtoruli namravlebi mocemulia E danarTSi (ix. `veqtorebis namravlebi~). magaliTad, 3.29 formulis SemTxvevaSi gveqneba ax × bx = ax bx ( × ) = 0 vinaidan ori erTeuli veqtori, da paraleluria, amitom maTi veqtoruli namravli nulis tolia. aseve gveqneba ax × by = ax by ( × ) = ax by × sididis gamosaTvlelad gamoviyeneT formula 3.27 (veqtorebs -s da -s aqvT erTeuli sidide da maT Soris kuTxea 90°). × -is mimarTulebis sapovnelad gamoviyeneT marjvena xelis wesi da davadgineT, rom is mimarTulia z RerZis dadebiTi mimarTulebiT (maSasadame -s mimarTulebiT). 3.29 formulaSi frCxilebis gaxsna mogvcems: × = (ay bz – by az) + (az bx – bz ax) + (ax by – bx ay) (3.30) veqtoruli namravlis gamoTvla, agreTve, SesaZlebelia determinantis (rogorc naC venebia E danarTSi) an kalkulatoris gamoyenebiT. imis dasadgenad, aris Tu ara xyz koordinatTa sistema marjvena sistema, × = veqto ruli namravlisTvis gamoiyene marjvena xelis wesi. Tu gaSlili ceri z RerZis dadebiTi mimarTulebiTaa, maSin es sistema marjvenaa. sakontrolo wertili 5. C da D veqtorebis sidideebia Sesabamisad, 3 da 4 erTeuli. ra iqneba C -s da D -s mimarTulebebs Soris kuTxe, Tu am veqtorebis veqtoruli namravli C × D aris 1) nuli da 2) 12 erTeuli? 77 amocanis amoxsnis nimuSi 3.8 3.21 suraTze veqtori xy sibrtyeSi Zevs, misi sididea 18 erTeuli, xolo daxriloba x RerZis dadebiTi mimarTulebis mimarT _ 250°. veqtoris sididea 12 erTeuli, xolo mimarTuleba z RerZis dadebiT mimarTulebas emTxveva. rogori iqneba veqtoruli namravli = × ? amoxsna: rodesac ori veqtori mocemulia sidide-kuTxuri saxiT, maT veqtorul namravls vpoulobT 3.27 formulis saSuale- veqtori (xy sibrtyeSi) veqtoruli namravlia. sur. 3.21. da veqtorebis biT: c = ab sin φ = (18)(12)(sin 90°) = 216 (pasuxi) aseTi Canaweris dros ori veqtoris Tuli. gaSlili ceri gviCvenebs mimarTulebas. maSasadame, rogorc suraTzea Zevs xy sibrtyeze. imis gamo, rom veqtoruli namravlis mimarTulebas vpou naCvenebi, lobT marjvena xelis wesiT. 3.21 suraTze misi mimarTuleba warmoidgine, rom Seni marjvena xeli kuTxe iqneba da veqtoridan -skenaa mimar- amocanis amoxsnis nimuSi Tu = = 3 – 4 da = –2 + 3 , ra iqneba × ? amoxsna: rodesac ori veqtori mocemulia erTeulovan-veqtoruli saxiT, maTi veqtoruli namravlis povna SeiZleba uRleba- -s perpendikularulia, 250° – 90° = 160° veqtorebis sibrtyis perpendikularulia, xolo TiTebi veqtoris (pasuxi) x RerZis dadebiTi mimarTulebisadmi. 3.9 jvena xelis wesiT. aq pirveli wevrisTvis or veqtors Soris φ kuTxe nulis tolia. danarCeni wevrebisTvis φ = 90°. vipovoT: = –6(0) + 9(– ) + 8(– ) – 12 = –12 – 9 – 8 (pasuxi) dobis kanonis gamoyenebiT. es niSnavs, rom veqtori SegviZlia davweroT: = (3 – 4 )×(–2 + 3 ) = 3 ×(–2 ) + 3 ×3 + (–4 )×(–2 ) + (–4 )×3 . Semdeg TiToeul wevrs gamoviTvliT 3.27 formuliT da mimarTulebas vipoviT mar- da veqtorebis perpendikularulia. amis damtkiceba SeiZleba, Tu vaCvenebT, rom navs, rom · = 0 da · = 0. es niS- veqtoris komponenti -s an -s mimarTulebis gaswvriv ar arsebobs. amocanis amoxsnis taqtika taqtika 5: dakavSirebuli domebi. veqtorul namravlTan gavrcelebuli Sec- veqtoruli namravlis povnis dros gavrcele bulia ramdenime Secdoma: 1) rodesac veqtorebi suraTze mocemulia ise, rom erTis saTave meores bolos emTxveva, maT sworad ar alageben. Tqven gonebaSi unda ganalagoT (ukeTesia gadaxazoT) veqtorebi ise, rom saerTo saTavidan gamodiodnen, magram ar unda SecvaloT maTi mimarTuleba. 2) marjvena xelis wesis arasworad gamoyeneba, rodesac xelSi kalkulatori an fanqari uWiravT. 3) zogjer namravlis pirvel veqtors ver mimarTaven meore veqtorisken, vinaidan es xelis saSinel daZabvas moiTxovs. zogjer igive xdeba, rodesac veqtors warmosaxviT abruneben marjvena xelis gamoyenebis nacvlad. 4) marjvena koordinatTa sistemasTan dakavSirebiT Secdoma xdeba, rodesac gaviwydeba, rogor aago aseTi sistema (ixile sur. 3.14). 78 veqtorebi Tavi 3 mimoxilva da Sejameba skalarebi da veqtorebi. skalarebs, mag- toruli komponentebi, xolo ax, ay da az – ska- aliTad temperaturas, mxolod ricxviTi laruli komponentebi. mniSvneloba aqvs. isini ganisazRvreba ricxviTi erTeuliT (10°C) da emorCileba Cveulebrivi ariTmetikisa da algebris kanonebs. veqtorebs, magaliTad, gadaadgilebas, veqtorebis Sekreba komponentebiT. veq torebis komponenturi formiT Sesakrebad, viyenebT wesebs: ricxviTi mniSvnelobac aqvs da mimarTule- rx = ax + bx , ry = ay + by , rz = az + bz (3.11- 3.13) bac (5 m, CrdiloeTiT). isini veqtoruli al- aq gebris kanonebs emorCileba. veqtorebis geometriulad Sesakrebad saWiroa isini ise davalagoT, rom erTis saTave meoris bolos emTxveodes. pirveli veqtoris saTavisa da meore veqtoris bolos damakavSirebeli veqtori aris am veqtor- -s rom Ta jami. gamovakloT, amisTvis sawinaaRmdego mimarTulebiT – -s misaRebad. Semdeg – Sesakrebi veqtorebia, xolo – veq- torTa jami. veqtorebis geometriuli Sekreba. da da SevabrunoT davumatoT -s. veqtorebis Sekreba emorCileba gadanacvlebadobisa da uRlebadobis kanonebs. veqtorebi da fizikuri kanonebi. nebismieri fizikuri veqtorebs mdgomareoba, Seexeba, SeiZleba romelic gamoisaxos bevr SesaZlo koordinatTa sistemaSi. Cveulebriv, virCevT umartives koordinatTa sistemas. Tumca, veqtorul sidideebs Soris kavSiri da fizikuri kanonebi ar aris damokidebuli Cvens mier arCeul koordinatTa sistemaze. skalaris namravli veqtorze. s skalarisa da veqtoris namravli axali veqto- veqtoris komponentebi. nebismieri organzomilebiani veqtoris ax da ay skalaru- ria, romlis sididea sν. mimarTuleba emTx- li komponentebi koordinatTa RerZebis Tu s uaryofiTia. gaswvriv moiZebneba veqtoris boloebidan koordinatTa RerZebze perpendikularebis daSvebiT. komponentebi moicema formulebiT: ax = a cos θ da ay = a sin θ , (3.5) sadac θ aris kuTxe x RerZis dadebiT mimarTulebasa da veva -s, Tu s dadebiTia, da sawinaaRmdegoa, -s s-ze gasayofad -s vam- ravlebT 1/s-ze. skalaruli namravli. da veqtorebis skalaruli namravli iwereba ase: · b da skalaruli sididea, romelic moicema formuliT: · veqtoris mimarTulebas = ab cos φ, (3.20) φ aris kuTxe -sa da -s mimarTule- Soris. komponentis algebruli niSani mis sadac mimarTulebas aCvenebs Sesabamisi RerZis bebs Soris. skalaruli namravli aris erTi mimarT. komponentebis saSualebiT SesaZle- veqtoris sididisa da am veqtoris mimar- veqtoris sididisa da mimarTulebis Tulebaze meore veqtoris skalaruli kom- belia dadgena: ponentis namravli. ay a = √ ax2 + ay2 da tg θ = a x (3.6) erTeulovan-veqtoruli Canaweri. erTeulovani veqtorebis , da -s sidideebi erTis tolia, xolo mimarTulia, Sesabamisad, marjvena koordinatTa sistemis x, y da z RerZebis dadebiT mimarTulebas emTxveva. veqtoris Cawera erTeulovani veqtorebis saSualebiT SeiZleba Semdegi saxiT: sadac ax , ay = ax + ay + az da az aris (3.7) veqtoris veq- erTelovan-veqtorul CanawerSi: · = (ax +ay +az )·(bx +by +bz ) (3.22) SegviZlia am formulis gavrcoba uRlebadobis kanonis gamoyenebiT. aRsaniSnavia, rom · = · . veqtoruli namravli. ori da veqtoris veqtoruli namravli iwereba ase: da aris veqtori, romlis sididea: c = ab sin φ, sadac · (3.27) φ aris -sa da -s mimarTulebebs So- 79 ris umciresi kuTxe. -s mimarTuleba veqtorul CanawerSi da × veqtorebis sibrtyis perpendikularulia da ganisazRvreba marjvena xelis wesiT, rogorc naCvenebia 3.20 suraTze. daimaxs- × ovre, rom = –( × ). erTeulovan- = (ax + ay + az )×(bx + by + bz ) (3.29) am gamosaxulebis gavrcobac SeiZleba uRle badobis kanonis gamoyenebiT. SekiTxvebi 1 floridis universitetis golfis gundma 5 unda iTamaSos mdeloze, romlis SuagulSi a) = + – , aris Tu ara + (– ) = + (– ); b) = (– ) + + xy koordinatTa sistemaSi. gundis wevrebma g) + (– ) = + burTi unda Caagdon xvrelSi, romlis koor- 6 3.23 suraTze naCvenebi ori veqtori Zevs xy sibrtyeSi. ra niSnebi eqneba x da y komponentebs a) 1 + 2-sTvis, b) 1 – 2-sTvis da g) 2 – 1-sTvis? y àligatoris ormoa~ amoTxrili. 3.22 suraTze naCvenebia golfis moednis zedxedi dinatebia (8 m, 12 m). magram burTis Cagdebis ufleba aqvT mxolod qvemoT mocemuli gadaadgilebebis erTxel an metjer gamoyenebiT: 1 = (8 m) + (6 m) , 2 = (6 m) , 3 Tu da ? = (8 m) àligatoris ormos~ koordinatebia (8 m, 6 m). 2 Tu gundis romelime wevri burTs am ormoSi x Caagdebs an ormoze gadaatarebs, es moTamaSe avtomaturad gadaecema floridis Statis universitets – mTavar mters. gadaadgilebebis ra mimdevroba unda gamoiyenos universitetis gundis wevrma, rom burTi ormoSi ar Cauvardes? y 1 sur. 3. 23. KkiTxva 6. 7 3.24 suraTze naCvenebi RerZebis romeli ganlagebaa marjvena koordinatTa sistema? rogorc wesi, RerZis simbolo mis dadebiT mimarTulebas aRniSnavs. ormo x z àligatoris ormos~ sur. 3.22 x kiTxva 1. 2 y SeiZleba Tu ara, rom ori veqtoris sidideebis jami am veqtorebis jamis sididis z (a) toli iyos? Tu ara, maSin ratom? Tu ki, maSin rodis? 3 3.2 formulaSi vxedavT, rom da dobis kanons. niSnavs Tu ara es, rom ? 4 gamosaxe a) + = b) + = g) + = da veqtorebi ise, rom da a + b = c; – ; da a2 + b2 = c2. – z y (b) (g) x z x veqtorebis Sekreba emorCileba gadanacvleba– y x x y y z = y (d) x z (e) sur. 3. 24. KkiTxva 7. (v) 80 veqtorebi Tavi 3 8 Tu · = · , aris Tu ara da erT- maneTis toli? 9 Tu veqtori da sxva oTxi veqtori, romelTac Tanabari = q( + ) da aris -s perpendikularuli, maSin rogori mimarTule-ba eqneba 10 3.26 suraTze naCvenebia -s 3.25 suraTze naCveneb sam mdgomareo- baSi, rodesac q mudmiva a) dadebiTia da sidide da sxvadasxva mimarTuleba aqvT. a) am oTxi veqtoridan romelTa skalaruli namravlia -ze erTmaneTis toli? b) romlis -ze skalaruli namravlia uaryofiTi? b) uaryofiTia? y y x x z z (1) θ θ θ θ x sur. 3. 26. KkiTxva 10. z (2) (3) sur. 3. 25. KkiTxva 9. amocanebi θ = 30°. gansazRvre veqtoris: a) x komponenti da b) y komponenti. SSM amoxsnebi mocemulia Student Solutions Manual-Si. www amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday •5 ILW Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley.com/ college/halliday. gemi sawyisi wertilidan 100 km-iT aRmosav- • – ••• wertilebis raodenoba amocanis sirTules gemma unda gacuros 120 km CrdiloeTis mimarTulebiT. moulodnelma qariSxalma leTiT gadaanacvla. a) ra manZili da b) ra gviCvenebs mimarTulebiT unda gacuros gemma daniSnulebis adgilamde misaRwevad? qveTavi 3-4. veqtoris komponentebi •1 ras udris xy sibrtyeze mdebare •6 3.28 veq- suraTze mZime meqanizmi 12,5 m (d = 12,5) manZilze aacures 20°-iT daxril (θ = 20°). toris a) x komponenti da b) y komponenti, Tu ficarze misi mimarTulebaa 250° x RerZis dadebiTi bidan ra simaRleze aacures meqanizmi? mimarTulebis mimarT, saaTis isris sawinaaR- b) horizontalurad ra manZilze gadaad mdego mimarTulebiT, xolo sididea 7,3 m? gilda igi? •2 a) sawyisi mdebareo- mocemuli kuTxeebi gamosaxe radianebSi: a) 20°, b) 50°, g) 100°. mocemuli kuTxeebi gadaiyvane gradusebSi: d) 0,33 radiani, e) 2,1 radiani, v) 7,7 radiani. •3 veqtoris x komponentia –25 m, xolo y komponentia +40 m. a) risi tolia sidide? b) ras udris kuTxe veqtoris θ veqtorsa da x RerZis dadebiT mimarTulebas Soris? SSM. •4 3.27 suraTze y ••7 gadaadgilebis veqtori xy sibrtyeSi Zevs. misi sididea 15 m, kuTxe ki sur. 3. 28. KkiTxva 6. θ sur. 3. 27. KkiTxva 4. oTaxis zomebia 3 m (simaRle) × 3,70 m × 4,3 m. buzi frenas erTi kuTxidan iwyebs, x Semoifrens oTaxis garSemo da Cerdeba diagonalurad mopirdapire kuTxeSi. a) risi tolia misi gadaadgilebis sidide? SeiZleba (30, 40), (bx , –70), (–20, cy), (–80,–70). oTxi gar- Tu ara misi traeqtoriis sigrZe b) naklebi benis mTliani gadaadgilebis xy komponente- iyos gadaadgilebis sidideze? g) meti iyos bia (–140, –20). risi tolia a) bx da b) cy? risi gadaadgilebis sidideze? d) toli iyos tolia mTliani gadaadgilebis g) sidide da gadaadgilebis sididisa? e) airCie Sesabamisi d) kuTxe (x RerZis dadebiTi mimarTulebis koordinatTa sistema da gamosaxe gadaadg- mimarT)? ilebis veqtoris komponentebi erTeulovan- •14 gvaqvs ori veqtori, veqtoruli saxiT. v) Tu buzi frenis nacvlad fexiT dadis, gamoTvale traeqtoriis umoklesi SesaZlebeli sigrZe (miniSneba: am kiTxvaze pasuxi gamoTvlis gareSe SeiZleba gaices. oTaxi yuTs hgavs. gaSale misi kedlebi da sibrtyeze ganalage). SSM WWW. qveTavi 3-5. veqtoris Sekreba komponentebiT •8 adamiani fexiT midis mocemuli marSru =(3m) +(4m) da b = (5 m) +(-2 m) . warmoadgine + b : a) erTeulo- van-veqtoruli saxiT, ipove misi b) sidide da g) kuTxe ( -s mimarT). axla warmoadgine b – d) erTeulovan veqtoruli saxiT da ipove misi e) sidide da v) kuTxe. •15 mocemulia ori veqtori: = (4 m) – (3 m) +(1 m) = (–1 m) +(1 m) +(4 m) da tiT: 3,1 km CrdiloeTiT, Semdeg 2,4 km dasav- erTeulovan-veqtoruli saxiT warmoadgine: leTiT, bolos ki 5,2 km samxreTiT. a) aage a) + , b) – b da g) mesame veqtori, rom- veqtorebi am moZraobisTvis. b) ra manZilze lisTvisac da g) ra mimarTulebiT gafrindeboda Citi •16 Sen akeTeb oTx wrfiv moZraobas brtyel wrfivad imave sawyisi wertilidan imave saboloo wertilamde. •9 adamianma ganizraxa mivides 3,4 km-iT daSorebul adgilze, romelic 35° Crdilo-aR- + c = 0. SSM. – zedapirze. moZraobas iwyeb xy koordinatTa sistemis aTvlis saTavidan da amTavreb wertilSi (–140m, 30m). moZraobebis x da y komponentebia (metrebSi): 20 da 60, bx da –70, –20 da dasavleTiT mimarTuli quCebi. ra minimal- cy , –60 da –70. risi tolia a) bx komponenti da b) cy komponenti? gamoTvale mTliani gadaadgilebis g) sidide da d) kuTxe (x Rer- uri manZilis gavlis Semdeg miaRwevs is dan- Zis dadebiTi mimarTulebis mimarT). iSnulebis adgils? •17 •10 manqana gadis 50 km-s aRmosavleTiT, Sem- didea 50 m da TiToeuli xy sibrtyeSi Zevs. x deg 30 km-s CrdiloeTiT, bolos ki 25 km-s 30° RerZis dadebiTi mimarTulebis mimarT maTi Crdilo-aRmosavleTiT. aage veqtorebi da kuTxeebia, Sesabamisad, 30°, 195° da 215°. ga- mosavleTiT mdebareobs. amisTvis man unda gaiaros Crdilo-samxreTiT an aRmosavleT- , da gansazRvre manqanis mTliani gadaadgilebis moTvale a) sidide da b) kuTxe. moTvale •11 gamoTvale a) erTeulovan-veqtorul CanawerSi ra iqneba jami + , Tu = (4 m) + (3 m) da (–13 m) +(7 m) ? gamoTvale + = -s b) sidide da g) mimarTuleba. SSM. •12 da gadaadgilebebis komponentebia c x = 7 , 4 , c y = – 3 , 8 , c z = – 6 , 1 ; d x= 4 , 4 , d y = – 2 , veqtorebidan TiToeulis si- + + -s: a) sidide da b) kuTxe. ga– + -s: g) sidide da d) kuTxe. veqtoris: e) sidide da v) kuTxe, romlisTvisac ( •18 + = + )–( + jamSi ) = 0. ILW. veqtoris sididea 12 m, xolo daxriloba x RerZis dadebiTi mimarTulebis mimarT aris 40°, saaTis isris moZraobis sawinaaRmdego mimarTulebiT. veqtoris sididea 15 m, xolo daxrilo- dz = 3,3. moZebne jamuri veqtoris a) x, b) y da g) z komponentebi. ba x RerZis uaryofiTi mimarTulebis mi- •13 texasuri mziT Sewuxebuli WianWvela xy dego mimarTulebiT. ras udris sibrtyeze darbis. oTxi mimdevrobiTi garbe- a) sidide da b) daxriloba (x RerZis dadebiTi nis x da y komponentebi santimetrebSi aseTia: marT 20° saaTis isris moZraobis sawinaaRm- mimarTulebis mimarT)? veqtoris 81 82 veqtorebi Tavi 3 •19 3.29 suraTze da veqtorebi Tana- pirveli xoWos axali mdebareobis adgilze? bari sididisaa — 10 m, xolo maTi kuTxeebia ••24 θ1 = 30° da θ2 = 105°. moZebne jamuri veqto- TovaSi moyva (Tova imdenad Zlieria, rom mi- -is sidide wis da zecis garCevac ki SeuZlebelia). mkvl- ris a) x da b) y komponentebi, g) bazaze dabrunebis dros mkvlevari da d) kuTxe x RerZis dadebiTi mimarTulebis evari apirebda 5,6 km-is gavlas CrdiloeTiT, mimarT. SSM ILW WWW. magram, rodesac amindi gamokeTda, aRmoaCina, rom 7,8 km gaiara Crdilo-aRmosavleTi- y dan 50° kuTxiT. a) ra manZilze da b) ra mimarTulebiT unda imoZraos mkvlevarma bazaze dasabruneblad? θ2 •••25 a) 3.30 suraTze a gverdis sigrZis kubi erTi kuTxiT exeba xyz koordinatTa sistemis aTvlis saTaves. diagonali aris θ1 0 x sur. 3. 29. KkiTxva 19. •20 a) risi tolia qvemoT mocemuli oTxi xazi, romelic erT kuTxes akavSirebs meoresTan centris gavliT. erTeulovan-veqtorul CanawerSi, ra iqneba sxeulis di- erTeulovan-veqtoru- agonali kuTxidan, romlis koordinatebia li saxiT? risi tolia am jamis b) sidide, a) (0,0,0), b) (a, 0, 0), g) (0, a, 0) da d) (a,a,0)? g) kuTxe gradusebSi da d) kuTxe radianeb- e) gamoTvale kuTxeebi diagonalsa da mimde- Si? bare gverdebs Soris. v) gansazRvre sxeulis veqtoris jami : 6 m + 0,9 radiani : 5 m — 75° : 4 m + 1,2 radiani : 6 m — 210° diagonalis sigrZe a-s mimarT. SSM. z •21 Wadrakis figurebi gadaadgildeba kvadratebze, romelTa gverdebi 1,0 sm-ia. mxedari gadaadgileba Semdegi gziT: a) ori kvadrati a win, erTi kvadrati marjvniv; 2) ori kvadrati marcxniv, erTi kvadrati win; 3) ori kvadrati y win, erTi kvadrati marcxniv. rogori iqneba x b) kuTxe (wina mimarTulebis mimarT) am sami sur. 3. 30. KkiTxva 25. svlis zedized gakeTebis Semdeg? ••22 A oazisi B oazisidan aRmosavleTiT 25 km manZilzea. aqlemi moZraobas iwyebs A oazisidan, gadis 24km-s samxreT-aRmosavleTidan 15° kuTxiT, Semdeg ki 8km-s CrdiloeTiT. ra manZiliTaa daSorebuli aqlemi B oazisidan? qveTavi 3-8. kanonebi • 26 3.31 or garbens da pirvelad 1,6 m-s garbis Crdilo-aRmosavleTidan 40° kuTxiT. rogori unda iyos xoWos meore garbenis a) sidide da b) mimarTuleba, Tu is unda dasruldes da suraTze 17 m sididis fizikis veqto- y y´ ay bils. pirveli xoWo garbis 0,5 m-s aRmosavleTidan 30° kuTxiT. meore xoWoc akeTebs veqtorebi ri mimarTulia x RerZis dadebiTi mimar- ••23 ori xoWo erTi wertilidan iwyebs sirleTiT, Semdeg ki 0,8 m-s Crdilo-aRmosav- a a mxedris mTliani gadaadgilebis a) sidide da x´ y´ x´ θ θ´ 0 θ´ ax sur. 3. 31. KkiTxva 26. x 83 y Tulebisadmi θ = 56° kuTxiT saaTis isris sawinaaRmdego mimarTulebiT. rogori iqneba veqtoris a) ax da b) ay komponentebi? meore koordinatTa sistema pirvelis mimarT daxrilia θ´ = 18° kuTxiT. rogori iqneba g) ax´ da d) ay´ komponentebi am koordinatTa sistemaSi? θ x qveTavi 3-8. veqtorebis gamravleba •27 veqtorebi xy sibrtyeze Zevs. maTi da sur. 3. 32. KkiTxva 35. sidideebia 4,5 da 7,3 erTeuli, xolo kuTxeebi _ 320° da 85° x RerZis dadebiTi mi- kuTxea θ = 30°. rogori iqneba veqtoris a) x marTulebis mimarT saaTis isris moZraobis da b) y komponentebi? rogori iqneba sawinaaRmdego mimarTulebiT. rogori iqne- ris g) x da d) y komponentebi? rogori iqneba · -isa da b) ba a) •28 da = 2 + 4 . moZebne a) g) ( + )· veqtoris e) x da v) y komponentebi? × -is mniSvnelobebi? =3 +5 mocemulia ori veqtori: da d) × , b) -s komponenti · , -s gas- = p + q , rogori iqneba z) p-s da T) q-s Tu mniSvnelobebi? ••36 mimebis Sexvedrisas, pirveli mimis wvriv (miniSneba: d)-sTvis ganixile formula gadaadgilebaa 3.20 da sur. 3.19). orisa – •29 mocemulia sami veqtori: = –1 – 4 + 2 a) · ( •30 ••31 · (b + =3 –2 +4 Tu tolia ( =2 +2 da ), b) × + 1 1 2 )·( =3 +3 –2 , ×4 1 ) + 1 . moZebne da g) × ( + ). da 2 )? =–5 +2 – , risi 2 gamosaTvlelad gamoiyene skalaruli nam- +3 + 3 da ••32 +4 + 6 = axbx + ayby+azbz. · faqti, rom 1 = ab cos θ da is · ravlis gansazRvreba =3 + = 2 + + 3 SSM ILW WWW. namravlSi =q × , q = 2, =2 + = 4 – 20 + 12 , Tu Bx = By , da rogori saxe eqneba veqtors erTeulovan- veqtorul CanawerSi? ••33 veqtoris sididea 6 erTeuli, veqtorisa – 7 erTeuli, amasTan, rogori iqneba kuTxe -sa da · 1 · -is 2 =(4m) +(5m) , =(–3m) +(4m) . gamoTvale a) 2 , g) ( 2 + 1 2 )· mimarTulebis 2 da d) · (2 × = –3 + 4 + 2 ••35 3.32 =7 –8 suraTze mocemuli veqtorebis sidideebia: a = 3 m, b = 4 m, c = 10 m, xolo × 1 , 2 1 komponenti (miniSneba: d)-sTvis ixile formula 3.20 da sur. 3.19). damatebiTi amocanebi kldis lis Si ori nawevi mopirdapire erTmaneTis suraTze A bamde ewodeba mimarT da B erTmaneTs bzars, mxare rom- warsul- wanacvlda. 3.33 wertilebi wanacvleemTxveoda. gadaadgileba AB wanacvlebis mTlian sibrtyezea. AB-s horizontaluri komponenti aris AC horizontaluri wanacvleba. AB-s komponenti, romelic wanacvlebis sibrtyis qvemoTaa mimarTuli, aris AD siRrmiseuli wanacvleba. a) gamoTvale AB mTliani gadaadgilebis horizontaluri wanacvleba )? =2 +3 –4 me- gaswvriv. mocemuli sami veqtorisTvis, risi toli iqneba 3 xolo 1 = 14. -s mimarTule- bebs Soris? ••34 b) 37 or mocemul veqtors Soris kuTxis veqto- siRrmiseuli wanacvleba φ wanacvlebis sibrtye sur. 3. 33. KkiTxva 37. 84 Tavi 3 veqtorebi sidide, Tu AC = 22 m, xolo AD = 17 m. b) Tu 43 3.35 suraTze mocemuli veqtorebisTvis wanacvlebis sibrtye horizontis mimarT ga- a = 4, b = 3 da c = 5. gamoTvale a) b) · da g) · . daixara φ = 52° kuTxiT, rogori iqneba AB-s vertikaluri komponenti? · , y 38 da veqtorebis komponentebia: ax= 3,2, ay = 1,6, bx = 0,5, by = 4,5. a) gamoTvale kuTxe -sa da -s mimarTulebebs Soris. xy sibrtyeze Zevs -s perpendikularuli ori veq- x tori, romelTagan TioToeulis sididea 5 m. veqtors aqvs dadebiTi x komponenti, xolo veqtors – uaryofiTi x komponenti. rogori iqneba veqtoris b) x da g) y komponentebi veqtoris d) x da e) y komponentebi? da P sur. 3. 35. Kamocanebi 43 da 44. 44 3.35 suraTze mocemuli veqtorebisTvis a = 4, b = 3 da c = 5. gamoTvale × -s a) si× dide da b) mimarTuleba. gamoTvale -s g) sidide da d) mimarTuleba. gamoTvale × -s e) sidide da v) mimarTuleba (z RerZi ar aris naCvenebi). P t1 momenti 45 10 erTeulis sididis t2 momenti Teuli sididis gansxvavdeba 60°-iT. ipove a) am veqtorebis sur. 3. 34. KkiTxva 39. 39 45 sm radiusis borbali horizontalur zedapirze migoravs (sur. 3.34). t1 momentSi borblis P wertili zedapirs exeba. t2 momentSi borbali naxevrad Semobrundeba. rogori iqneba P-s gadaadgilebis a) sidide da b) kuTxe (zedapiris mimarT)? 40 momitinge Tavisi plakatiT moZrao- bas iwyebs xyz koordinatTa sistemaSi xy sibrtyeze. is gadis 40 m-s x RerZis uaryofiTi mimarTulebiT, Semdeg 20 m-s marjvniv, perpendikularuli traeqtoriiT, bolos ki 25 m-s zeviT, wyalsaqaCi koSkis mimarTulebiT. a) rogori iqneba plakatis mTliani gadaadgileba erTeulovan-veqtorul CanawerSi? b) momitingem plakati wyalsaqaC koSkTan daagdo. rogori iqneba plakatis sruli gadaadgilebis sidide misi moZraobis dasawyisidan dasasrulamde? 41 Tu Tu -s, miviRebT 6 + . -s davumatebT -s gamovaklebT risi tolia -s, miviRebT –4 + 7 . -s sidide? 42 Tu B -s davumatebT =3 +4 -s, miviRebT veqtoris sididisa da y RerZis dadebiTi mimarTulebis veqtors. risi tolia dide? veqtori da 6 er- veqtori erTmaneTisgan -s si skalaruli namravli da b) × veqtoruli namravlis sidide. 46 -s sididea 12 m, xolo kuTxe RerZis dadebiTi mimarTulebisadmi _ 60° saaTis isris moZraobis sawinaaRmdego mimarTulebiT. koordinatTa imave sistemaSi = (12 m) + + (8m) . koordinatTa sistema aTvlis saTavis garSemo SemovabrunoT saaTis isris moZraobis sawinaaRmdego mimarTulebiT 20° kuTxiT. miviRebT xý´ koordinatTa sistemas. rogori iqneba axal sistemaSi a) b) veqtorebi da erTeulovan-veqtorul CanawerSi? 47 ialqniani navi eris tbis amerikuli napiridan kanaduri napiris im wertilisken micuravs, romelic CrdiloeTiT 90 km-zea. Tumca, mezRvauri Cerdeba sawyisi wertilidan 50 km-iT aRmosavleTiT. a) ra manZili da b) ra mimarTulebiT unda gascuros mezRvaurma daniSnulebis adgilze misaRwevad? 48 ipove mocemuli oTxi veqtoris jami a) erTeulovan-veqtorul CanawerSi, misi b) sidide da g) +x-Tan kuTxe. P : 10 m, +x-dan 25° saaTis isris moZraobis sawinaaRmdego mimarTulebiT 85 Q : 10 m, +y-dan 10° saaTis isris moZraobis a) sawinaaRmdego mimarTulebiT Soris? R : 8 m, –y-dan 20° saaTis isris moZraobis 53 mocemuli gvaqvs sami gadaadgileba me- mimarTulebiT trebSi: S : 90 m, –y-dan 40° saaTis isris moZraobis 1 2 3 sawinaaRmdego mimarTulebiT 49 mocemulia ori veqtori: = (4 m) -(3 m) ras udris = (6 m) +(8 m) da veqtoris a) da veqtoris g) sidide da d) kuTxe? ras udris + e) sidide da v) kuTxe? ras udris – -s – -s z) sidide da T) kuTxe? ras udris -s i) sidide da k) kuTxe? l) rogori iqneba kuTxe – -s · , b) 2 1 × 2 – -s mimarTulebebs Soris? da g) kuTxe -sa da -s 1 2 =4 +5 –6 =–1 +2 +3 da =4 +3 +2 = a) rogori iqneba sidide b) kuTxe ( -sadmi)? ras udris 1 1 – 2 + ? 3 b) rogori iqneba kuTxe -sa da z RerZis dadebiT mimarTulebas Soris? g) risi tolia komponenti -is mimarTulebis gaswvriv? 2 d) risi tolia 2 -is da -is 1 1 -is komponenti, romelic mimarTulebis perpendikularulia -is sibrtyezea? (miniSneba: 2 -isa da 1 g)-sTvis gaiTvaliswine formula 3.20 da sur. 50 cecxlovani WianWvela piknikis teri- 3.19; d)-sTvis gaiTvaliswine formula 3.27). toriaze cxeli sousis mosaZebnad midis 54 da horizontalur zedapirze akeTebs sam = 0,4 m samxreT-dasavle= 0,5 m aRmosavleTiT, da 3 = 0,6 m 2 gadaadgilebas: TiT, 1 Crdilo-aRmosavleTiT dadebiTi mimarTuleba 60°-iT. x iyos RerZis aRmosavle- Ti, xolo y RerZis dadebiTi mimarTuleba – CrdiloeTi. rogori iqneba b) y komponentebi? 2 -is a) x da 1 -is g) x da d) y komponen- -is e) x da v) y komponentebi? rogo- tebi? 3 ri iqneba WianWvelas sruli gadaadgilebis z) x komponenti, T) y komponenti, i) sidide da k) mimarTuleba? Tu WianWvela pirdapir sawyis wertils daubrundeba, l) ra manZilze da m) ra mimarTulebiT unda imoZraos? 51 qali gadis 250 m-s Crdilo-aRmosavle TiT 30°-iT, Semdeg ki 175 m-s aRmosavle- veqtorebi xy sibrtyeze Zevs. da -s sididea 8, xolo kuTxe aris 130°. -s komponentebia Bx= -7,72 da By= -9,2. gamoTvale kuTxeebi y RerZis uaryofiT mimarTulebas Soris, (b) g) 55 ×( da da × (a) -s mimarTulebas -s mimarTulebas Soris, + 3 ) mimarTulebas Soris. veqtorebi xy sibrtyeze Zevs. sididea 8.00, xolo kuTxe – 130°. ponentebia Bx = 5 · -s -s kom- –7,72 da By = –9,2. (a) ipove . rogori iqneba 4 ×3 (b) erTeu- lovan-veqtorul CanawerSi da (g) sididekuTxur CanawerSi sferuli koordinatebiT (ixile sur. 3.36)? (d) ipove kuTxe -sa da 4 × 3 -is mimarTulebebs Soris (miniSneba: dafiqrdi, sanam gamoTvlebs daiwyeb). rogoz TiT. gamoTvale qalis sruli gadaadgilebis a) sidide da b) kuTxe. g) gamoTvale mis mier ganvlili manZili. d) ra ufro metia, ganvlili manZili Tu gadaadgilebis sidide? SSM. 52 1 gadaadgileba Zevs yz sibrtyeze, φ y RerZis dadebiTi mimarTulebisadmi 63° kuTxiTaa daxrili, aqvs dadebiTi z komponenti da 4,5 m sidide. 2 y gadaadgileba Zevs xz θ sibrtyeze, x RerZis dadebiTi mimarTulebisadmi 30° kuTxiTaa daxrili, aqvs dadebiTi z komponenti da 1,4 m sidide. rogori iqneba x sur. 3. 36. Kamocama 55. 86 veqtorebi Tavi 3 ri iqneba ×3 (e) erTeulovan-veqtorul + lia -s (a) sidide da b) mimarTuleba? – CanawerSi da (v) sidide-kuTxur CanawerSi risi tolia sferuli koordinatebiT? Tuleba? TiToeuli kombinaciisTvis aage 56 1 veqtori mimarTulia y RerZis uar -s g) sidide da d) mimar- veqtoruli diagrama. veqtori 63 golfis moTamaSe burTs xvrelSi sam – x RerZis dadebiTi mimarTulebiT. ra mimar dartymaSi agdebs. pirveli dartyma burTs yofiTi mimarTulebiT, xolo /(–4)-s? rogori iqneba namravlis sidide (g) 1 · 2-sTvis da d) 1 · ( 2/4)-sTvis? rogori iqneba Sedegad miRebuli veqtoris mimarTuleba (e) 1× -sTvis da (v) 2 × 1-sTvis? rogori iqneba 2 Tuleba eqneba (a) 2 /4-s da (b) 2 1 veqtoruli namravlis sidide (z) (e)-Si da (T) (v)-Si? rogori iqneba 1 ×( 2 /4)-is (i) sidide da (k) mimarTuleba? 57 veqtori mimarTulia samxreTiT da misi sididea 3 m. ras udris 5 veqtoris (a) sidide da (b) mimarTuleba? ras udris –2 veqtoris (g) sidide da (d) mimarTuleba? 58 Tu 1 + 2 =5 3 , 1 – maSin rogori iqneba (a) =3 2 3 da da (b) 1 2 =2 +4 3 erTeulo- van veqtorul CanawerSi? 59 gamoTvale mocemuli oTxi veqtoris jami a) erTeulovan veqtorul CanawerSi, misi b) sidide da g) kuTxe. xolo mesame dartyma – 0,91 m samxreT-dasavleTiT. risi tolia erTi dartymiT burTis Casagdebad saWiro gadaadgilebis (a) sidide da (b) mimarTuleba? 64 wertili sibrtyeze samjer gadaad gildeba: 1 – 4 m samxreT-dasavleTiT, Semdeg 2 – 5 m aRmosavleTiT da bolos koordinatTa sistema, sadac y RerZi aCvenebs CrdiloeTs, xolo x RerZi – aRmosavleTs. risi tolia 1 nenti da d) y komponenti? risi tolia unda davumatoT 7.0 m sididis veqtori veqtors. jami gvaZlevs mesame veqtors, romelic mimarTulia y RerZis gaswvriv da misi sidide 3-jer metia -s sidideze. rogori iqneba -s sidide? 61 sami veqtori mocemulia metrebSi: tolia wertilis mTliani gadaadgilebis (z) x komponenti, (T) y komponenti, (i) sidide veqtori yz sibrtyeze Zevs, y RerZis dadebiTi mimarTulebisadmi daxrilia 63°-iT, aqvs dadebiTi z komponenti da 3,2 erTeulis toli sidide. iTaa daxrili, aqvs dadebiTi z komponenti da 1,4 erTeulis toli sidide. ipove (a) (b) × da (g) kuTxe -s da -s Soris. 66 (a) erTeulovan-veqtorul =3 +3 +2 1 2 =–2 –4 +2 =5 +4 –6 , 3 =2 +3 +1 = 4 + 3 × 2 ), (g ) 3 62 5 m sididis ·( 2 ×( 1 + werSi veqtori samxreTiT. 4 m sididis rogori + 2 iqneba = – + , · , Cana Tu = –2 + 2 + 3 da (b) gamoTvale kuTxe -sa da z RerZis dadebiT mimarTulebas Soris. ), 3 + 2 veqtori Zevs xz sibrtyeze, x RerZis dadebiTi mimarTulebisadmi 48° kuTx- ·( 1 -is 3 (e) x komponenti da (v) y komponenti? risi 65 60 x RerZis gaswvriv mimarTuli (b) -is (g) x kompo- 2 imoZraos? : 5m, –235° 1 -is (a) x komponenti da (b) y komponenti? risi tolia unda gaiaros da (m) ra mimarTulebiT unda = (–4 m) – (6 m) ipove Sedegebi: (a) 3 – 6 m Crdilo-aRmosavleTiT 60°-iT. airCie bareobaSi brundeba, maSin (l) ra manZili : 4 m, +65° dartyma – 1,83 m samxreT-aRmosavleTiT, da (k) mimarTuleba? Tu wertili sawyis mde- = (2 m) + (3 m) gadaaadgilebs 3,66 m CrdiloeTiT, meore ). 3 mimarTulia veqtori mimarTu- lia Crdilo-dasavleTidan 35°-iT. risi to- (g) rogori iqneba -s komponenti Tulebis gaswvriv? (d) ra iqneba nenti da -s mimar-s kompo- -s perpendikularulad, magram -sa -s sibrtyeze? (miniSneba: (g)-sTvis ixile formula 3.2 da sur. 3.19. (d)-sTvis ixile adgilebis veqtorebi mTliani gadaadgilebis formula 3.27). Semdegi sidideebis misaRebad: (a) 7 m, (b) 1 m da 67 8 m sididis veqtori emateba veqtors, romelic x RerZis gaswvriv aris mimarTuli. 75 bostonis centrSi banki gaZarcves (ixile am ori veqtoris jami gvaZlevs axal veqtors, romelic y RerZis gaswvrivaa mimarTuli da aqvs -ze orjer meti sidide. risi tolia -s sidide? 68 ganvixiloT x RerZis dadebiTi mimarveqtori, y RerZis dadebiTi Tulebis veqtori da d skalari. ro- mimarTulebis /d-s mimarTuleba, Tu d (a) da gori iqneba debiTia da (b) uaryofiTia? risi tolia (g) · · -s da (d) /d-s sidide? ra mimar× Tuleba eqneba veqtors (e) da (v) (g) 5 m. ruka 3.37 suraTze). policiisgan Tavis dasaRwevad mZarcvelebi TviTmfrinaviT gafrindnen da Seasrules sami frena: 32 km samxreTiT dasavleTis mimarT 45°-iT, 53 km dasavleTiT CrdiloeTis mimarT 64°-iT da 26 km samxreT- · gadafrenis Si moxda es? (ruqaze gadaadgilebebis Sesakrebad gamoiyene geometriuli meTodi). bostoni da misi Semogareni × ? salemi voburni leqsingtoni /d-s T) sidide da (i) mimarTuleba, Tu d dadebiTia? 69 mesame bolos mZarcvelebi daakaves. romel qalaq- (z) rogori iqneba veqtoruli namravlis sidide (e)-Si? rogori iqneba 18°-iT. aRmosavleTiT arlingtoni banki medfordi veqtoris sididea 2,5 m, xolo mimar valthami Tuleba – CrdiloeTi. risi tolia 4 -s niutoni (a) sidide da (b) mimarTuleba? risi tolia velisli 3 -s (g) sidide da (d) mimarTuleba? bostoni bruklini 70 kaci seirnobas iwyebs xyz koordinatTa dedhami quinsi sistemis saTavidan. xy sibrtye horizonta- veimousi luria, xolo x RerZi mimarTulia aRmosav- uolpoli leTisken. kacma gaiara 1000 m aRmosavleTiT, Semdeg 2000 m CrdiloeTiT, bolos ki sur. 3. 37. Kamocana 75. erTcentiani moneta 500 m simaRlis kldidan gadauvarda. (a) erTeulovan-veqtorul Cana werSi, risi tolia erTcentiani monetis ga daadgileba kacis moZraobis dawyebidan dacemamde? (b) rodesac kaci aTvlis saTavesTan brundeba, risi tolia misi mTliani 76 a) aCvene, rom lia nebismieri · ( × a ) nulis toda ×( × b) risi tolia veqtorisTvis. ) sidide, Tu -sa da -s mimarTulebebs Soris kuTxea φ? gadaadgilebis sidide? 77 aCvene, rom 71 moZebne a gverdis sigrZis kubSi diago- xazs Soris moqceuli samkuTxedis (sur. 3.38) nalebis TiToeul wyvils Soris kuTxe. ixile farTobia 1 | 2 amocana 25. 72 Tu – =2 , Sin risi tolia (a) 73 davuSvaT, ken, + = 4 da da (b) ? |. φ mimarTulia aRmosavleTis – CrdiloeTisken, xolo – zeviT. risi · (– ) sidide? saiTaa mimarTuli namravlebi (d) veqtorebsa da wiTel = 3 + 4 , ma- tolia namravlebis (a) ( · ) (b) (– ) · (– ) da (g) × da × , (e) (– ) · (– ) da (v) (– ) × (– )? 74 ganixileba 3 m da 4 m sididis ori gadaadgileba. aCvene, rogor unda dajgufdes gada sur. 3. 38. Kamocana 77. 78 aCvene, rom formula 3.29 gardaiqmneba 3.30-ad. 79 aCvene, rom formula 3.22 daiyvaneba 3.23-ze. 87 4 organzomilebiani da samganzomilebiani moZraoba robi niveli `didi kanionis~ erTi napiridan meoreze gadaxta da motocikliT xtomis gasaocari rekordi daamyara _ 68 metri. robi, ra Tqma unda, kanionis meore napirze daSvebas eswrafoda da sulac ar undoda, rom Tundac umciresi Secdomis gamo kanionis fskerze aRmoGeniliyo. rogor unda gansazRvros motocikliT mxtomelma naxtomis sawyisi siCqare? pasuxs am TavSi SeityobT. 4.1 ra aris fizika? am TavSi ganvagrZobT fizikis im nawilis Seswavlas, romelic moZraobas Seexeba, Tumca amjerad ganvixilavT or an samganzomilebian moZraobas. magaliTad, medicinis mkvlevrebi da aviainJinrebi did yuradRebas uTmoben or da samganzomilebian brunviT moZraobebs, romlebic pilotebs sahaero brZolebSi sWirdebaT (moZraobis romeli parametrebi gamoiwvevs pilotis mier gonebis dakargvas?). sportuli inJinrebisTvis mTavaria, magaliTad, kalaTburTTan dakavSirebuli sakiTxebi (rodesac moTamaSe samquliani xazidan tyorcnis, ra siCqariT da kuTxiT unda gafrindes burTi, raTa kalaTSi moxvdes?). samganzomilebiani moZraoba ioli gasaazrebeli ar aris. magaliTad, manqanas trasaze albaT kargad marTavT (erTganzomilebiani moZraoba), magram asafren bilikze TviTm- frinavis dasma (samganzomilebiani moZraoba) varjiSis gareSe Zalian gagiWirdebaT. organzomilebiani da samganzomilebiani moZraobis Seswavlas mdebareobiT da gadaadgilebiT viwyebT. 4.2 mdebareoba da gadaadgileba wertilis mdebareobis gansazRvra mdebareobis veqtoriT xdeba. mdebareobis veqtori CvenTvis cnobil mdebareobas (Cveulebriv, koordinatTa sistemis aTvlis saTaves) am 88 89 4.2 mdebareoba da gadaadgileba y wertilTan akavSirebs. erTeulovan-veqtorul CanawerSi veqtori iqneba: =x +y +z , sadac x , y da z (2 m) (-3m) (5 m) (4 .1 ) 0 -is veqtoruli komponentebia, x xolo x, y da z koeficientebi – -is skalaruli komponentebi. x, y da z koeficientebi gviCvenebs wertilis mdebareobas koordinatTa RerZebis gaswvriv da aTvlis saTavis mimarT, anu wertils gaaCnia marTkuTxa koordinatebi (x,y,z). magaliTad, 4.1 suraTze naCvenebia z sur. 4.1 wertilis mdebareobis veqtori aris wertilis veqtoruli komponentebis jami. wertili mdebareobis veqtoriT: = (–3m) + (2m) + (5m) da marTkuTxa koordinatebiT (-3m, 2m, 5m). x RerZis gaswvriv wertili 3 metriTaa daSorebuli aTvlis saTavidan (– ) mimarTulebiT. y RerZis gaswvriv wertili 2 metriTaa mimarTulebiT. z RerZis gaswvriv wertili 5 metriTaa daSorebuli aTvlis saTavidan + daSorebuli aTvlis saTavidan + mimarTulebiT. wertilis moZraobasTan erTad misi mdebareobis veqtori ise icvleba, rom am wertils yovelTvis aTvlis saTavesTan akavSirebs. Tu mdebareobis veqtori drois garkveul monakveTSi Seicvala 1 -dan 2 -ze, maSin wertilis D gadaadgileba drois am monakveTSi iqneba: Dr = 2 – 1 . (4.2) 4.1 formulis erTeulovan-veqtoruli Canaweris gamoyenebis Sedegad es gadaadgileba Semdeg saxes miiRebs: D = (x2 + y2 an sadac + z2 ) – (x1 + y1 D = (x2 – x1) + (y2 – y1) (x1,y1,z1) koordinatebi Seesabameba dinatebi – 2 1 + z1 ) + (z2 – z1) , (4 .3 ) mdebareobis veqtors, xolo (x2,y2,z2) koor- mdebareobis veqtors. gadaadgilebis Cawera aseve SeiZleba, Tu (x2 _ x1)-is nacvlad CavsvamT Δx-s, (y2 – y1)-is nacvlad Δy-s da (z2 – z1)-is nacvlad Δz-s: D 1 = Dx + Dy + Dz . amocanis amoxsnis nimuSi (4.4) 4.1 4.2 suraTze wertilis sawyisi mdebareobis y veqtoria 1 = (–3 m) + (2 m) + (5 m) , 0 xolo momdevno mdebareobis: 2 -dan 2 D gadaadgileba z -mde? amoxsna: D sur. 4.2 D = gadaadgileba x 2 = (9 m) + (2 m) + (8 m) . rogori iqneba wertilis D 1 1 miiReba 2 sab- – 1 gadaadgilebis veqtori 2 veqtoris bolos aerTebs 2 veqtoris bo1 losTan. 90 Tavi 4 organzomilebiani da samganzomilebiani moZraoba oloo mdebareobis veqtoridan 1 -is gamo klebiT: D = 2 – 1 = [9–(–3)] + [2 – 2] + [8 – 5] = (12 m) + (3 m) (pasuxi) gadaadgilebis veqtori xz sibrtyeze Zevs, vinaidan y komponenti ara gvaqvs. es faqti sakontrolo wertili 1. a) Tu Ramura (–2m, 4m, –3m) wertilidan frindeba da (6m, –2m, –3m) wertilSi gadadis, rogori iqneba misi D gadaadgileba er Teulovan-veqtorul CanawerSi? b) aris Tu ara D koordinatTa romelime sibrtyis paraleluri? Tu aris, maSin romeli sibrtyis? ricxobriv SedegSi ufro kargad Cans, vidre 4.2 suraTze. 4.2 amocanis amoxsnis nimuSi melzec, raoden gasakviric ar unda iyos, amoxsna: 4.5 da 4.6 formulebis Tanaxmad, kurdRlis mdebareobis x da y koordi- koordinatTa RerZebia gavlebuli. kurd- natebi Rlis mdebareobis koordinatebi, rogorc komponentebia. maSasadame, SegviZlia davw- t drois funqcia, mocemulia formulebiT: eroT: kurdReli da garbis avtosadgomze, ro x = – 0,31 t2 + 7,2t + 28 (4.5) y = 0,22 t2 – 9,1t + 30 (4.6) (t) = x(t) + y(t) ( -is nacvlad a) rodesac t = 15 wm, rogori iqneba kur- (t)-s imis gamo davwereT, -ic t veqtori erTeulo- drois funqciaa). rodesac skalaruli kompo x = (–0,31) (15)2 + (7,2) (15) + 28 = 66 CanawerSi? y t = 15 wm, nentebia: van-veqtorul CanawerSi da sidide-kuTxur m (4.7) rom komponentebi da, maSasadame, t izomeba wamebSi, xolo x da y _ metrebSi. dRlis mdebareobis mdebareobis veqtoris skalaruli ym 40 40 t = 0 wm 20 20 -41 0 20 40 60 x 80 m 0 20 -20 40 60 xm 80 5 wm -20 -40 r 10 wm -40 -60 15 wm -60 (a) 25 wm 20 wm (b) sur. 4.3 a) kurdRlis mdebareobis veqtori, rodesac t =15 wm. -is skalaruli komponentebi naCvenebia RerZebis gaswvriv. b) kurdRlis traeqtoria da misi mdebareobebi t-s xuTi mniSvnelobisTvis. 91 4.3 saSualo siCqare da myisi siCqare da y = (0,22) (15)2 – (9,1) (15) +30 = – 57 -is komponentebis niSnebis Seswavlam gvaCvena, rom sasurveli kuTxe meoTxe meoTxedSia, maSasadame 139°–180° = – 41°). b) aage kurdRlis traeqtoriis grafiki t = 0-dan t = 25 wm-mde. maSasadame, rodesac t = 15 wm, (t) = (66 m) – (57 m) , (pasuxi) rac gamosaxulia 4.3a suraTze. -is sididisa da daxrilobis gamo saTvlelad gamoviyenoT veqtoruli kalkulatori an 3.6 formulis Tanaxmad davweroT: = r= = 87 m, da amoxsna: t-s ramdenime mniSvnelobisTvis SegviZlia a) nawils davubrundeT da avagoT grafiki. 3.4.b suraTze asaxulia grafiki t-s xuTi mniSvnelobisTvis. aseve SegviZlia gamoviyenoT grafikuli kalkulatori da avagoT parametruli anu y-is x-ze damokidebulebis grafiki, romlis koordinatebi drois funqciebis saxiT mocemulia 4.5 da 4.6 formulebiT. (pasuxi) (–57 m) y θ = tg–1 = tg–1 = – 41° x 66 m (pasuxi) (miuxedavad imisa, rom kuTxeebis 139° da –41° tangensebi erTmaneTis tolia, 4.3 saSualo siCqare da myisi siCqare rodesac wertili moZraobs erTi adgilidan meorisken, SeiZleba misi siCqaris gageba dagvWirdes. rogorc me-2 TavSi vTqviT, siCqares ori sidide asaxavs: saSualo siCqare da myisi siCqare. Tumca, am SemTxvevaSi es sidideebi unda ganvixiloT, rogorc veqtorebi da gamoviyenoT veqtoruli Canaweri. Tu wertili D gadaadgilebas asrulebs Δt drois SualedSi, maSin misi siCqarea: saSualo siCqare = anu saS = es formula gvauwyebs, rom saS-s saS saSualo gadaadgileba drois Sualedi D Δt (4.8) (veqtori 4.8 formulis marcxena mxares) mimar- Tuleba unda daemTxves D -is mimarTulebas (veqtori formulis marjvena mxares). 4.4 formulis gamoyenebiT 4.8 formula SeiZleba ase davweroT: saS = Dx + Dy Δt + Dz = Dx Δt + Dy Δt + Dz Δt (4.9) magaliTad, Tu wertili amocanis 4.1 nimuSSi sawyisi mdebareobidan momdevno mdebareobamde misvlas 2 wams andomebs, maSin misi saSualo siCqare iqneba: saS = (12 m) + (3 m) D Δt = 2 wm = (6 m/wm) + (1,5 m/wm) anu saSualo siCqaris (veqtoruli sididis) komponentebia 6 m/wm x RerZis gaswvriv da 1,5 m/wm z RerZis gaswvriv. Cveulebriv, rodesac wertilis siCqareze vsaubrobT, vgulisxmobT wertilis siCqares garkveul momentSi. aris sidide, romelsac sac Δt intervali 0-mde mcirdeba. saS myis zRvrulad uaxlovdeba, rode- SegviZlia davweroT warmoebulis saxiT: 92 Tavi 4 organzomilebiani da samganzomilebiani moZraoba = d . dt (4.10) 4.4 suraTze naCvenebia xy sibrtyeze moZravi werti- y lis traeqtoria. rodesac wertili gadaadgildeba Tuli. Δt intervalSi mdebareobis veqtori icvleba 2 D 1 marjvniv, misi mdebareobis veqtori marjvnivaa mimar- mxebi 1 1 2 -dan 2 -mde da wertilis gadaadgilebaa D . traeqtoria wertilis myisi siCqaris sapovnelad, magaliTad, t1 mo- x mentSi (rodesac wertili imyofeba pozicia 1-Si), Δt in- 0 sur. 4.4 wertilis D gadaadgileba Δt tervals nulamde vamcirebT. am dros sami movlena xdeba: mdebareointervalSi, t1 momentSi 1 bis veqtoridan t2 momentSi 2 mdebareobis veqtoramde. pozicia 1-Si wertilis traeqtoriis mxebia naCvenebi. 1) 2 1 mdebareobis veqtori 4.4 suraTze moZraobs -sken ise, rom D mcirdeba nulamde. 2) D /Δt-s (da, maSasadame, saS-s) mimarTuleba uaxlovdeba wertilis 1 poziciaze traeqtoriis mxebis mimarTulebas). 3) saS saSualo siCqare uaxlovdeba roca Δt → 0, gveqneba saS myis siCqares t1 momentSi. → , da, rac yvelaze mniSvnelovania, saS iRebs mxebis mi- marTulebas, maSasadame, -sac igive mimarTuleba aqvs: wertilis myisi siCqaris mimarTuleba yovelTvis traeqtoriis mxebia konkretul momentSi. sami ganzomilebis SemTxvevaSi Sedegi igivea: yovelTvis wertilis traeqtoriis mxebia. 4.10 formulis erTeulovan-veqtoruli saxiT dasawerad 4.1 formulidan CavsvaT . d = (x +y +z ) = dx + dy + dz dt dt dt dt es formula SeiZleba ase gamartivdes: = vx + vy sadac + vz , (4.11) dx dx dz , v = da vz = . dt y dt dt (4.12) -s skalaruli komponentebia: vx = magaliTad, dx/dt aris -s skalaruli komponenti x RerZis gaswvriv. maSasadame, -s skalaruli komponentebis povna -is skalaruli komponentebis diferencirebiT SegviZlia. 4.5 suraTze naCvenebia siCqaris veqtori da misi x da y skalaruli komponentebi. gaiTvaliswine, rom y mxebi aris wertilis traeqtoriis mxebi konkretul mo- vy mentSi. aRsaniSnavia, rom mdebareobis veqtori (4.1- vx 4.4 suraTebi) aris isari, romelic akavSirebs erT mdebareobas (àq~) meoresTan (ìq~). siCqaris veqtori 0 traeqtoria sur. 4.5 wertilis li komponentebiT. x siCqare skalaru- mdebareobebs ar akavSirebs, is ufro wertilis moZraobis myis mimarTulebas gviCvenebs, misi sigrZe ki siCqaris sidides asaxavs. 93 4.3 saSualo siCqare da myisi siCqare 2 y suraTze naCvenebia wertilis moZraobis wriuli traeqtoria. Tu wertilis myisi siCqa- sakontrolo wertili rea = (2 m/wm) – (2 m/wm) , romel meoTxedSia wertili am momentSi, Tu is wris garSemo moZraobs a) saaTis isris moZraobis mimarTulebiT da b) saaTis isris moZraobis sawinaaRmdego mimarTulebiT? orive SemTxvevisTvis grafikze gaavle x veqtori. amocanis amoxsnis nimuSi 4.3 y (m ) amocanis 4.2 nimuSSi naxsenebi kurdRlisTvis erTeulovan-veqtorul CanawerSi da sidide-kuTxur CanawerSi ipove siCqare, 40 rodesac t =15 wm. amoxsna: amocanis amoxsna wilisgan Sedgeba: 1) kurdRlis ori na siCqaris 20 povna SegviZlia, Tu Tavdapirvelad mis komponentebs vipoviT; 2) am komponentebis 0 povna SesaZlebelia kurdRlis mdebareobis 20 40 60 80 x (m ) veqtoris komponentebis gawarmoebiT. 4.12 formul-is vx nawilSi 4.5-is CasmiT miviRebT -s x komponents: vx = dx d = (–0,31t2 + 7,2t + 28) dt dt = –0,62t + 7,2. -20 -40 (4.13) rodesac t = 15 wm, vx = –2,1 m/wm. aseTive gziT vipoviT y komponents: vy = x -60 dy d = (0,22t2 – 9,1t + 30) dt dt = 0,44t – 9,1. -130 o (4.14) rodesac t = 15 wm, vy = –2,5 m/wm. amis Semdeg 4.11 formula mogvcems: = (–2,1 m/wm) rogorc 4.6 + (–2,5 m/wm) suraTzea naCvenebi, siCqare, rodesac t = 15 wm. si Cqaris veqtori kurdRlis mdebareobis mxebia am mo mentSi. aqve naCvenebia -s skalaruli komponentebi. sur. 4.6 kurdRlis (pasuxi) aris kurdRlis moZraobis traeqtoriis mxebi da igive mimarTuleba aqvs, rogoric kurdRlis siCqares, rodesac t = 15 wm. da θ = tg–1 (–2,5 m/wm) vy = tg–1 vx (–2,1 m/wm) = tg–1 1,19 = – 130° (pasuxi) (miuxedavad imisa, rom 50° da –130° kuTx- -s sididisa da kuTxis gamosaTvlelad eebis tangensebi tolia, siCqaris kompo- gamoviyenoT veqtoruli kalkulatori an nentebis niSnebis gamokvleva aCvenebs, rom formula 3.6: sasurveli kuTxe mesame meoTxedSia, maSa- v= sadame 50° – 180° = –130°. = = 3,3 m/wm. (pasuxi) 94 organzomilebiani da samganzomilebiani moZraoba Tavi 4 4.4. saSualo aCqareba da myisi aCqareba rodesac wertilis siCqare Δt intervalSi icvleba -dan 1 -mde, misi 2 saS saSualo aC- qareba am SualedSi iqneba: saSualo siCqare = anu saS siCqaris cvlileba drois Sualedi = 2 – Δt 1 = Δv Δt . (4.15) Tu Δt-s romelime momentisTvis 0-s mivuaxlovebT, maSin am momentSi saS zRvrulad uaxlovdeba myis aCqarebas (an ubralod aCqarebas), anu = d . dt (4.16) Tu siCqaris sidide an mimarTuleba (an orive erTad) icvleba, wertils aucileblad unda hqondes aCqareba. 4.16 formulis Cawera erTeulovan-veqtoruli saxiTac SeiZleba, Tu -s nacvlad CavsvamT 4.11 formulas: d = dt (vx +vy +vz ) dv = dtx dv + dty dv + dtz es gadaiwereba Semdegi saxiT: = ax +ay +az , (4.17) dvy dvx dvz , ay = dt da az = dt dt (4.18) sadac -s skalaruli komponentebia: ax = y maSasadame, -s skalaruli komponentebis diferencirebiT Sei- ax Zleba. 4.7 suraTze naCvenebia ay traeqtoria x O sur. 4.7. wertilis aCqareba da misi skalaruli komponentebi. ravi wertilisTvis. aRsaniSnavia, rom 4.7 suraTze aCqarebis veqtori mdebareobebs ar akavSirebs. is mxolod wertilis aCqarebis mimarTulebas gviCvenebs, misi sigrZe ki aCqarebis sidides asaxavs. ipove amocanis amoxsnis 4.2 da 4.3 ninaxsenebi kurdRlis aCqarebis veqtori da misi skalaruli komponentebi or ganzomilebaSi moZ- amocanis amoxsnis nimuSi muSebSi -s skalaruli komponentebis povna aCqareba, 4.4 amoxsna: amocanis amoxsna ori nawilisgan Sedgeba: 1) kurdRlis aCqarebas vipoviT, rodesac t = 15 wm. pasuxi Cawere erTeulo- Tu Tavdapirvelad aCqarebis komponentebs van-veqtoruli da sidide-kuTxuri saxiT. gavigebT; 2) am komponentebis povna Sesa- 95 4.4. saSualo aCqareba da myisi aCqareba Zlebelia kurdRlis siCqaris komponentebis gawarmoebiT. 4.18 formulis ax nawilis formula 4.13-Si CasmiT miviRebT y (m ) 40 aCqare- bis x komponents: dv d ax = dtx = dt (–0,62t + 7,2) = –0,62 m/wm2. 20 aseTive gziT vipoviT y komponents: dv d ay = dty = dt (0,44t – 9,1) = 0,44 m/wm2. 0 x (m ) 20 40 60 80 rogorc vxedavT, aCqareba droSi ar icvleba (Tanabaria), vinaidan t drois cvladi arc -20 erTi komponentis formulaSi ar gamoCnda. amis Semdeg formula 4.17 mogvcems: = (–0,62 m/wm2) + (0,44 m/wm2) , (pasuxi) -40 a rac 4.8 suraTzea naCvenebi kurdRlis moZraobis traeqtoriasTan erTad. -s sididisa da kuTxis gamosaTvlelad 145 o x -60 gamoviyenoT veqtoruli kalkulatori an formula 3.6. sididisTvis gveqneba: a= sur. 4.8 kurdRlis a aCqareba, rodesac t = 15 wm. kurdRels traeqtoriis yvela wertilSi erTi da igive aCqareba aqvs. = = 0,76 m/wm2 (pasuxi) xolo kuTxe iqneba: θ = tg–1 0,44 m/wm ay –1 = tg = – 35° ax (–0,62) m/wm kalkulatorze naCvenebi uwyebs, rom 4.8 suraTze kuTxe gva mimarTulia marjvniv da qveviT. magram komponentebidan viciT, rom mimarTuli unda iyos marcxniv da zeviT. im kuTxis sapovnelad, rom- sakontrolo wertili 3. mocemulia xy sibrtyeze moZravi hokeis Saibis mdebareobis (metrebSi) aRwerebi: 1) x = –3t2 + 4t – 2 da y = 6t2 – 4t 2) x = –3t2 – 4t da y = –5t2 + 6 lis tangensi –35°-is tangensis toli iqneba, 3) = 2t2 – (4t + 3) –35°-s davumatoT 180°. miviRebT: 4) = (4t3 – 2t) + 3 –35° + 180° = 145° es pasuxi ponentebs. ar ewinaaRmdegeba gaiTvaliswine, (pasuxi) -s rom kom -s sidide da mimarTuleba emTxveva kurdRlis gadaadgilebas, vinaidan aCqareba TiToeuli aRwerisTvis gansazRvre, aris Tu ara Saibis aCqarebis x da y komponentebi mudmivi da aris Tu ara aCqareba Tanabari. Tana baria. amocanis amoxsnis nimuSi wertilis siCqarea 0 = –2 + 4 . rodesac 2 t = 0, is iwyebs a = 3 m/wm TanabaraCqarebul moZraobas x RerZis dadebiTi mimarTu 4.5 da sidide-kuTxur CanawerSi, rodesac t = 5 wm? amoxsna: Tavdapirvelad SevniSnoT, lebisadmi θ = 130° kuTxiT. rogori iqneba mocemuli gvaqvs xy sibrtyeze wertilis siCqare erTeulovan-veqtorul milebiani moZraoba. amocanis rom organzo amoxsna ori 96 organzomilebiani da samganzomilebiani moZraoba Tavi 4 nawilisgan Sedgeba: 1) vinaidan aCqareba Tanabaria, SegviZlia gamoviyenoT formula 2.11 (v = v0 + at); 2) vinaidan formula 2.11 miesadageba mxolod wrfiv moZraobas, is unda gamoviyenoT x da y RerZebisadmi paraleluri moZraobebisTvis cal-calke. anu, vx da vy siCqaris komponentebi unda vipovoT formulebidan: is x da y komponentebia, xolo ax da ay – komponentebi. ax-isa da ay-is sa- veqtoruli kalkulatoris sa- SualebiT davagegmiloT Semdeg = (–12 m/wm) + (16 m/wm) . (pasuxi) veqtoruli kalkulatoris an 3.6 formulis gamoyenebiT vipoviT v= am formulebSi v0x (= –2 m/wm) da v0y ( = 4 m/wm)-s maSasadame, rodesac t = 5 wm, damrgvalebis -s sididesa da kuTxes: vx = v0x + ax t da vy = v0y + ax t povnelad vy = 4 m/wm + (2,3 m/wm2) (5 wm) = 15,5 m/wm. an gamoviyenoT formula 3.5: = 19,4 m/wm ≈ 19 m/wm (pasuxi) da θ = tg–1 vy = 127° ≈ 130° (pasuxi) vx Seamowme, ra pasuxs mogcems kalkulatori, 127°-s Tu –53°-s. axla aage veqtori Tavisi komponentebiT, raTa naxo, romeli kuTxea swori. ax = a cos θ = (3 m/wm )(cos 130°) = –1,93 m/wm2 2 ay = a sin θ = (3 m/wm2) (sin 130°) = 2,3 m/wm2 rodesac es mniSvnelobebi Caismeba vx da vy formulebSi, miviRebT, rom, rodesac t = 5 wm, sakontrolo wertili 4. mawanwa- las mdebareoba mocemulia formuliT: = (4t3 – 2t) + 3 , sadac metrebSia, xolo t wamebSi. rogori unda iyos koeficientebis: 4, –2 da 3 erTeulebi? vx = – 2 m/wm + (–1,93 m/wm2) (5 wm) = –11,65 m/wm 4.5 am ani Wurvisebri moZraoba TavSi ganvixilavT moZraobis organzomilebi- gansakuTrebul SemTxvevas. wertili vertikalur sibrtyeze moZraobs garkveuli 0 siCqariT da misi aCqareba qve- moT mimarTuli Tavisufali vardnis aC- qarebaa. aseT wertils ewodeba Wurvi (rac imas niSnavs, rom wertili isroles), xolo mis moZraobas – Wurvisebri moZraoba. Wurvi SeiZleba iyos CogburTis (sur. 4.9.) an beisbolis burTi, magram ara TviTmfrinavi sur. 4.9 myari zedapiridan asxletili CogburTis burTis stroboskopuli foto. Sejaxebebs Soris burTi Wurvisebr moZraobas asrulebs. an ixvi. Cveni mizania aseTi moZraobis gaanalizeba 4.2-4.4 qveTavebSi mocemuli organzomilebiani moZraobis aRweris daxmarebiT. vuSvebT, rom haeri Wurvze zegavlenas ar axdens. 4-10 suraTze (gaanalizebulia Semdeg qveTavSi) naCvenebia sxeulis traeqtoria haeris zemoqmedebis gareSe. sxeuli gauSves 0 sawyisi siCqariT, romelic asec SeiZleba daiweros: 0 0x da 0y = v0x + v0y komponentebis povna SesaZlebelia, Tu viciT θ0 kuTxe dadebiT mimarTulebas Soris: (4.19) 0 -sa da x RerZis 97 4.5 Wurvisebri moZraoba v0x = v0 cos θ0 da v0y = v0 sin θ0 organzomilebiani moZraobis dros, sxeulis tori mudmivad icvleba, magram mdebareobis veqtori da (4.20) siCqaris veq- aCqarebis veqtori ucvlelia da yovelTvis mimarTulia vertikalurad qveviT. sxeuls horizontaluri aCqareba ara aqvs. Wurvisebri moZraoba, magaliTad, 4.9 da 4.10 suraTebze, sakmaod rTuli Cans, magram arsebobs erTi aspeqti, romelic mas sakmaod amartivebs (dadgenilia eqsperimentebis Sedegad): Wurvis moZraobisas horizontaluri da vertikaluri moZraoba erTmaneTisgan damoukidebelia anu arc erTi ar axdens gavlenas meoreze. es Tviseba saSualebas gvaZlevs, davSaloT organzomilebiani moZraobis amocana or erTganzomilebian moZraobad – erTi iqneba horizontaluri moZraoba (nulovani aCqarebiT), xolo meore iqneba vertikaluri moZraoba (TanabaraCqarebuli, qveviT mimarTuli aCqarebiT). qvemoT mocemulia ori eqsperimenti, romliTac mtkicdeba, rom horizontaluri da vertikaluri moZraobebi erTmaneTisgan damoukidebelia. golfis ori burTi 4.11 suraTze naCvenebia golfis ori burTis stroboskopuli foto. erTs ubralod xeli gauSves, meore ki horizontalurad gaisroles. golfis orive burTi vertikalurad TanabaraCqarebulad moZraobs g aCqarebiT da erTsa da imave manZils gadis drois tol – sur. 4.10. da y0 = 0, Wurvis 0 traeqtoria wertilidan x0 = 0 sawyisi siCqariT. Wurvis traeqtoriaze naCvenebia sawyisi da momdevno siCqareebi komponentebTan erTad. rogorc vxedavT, horizontaluri siCqaris komponenti mudmivia, xolo vertikaluri siCqaris komponenti icvleba. R aris horizontaluri manZili, sur. 4.11. erT burTs xeli gauSves, xolo romelic Wurvma gaiara gasrolis wertilis simaRlis meore horizontalurad marjvniv gaisroles. maTi vertikaluri moZraobebi erTnairia. miRwevamde. 98 organzomilebiani da samganzomilebiani moZraoba Tavi 4 monakveTSi. is, rom erTi burTi vardnis dawyebamde horizontalurad moZraobs, ar axdens gavlenas mis vertikalur moZraobaze, anu horizontaluri da vertikaluri moZraobebi erTmaneTze ar aris damokidebuli. mSvenieri gasamxneveli studentebisTvis ro m g 0 ia r iy to l aq fi e o tr y o 4.12 suraTs fizikis bevri leqcia gamoucocxlebia. aq gamosaxulia patara mili G, romlidanac sulis SeberviT burTs isvrian. samiznea M magnitze Camokidebuli yuTi da Tofic pirdapir maszea damiznebuli. eqsperimenti isea dagegmili, rom, rogorc ki milidan burTi gamoityorcneba, yuTi agnits mowydeba. g (Tavisufali vardnis aCqareba) rom nulis toli yofiliyo, burTi 4.12 suraTze naCvenebi wrfivi traesur. 4.12. nasroli burTi Camovardnil yuTs yovelTvis xvdeba. TiToeuli gadis h manZils im adgilidan, sadac Sexvdebodnen, Tavisufali vardnis aCqareba rom ar arsebobdes. qtoriiT gadaadgildeboda, xolo yuTi erT adgilze ilivlivebda haerSi. aseT SemTxvevaSi burTi aucileblad moxvdeboda yuTs. asea Tu ise, g ar aris nulis toli, magram burTi mainc xvdeba yuTs! rogorc 4.12 suraTze moCans, burTis frenis ganmavlobaSi burTic da yuTic „nulovani g-s“ Sesabamisi poziciidan Tanabar h manZils gadian. rac ufro Zlierad uberavs Tofs demonstratori, miT ufro didia burTis sawyisi siCqare, miT ufro naklebia frenis dro da h-is sididec mcirea. 4.6 Wurvisebri moZraobis analizi axla mzad varT gavaanalizoT Wurvisebri moZraoba horizontalurad da ver tikalurad. horizontaluri moZraoba rogorc 4.13 suraTze Cans, horizontaluri aCqarebis arqonis gamo Wurvis siCqaris vx komponenti mTeli moZraobis manZilze ucvlelia da v0x sawyisi sididis tolia. nebismier t momentSi Wurvis x – x0 horizontaluri gadaadgileba sawyisi x0 mdebareobidan mocemulia formuliT 2.15. vinaidan a = 0, amitom es formula SeiZleba daiweros Semdegi saxiT: x – x0 = v0x t rogorc viciT, v0x = v0 cosθ0, amitom gveqneba: x – x0 = (v0 cosθ0)t (4.21) sur. 4.13. skeitbordistis siCqaris vertikaluri komponenti icvleba, magram horizontaluri komponenti, romelic skeitbordis siCqares emTxveva, ucvlelia. amis Sedegad skeitbordi sportsmenis qvemoT rCeba da am ukanasknels masze daxtomis saSualeba aqvs. 99 4.6 Wurvisebri moZraobis analizi vertikaluri moZraoba Tavisuflad vardnili sxeulis vertikaluri moZraoba 2.9 qveTavSi ganvixileT. yvelaze mniSvnelovania, rom am dros aCqareba Tanabaria. maSasadame, 2.1 cxrilis formulebi gamogvadgeba. a-s nacvlad davweroT – g, x ki SevcvaloT y-iT. magaliTad, formula 2.15-sTvis miviRebT: y – y0 = v0yt – 12 gt2 = (v0 sin θ0)t – 12 gt2 (4.22) aq sawyisi vertikaluri siCqaris v0y komponenti Canacvlebulia v0 sin θ0-iT. analogiurad gardaiqmneba formulebi 2.11 da 2.16: vy = v0 sin θ0 – gt vy2 = (v0 sin θ0)2 – 2g(y – y0). da (4.23) (4.24) rogorc 4.10 suraTze da 4.23 formulaSi Cans, vertikaluri siCqaris komponenti ise icvleba, TiTqos burTi vertikalurad maRla aisroles. is mimarTulia maRla da misi sidide nuls uaxlovdeba traeqtoriis umaRles wertilamde. amis Semdeg, vertikaluri siCqaris komponenti mimarTulebas icvlis da misi sididec drois mixedviT izrdeba. traeqtoriis gantoleba Wurvis traeqtoriis gantolebis gamoyvana SesaZlebelia, Tu 4.21 da 4.22 formulebSi gamovricxavT t cvlads. 4.21 formulidan gavigoT t da CavsvaT 4.22-Si. mcireodeni gardaqmnebis Semdeg miviRebT: y = (tg θ0)x – gx2 2(v0 cos θ0)2 (traeqtoria) (4.25). es formula 4.10 suraTze naCvenebi traeqtoriis gantolebaa. simartivisTvis davuSvaT, rom 4.21 da 4.22 formulebSi x0 = 0 da y0 = 0. vinaidan g, θ0 da v0 mudmivebia, 4.25 formula miiRebs aseT saxes: y = ax + bx2, sadac a da b mudmivebia. miviReT parabolis formula, maSasadame, traeqtoria parabolaa. y cxrili 4.1 ori burTis traeqtoria a II I v0 traeqtoria I traeqtoria II (haerSi) (vakuumSi) 60o areali (frenis siSore) 98,5 m 177 m 53 m 76,8 m 6,6w m 7,9 wm x maqsimaluri simaRle sur. 4.14 (I) burTis frenis traeqtoria haeris winaaRmdegobis gaTvaliswinebiT. (II) am TavSi ganxiluli meTodebiT gamoTvlili burTis frenis traeqtoria vakuumSi. Sesabamisi monacemebisTvis ixile cxrili 4.1. frenis dro a ixile sur. 4.14. gasrolis kuTxea 60°, xolo sawyisi siCqare _ 44,7m/wm. 100 Tavi 4 organzomilebiani da samganzomilebiani moZraoba horizontaluri areali rogorc 4.10 suraTze naCvenebi, Wurvis R horizontaluri areali aris Wurvis mier ganvlili manZili sawyis simaRleze dabrunebamde. R arealis sapovnelad, 4.21 formulaSi CavsvaT x – x0 = R, xolo 4.22-Si – y – y0 = 0. miviRebT: R = (v0 cos θ0)t da 0 = (v0 sin θ0)t – 1 2 gt2 formulebSi t-s gamoricxva mogvcems: R= 2v02 sin θ0 cos θ0 g Tu gavixsenebT, rom sin 2θ0 = 2 sin θ0 cos θ0 (ixile danarTi E), miviRebT: R= v02 sin 2θ0 g (4.26) yuradRebiT: es formula ar gvaZlevs Wurvis mier ganvlil horizontalur manZils, rodesac saboloo simaRle ar emTxveva sawyis (srolis) simaRles. aRsaniSnavia, rom 4.26 formulaSi R maqsimaluri iqneba, rodesac 2θ0 = 90°, ese igi θ0 = 45°. sin 2θ0 = 1, anu R horizontaluri areali maqsimaluria srolis 45° kuTxisTvis. haeris zemoqmedeba aqamde vuSvebdiT, rom Wurvis moZraobaze haeri zemoqmedebas ar axdens. Tumca, gamoTvlebsa da Wurvis realur moZraobas Soris sxvaoba SeiZleba sakmaod didi iyos, vinaidan haeri moZraobas ewinaaRmdegeba. magaliTad, 4.14 suraTze naCvenebia burTis frenis ori traeqtoria, rodesac burTi moZraobas iwyebs 60° kuTxiT da 44,7 m/wm sawyisi siCqariT. traeqtoria I gamoTvlilia haerSi frenis realur pirobebTan miaxloebiT. traeqtoria II ki burTis vakuumSi frenis traeqtoriaa. sakontrolo wertili 5. burTi moednis farglebs gareT gaisroles. ra mouva frenis dros (haeris zemoqmedeba ugulvebelyavi) mis a) horizontalur da b) vertikalur komponentebs? ra iqneba misi aCqarebis g) horizontaluri da d) vertikaluri komponentebi, zeviT moZraobis, qveviT moZraobisa da frenis umaRlesi wertilis miRwevis dros? amocanis amoxsnis nimuSi 4.15 suraTze maSveli 4.6 TviTmfrinavi y mifrinavs 198km/sT (55m/wm) siCqariT da sazRvao katastrofis msxverplisgan 500 m simaRleze imyofeba. pilots surs maSveli kafsulis CaSveba msxverplTan rac SeiZle- 0 mifrinavs Tanabari siCqariT, agdebs gadamrCenel kapsulas. kapsulis vardnisas misi horizontaluri siCqaris komponenti rCeba TviTmfrinavis siCqaris toli. tr φ ba axlos. sur. 4.15. TviTmfrinavi, romelic horizontalurad x 0 h mx ed ve lo bi ae qt sa or ia re θ x 4.6 Wurvisebri moZraobis analizi 101 (a) rogori φ kuTxiT unda iyos TviTmfri- –500 m = (55 m/wm)(sin0°)t – 12 (9,8 m/wm)t 2 navi msxverplis mimarT, rodesac piloti t-s amoxsna mogvcems t = 10,1 wm. am Sedegis kapsulas gadaagdebs? 4.28 formulaSi CasmiT miviRebT: amoxsna: gadagdebis Semdeg kafsula Wurvi xdeba. maSasadame, misi horizontaluri da vertikaluri moZraoba erTmaneTisgan x – 0 = (55 m/wm)(cos0°)(10,1 wm) x = 555,5 m anu Semdeg formula 4.27 mogvcems:L damoukidebelia da SesaZlebelia maTi calcalke ganxilva (kafsulis namdvili traeq- φ = tg–1 toria ar gvWirdeba). 4.15 suraTze naCvenebia koordinatTa sistema, romlis aTvlis saTave kapsulis gadagdebis wertilia. rogorc vxedavT, φ moicema formuliT: φ = tg–1 x , h (4.27) sadac x aris msxverplis (da wyalze dacemuli kafsulis) horizontaluri koordinati, xolo h – TviTmfrinavis frenis simaRle. simaRle 500 m-ia, maSasadame φ-s sapovnelad mxolod x gvWirdeba. x-is povna SesaZlebelia 4.21 formulaiT: x – x0 = (v0 cos θ0)t (4.28) kafsula gadaagdes, ar usvriaT, amitom misi sawyi si 0 siCqare TviTmfrinavis siCqaris tolia. maSasadame, sawyisi siCqarea v0= 55 m/wm, xolo kuTxe θ0 = 0° (x RerZis dadebiTi mi marTulebisadmi). miuxedavad yvelafrisa, kafsulis gadagdebidan wyalze dacemamde kafsulis amoxsna: kafsulis horizontaluri da ver tikaluri moZraobebi erTmaneTisgan damo ukidebelia. vx horizontaluri komponenti ar icvleba da v0x = v0 cos θ0 sawyisi mniSvnelobis tolia, vinaidan horizontaluri aCqareba ara gvaqvs. maSasadame, rodesac kafsula wyalze daecema: vx = v0 cos θ0 = (55 m/wm)(cos 0°) = 55 m/wm. vy vertikaluri komponenti icvleba sawyisi v0y = v0 sin θ0 mniSvnelobidan, vinaidan vertikaluri aCqareba arsebobs. 4.23 formulisa da kafsulis dacemis t = 10.1 wm drois gamoyenebiT gavigebT rom, rodesac kafsula wyalze ecema, vy = v0 sin θ0 – gt t-s sapovnelad ganvixiloT vertikal- = (55 m/wm)(sin 0°) – (9,8 m/wm2)(10,1 wm) = –99 m/wm. uri moZraoba da, kerZod, formula 4.22: (4.29) maSasadame, wyalze dacemis dros kafsulis siCqare iqneba: aq kafsulis vertikaluri gadaadgileba y – y0 = – 500 m (uaryofiTi niSani kafsulis qvemoT moZraobas asaxavs). 4.29 formulaSi am da sxva cnobili sidideebis CasmiT miviRebT: amocanis amoxsnis nimuSi siCqare erTeulovan-veqtorul da sidide-kuTxur CanawerSi? gasuli t dro ar viciT. y – y0 = (v0 sin θ0 t) – 12 gt2 (pasuxi) (b) wyalze dacemis dros, rogori iqneba vinaidan aTvlis saTave kapsulis gadagdebis wertilia, amitom x0 = 0. 555,5 m = 48° 500 m v = (55 m/wm) – (99 m/wm) (pasuxi) formula 3.6-is an veqtoruli kalkula toris gamoyenebiT vipovoT -s sidide da kuTxe: v = 113 m/wm da θ = –60,9° (pasuxi) 4.7 4.16 suraTze naCvenebia mekobreTa xomaldi, a) horizontaluri xazidan ra θ0 kuTxiT romelic kunZulze mdgari cixesimagridan unda gaisrolon Wurvi, gems rom moxvdes? 560 m manZilzea. qvemexi, romelic zRvis doneze dgas, Wurvebs v0 = 82 m/wm sawyisi amoxsna: cxadia, qvemexis Wurvs Wurvise- siCqariT isvris. bri moZraobis formulebi Seesabameba. Cven 102 organzomilebiani da samganzomilebiani moZraoba Tavi 4 gvWirdeba srolis θ0 kuTxesTan dakavSirebuli formula, rodesac tyvia horizontalurad gadaadgildeba qvemexidan xomaldisken. vinaidan qvemexi da gemi erT simaRleze arian, amitom areali iqneba horizontalu ri gadaadgileba. Cven SegviZlia srolis θ0 kuTxe R arealTan davakavSiroT 4.26 form mulis saSualebiT: R= 2v02 sin 2θ0 , g sur. 4.16 aseT manZilze gems Wurvi ori kuTxiT gasrolis SemTxvevaSi moxvdeba. (4.30) rac mogvcems: 2 gR –1 (9,8 m/wm )(560 m) 2 = sin (82 m/wm)2 v0 2θ0 = sin–1 b) ra manZilze unda daSordes meko- = sin–1 0,816. sin–1 Sebrunebul funqcias ori (4.31) breebis gemi zarbazans, rom Wurvi ar mox- pasuxi amoxsna: rogorc ukve viciT, Wurvis gas- vdes? aqvs. erT maTgans (am SemTxvevaSi 54,7°) rolis maqsimaluri areali Seesabameba 45° kalkulatori aCvenebs. meore pasuxis gas- kuTxes. 4.30 formulaSi θ0 = 45° kuTxis Cas- agebad es kuTxe 180°-s gamovakloT da mi- miT miviRebT: viRebT 125,3°-s. maSasadame, formula 4.31 R= mogvcems: (82 m/wm)2 2v02 sin 2θ0 = sin (2×40°) 9,8 m/wm2 g = 686 m ≈ 690 m. (pasuxi) θ0 = 12 (54,7°) ≈ 27° (pasuxi) θ0 = 12 (125,3°) ≈ 63° (pasuxi) srolis orive kuTxe θ0 = 45° kuTxes uax- cixesimagris komendants SeuZlia zarbazani lovdeba. am kuTxis dros gemi 690 m man- am ori kuTxiT awios da Wurvi mekobreebis Zilze iqneba. amaze ufro Sors gems aRa- gems moartyas (Tu haeri ar SeuSlis!) raferi emuqreba. da roca jeison renis mier damyarebuli motocikliT xtomis rekordi. nel-nela kunZuls Sordeba, 4.8 amocanis amoxsnis nimuSi 4.17a suraTze naCvenebia 2002 wels gemi asafreni tramplini dasaSvebi tramplini asax- tomi tramplinebis simaRlea H = 3 m, kuTxe manZili θR = 12°, xolo D = 77 m. motociklisti maT Soris davuSvaT, meore rom tramplinis SuagulSi daeSva. haeris zemoqmedeba ugulebelvyoT. gamoTvale motociklistis siswrafe pirveli tramplinidan gadaxtomis dros. amoxsna: reni da misi motocikli Wurvia. Tu maT ganvixilavT, rogorc erT wertils, maSin SegviZlia gamoviyenoT Wurvis formulebi. sur. 4.17 a) asafreni da dasaSvebi tramplinebi. afrenis b) afrenis siCqaris 0 (θ0 kuTxisas) x komponentia v0 cos θ0, xolod y komponenti _ v0 sin θ0; g) daSvebis wertili tramplinis SuagulSia. sawyisi ; 0 103 4.6 Wurvisebri moZraobis analizi xy sistema dro. vinaidan horizontalur RerZze jer- ise gavavloT, rom aTvlis saTave pirveli jerobiT verafers mivaRwieT, amitom ver- tramplinis Zirs daemTxvas. renis koordi- tikalur RerZze gadavideT. 4.22 formula natebi naxtomis dawyebisas (rodesac t = 0) v0-s akavSirebs y – y0 gadaadgilebasTan: gadaxtoma. koordinatTa iqneba x0 = 0 da y0 = H = 3 m. θ0 gadaxtomis y – y0 = (v0 sin θ0)t – 12 gt2 kuTxe tramplinis θR daxrilobis toli 1,5 m – 3 m = (v0 sin 12°)t – 12 (9,8 m/wm2)t2. iqneba (sur. 4.17b): θ0 = θR = 12°. daSveba. 4.17g suraTze tramplinis sigrZe L marTkuTxa samkuTxedis hipotenuzas qmnis. reni tramplinis SuagulSi anu wverodan L/2 manZilze wertilSi daeSva. SegviZlia vaCvenoT, rom daSvebis wertili mdebareobs tramplinis H simaRlisa da d horizontaluri sigrZis SuagulSi, ma- kvlav dabrkolebas mivadeqiT, vinaidan t-s gareSe v0-s ver gavigebT. Tumca, rogorc vxedavT, 4.32 da 4.33 formulebSi ucnobTa erTi da igive wyvilia: v0 (sasurveli) da t (arasasurveli). maSasadame formula 4.32is t-sTvis amoxsna SesaZlebelia: Sasadame, daSvebis wertilis vertikaluri t= koordinatia H 3m = = 1,5 m 2 2 daSvebis wertilis x horizontaluri y= koordinatis sapovnelad rZe: d= x – x0 v0 cos θ0 Semdeg CavsvaT es Sedegi 4.33 formulaSi da gamovricxoT t: y – y0 = (v0 sin θ0) gamovTvaloT dasaSvebi tramplinis horizontaluri sig- (4.33) = (x – x0) tg θ0 – ( ) ( x – x0 x – x0 – 1g 2 v0 cos θ0 v0 cos θ0 ) 2 g(x – x0)2 , 2(v0 cos θ0)2 sadac gamoviyeneT igiveoba: tg θ0 = (sinθ0)/ 3m H = = 14,11 m tg 12° tg θR (cos θ0). CavsvaT cnobili monacemebi da mi- Semdeg vipovoT horizontaluri koordi- viRebT: nati: 1,5 m – 3 m = (84,1 m)(tg 12°) d 14,11 m x = D + = 77 m + = 84,1 m 2 2 – (9,8 m/wm2)(84,1 m)2 2 2 v0 (cos 12°)2 v0 gadaxtomis siCqaris povna. x – x0 gadaadgilebis da v0-is dasakavSireblad viyenebT anu 4.21 formulas: napiridan aseTi siswrafiT gadaxtoma al- x – x0 = (v0 cos θ0)t (4.32) 84,1 m – 0 = (v0 cos 12°)t. v0-sTvis formulis amoxsna SeuZlebelia, radgan ar viciT renis haerSi yofnis t v0 = 43,2 m/wm ≈ 156 km/sT (pasuxi) baT sulisSemZvrelia. magram renis 156 km/ sT-ze meti siCqare dasWirda meore napirze dasaxtomad, vinaidan haeri mas da motocikls mniSvnelovnad Seanelebda. amocanis amoxsnis taqtika taqtika 1 : algebra ricxvebis nacvlad damrgvalebasa da sxva cdomilebebis- amomxsnelebi swored ase iqcevian. pirvel gan Tavis dasazRvevad amocanebi algebru- TavebSi vamjobineT amocanebi nawil-nawil lad amoxseniT da ricxvebi mxolod bolos amogvexsna, raTa ufro kargad Cawvdomod- CasviT. amocanebis amoxsnis 4.6 da 4.8 ni- iT maT mimdinareobas. SemdegSi met yur- muSebSi es sakmaod advilia da gamocdili adRebas algebras davuTmobT. 104 Tavi 4 4.7. organzomilebiani da samganzomilebiani moZraoba Tanabari wriuli moZraoba wertilis moZraobas Tanabari wriuli moZraoba ewodeba, roca wertili wrewirze an wrewiris rkalze moZraobs Tanabari siCqariT. miuxedavad imisa, rom siCqare ar icvleba, wertils gaaCnia aCqareba. es SeiZleba gagvikvirdes, vinaidan aCqareba (siCqaris cvlileba) warmogvidgenia, rogorc siCqaris zrda an kleba. Tumca, gaixseneT, rom siCqare veqtoria da ara skalari. maSasadame, maSinac ki, roca siCqare mxolod mimarTulebas icvlis, aCqareba mainc arsebobs. zustad ase xdeba Tanabari wriuli moZraobis dros. sur. 4.18. saaTis isris moZraobis sawinaaRmdego mimarTulebiT Tanabarwriulad moZravi wertilis siCqarisa da aCqarebis veqtorebi. maTi sidide ucvlelia, mimarTuleba ki mudmivad icvleba. 4.18 suraTze naCvenebia siCqarisa da aCqarebis veqtorebs Soris kavSiri Tanabari wriuli moZraobis sxvadasxva periodebSi. orive veqtoris sidide moZraobis dros mudmivia, magram maTi mimarTulebebi icvleba. siCqare yovelTvis wrewiris mxebia moZraobis mimarTulebiT. aCqareba ki yovelTvis mimarTulia wrewiris centrisken. amis gamo, Tanabar wriul moZraobasTan dakavSirebul aCqarebas centriskenul aCqarebas uwodeben. rogorc SemdegSi vnaxavT, aCqarebis sididea: a= v2 r sadac r wrewiris radiusia, xolo (centriskenuli aCqareba), ν (4.34) – wertilis siCqare. garda amisa, am aCqarebis mqone wertili wrewiris garSemo srul bruns asrulebs anu 2πr manZils gadis Semdeg droSi: T= 2πr v (periodi) (4.35) T-s brunvis periodi an, ubralod, periodi ewodeba. esaa dro, romelic wertils sWirdeba erTi sruli brunis gasakeTeblad. 4-34 formulis damtkiceba Tanabari wriuli moZraobisas aCqarebis sididisa da mimarTulebis sapovnelad ganvixiloT sur. 4.19. 4.19a suraTze p wertili Tanabari ν siCqariT moZraobs r radiusis mqone wrewirze. mocemul momentSi p-s koordinatebia xp da yp. 4.3 qveTavidan gavixsenoT, rom moZravi wertilis siCqare misi traeqtoriis mxebia. 4.19a suraTze es niSnavs, rom siCqare wertilis mdebareobaze gavlebuli radiusis perpendikularulia. maSin θ kuTxe -sa da p wertilze gavlebul vertikals Soris r radiussa da x RerZs Soris θ kuTxis tolia. -s skalaruli komponentebi 4.19b suraTzea naCvenebi. maTi saSualebiT SeiZleba Semdegi saxiT Caiweros: = vx + vy = (–v sin θ) + (v cos θ) (4.36) 105 4.7. Tanabari wriuli moZraoba y y y v θ θ vy ax vx r a yp θ x x φ ay x xp sur. 4.19. p wertili wreze moZraobs saaTis isris sawinaaRmego mimarTulebiT. (a) misi mdebareoba da qare mocemul momentSi. (b) siCqare da misi komponentebi. (g) wertilis siC- aCqareba da misi komponentebi. axla, 4.19 suraTze mocemuli marTkuTxa samkuTxedis gamoyenebiT, sin θ SevcvaloT yp/r-iT, xolo cos θ – xp /r-iT, gveqneba: vy = – p r ( ) ( ) vxp r + . (4.37) p wertilis aCqarebis sapovnelad es formula gavawarmooT drois mimarT. vinaidan ν siCqare da r radiusi droSi ar icvleba, miviRebT: ( d v dyp = dt = – r dt ) ( + v dxp r dt ) . (4.38) axla gaviTvaliswinoT, rom yp-s cvlilebis tempi dyp /dt siCqaris νy komponentis tolia. analogiurad, dxp /dt = νx . 4.19b suraTidan vxedavT, rom νx = –ν sin θ da νy = ν cos θ. CavsvaT es yvelaferi 4.38 formulaSi da miviRebT: ( = – v2 cos θ r ) ( + – v2 sin θ r ) . (4.39) 4.19g suraTze es veqtori da misi komponentebia naCvenebi. 3.60 formulis Tanaxmad, miviRebT: a= = v2 r risi damtkicebac gvindoda. = v2 r = v2 , r -s mimarTulebis sapovnelad unda vipovoT φ kuTxe: tg φ = maSasadame, φ = θ. es niSnavs, rom ay –(v2/r) sin θ = = tg θ. ax –(v2/r) cos θ mimarTulia 4.19a suraTze naCvenebi r radiusis gaswvriv wrewiris centris mimarTulebiT. swored amis damtkiceba gvindoda. sakontrolo wertili 6. sxeuli Tanabari siCqariT moZraobs wriulad xy sibrtyeze, romlis aTvlis saTave wrewiris centrs emTxveva. rodesac sxeuli imyofeba x = – 2 m wertilze, misi siCqarea –(4 m/wm) . gamoTvale sxeulis a) siCqare da b) aCqareba, rodesac is imyofeba y = 2 m wertilSi. 106 organzomilebiani da samganzomilebiani moZraoba Tavi 4 4.9 amocanis amoxsnis nimuSi samxedro reaqtiuli TviTmfrinavebis pilotebi nicdian dros garkveul mkveTrad pilotis diskomforts moxvevis sxeuli ga dros. aseT ganicdis centriskenul aCqarebas, Tavi ki mimarTulia mrudis centrisken. tvinSi sisxlis wneva mcirdeba, rac tvinis funqciis dakargvas iwvevs. pilotis gasafrTxileblad ramdenime niSani arsebobs. rodesac centriskenuli aCqarebaa 2g an 3g, piloti Tavs damZimebulad grZnobs. 4g-s dros pilotis mxedveloba Sav-TeTri xdeba da mniSvnelovnad izRudeba. Tu aCqarebam kidev moimata, mxedveloba qreba da piloti gonebas kargavs. am mdgomareobas g-LOC anu `g-Ti gamowveuli gonebis dakargva~ ewodeba. pilotis aCqareba g-ebSi, Tu F- 22 mifrinavs ν = 2500 km/sT (694 m/wm) siCqariT wreze, romlis radiusia r = 5,8 km? amoxsna: miuxedavad imisa, rom wriul traeqtoriaze moZraobisas pilotis siCqaris sidide mudmivia, siCqaris mimarTulebis cvlilebis gamo warmoiqmneba centriskenuli aCqareba, romlis sidide mocemulia 4.34 formulis saSualebiT: v2 (694 m/wm)2 a= = = 83 m/wm2 = 8,5g (pasuxi) 5800 m r Tu uyuradRebo piloti sahaero brZo- lis dros TviTmfrinavs mkveTrad Semoabrunebs, is momentalurad gadava g-LOC mdgomareobaSi da ver SeniSnavs gamafrTxilebel signalebs. rogori iqneba centriskenuli 4.8 erTganzomilebiani fardobiTi moZraoba warmoidgine, rom xedav ixvs, romelic CrdiloeTiT mifrinavs 30 km/sT siCqariT. meore ixvisTvis, romelic imave siCqariT mifrinavs CrdiloeTiT, is uZravia. sxvagvarad rom vTqvaT, wertilis siCqare damokidebulia damkvirveblis aTvlis sistemaze. Cveni miznebidan gamomdinare, aTvlis sistema fizikuri obieqtia, romelsac koordinatTa sistemas vukavSirebT. yoveldRiur cxovrebaSi es obieqti miwaa. magaliTad, sagzao niSanze miTiTebuli daSvebuli siCqare yovelTvis gazomilia miwis mimarT. policiis oficris mimarT siCqare sxvadasxva iqneba, vinaidan policieli siCqaris gazomvis dros imoZravebs. warmoidgine, rom aleqsi (4.20 suraTze A y y sistemis aTvlis saTaveSi) gaCerebulia trasis ganapiras da P CarCo manqanis (wertilis) swrafad moZraobas akvirdeba. barbara (B sistemis aTv- A CarCo lis saTaveSi) trasaze mudmivi siCqariT midis B P BA xBA xPB x xPA = xPB + xBA da isic P manqanas uyurebs. davuSvaT, orive maTgani adgens manqanis mdebareobas mocemul x sur. 4.20. aleqsi (naxazi A) da barbara (naxazi B) P manqanas adevneben Tvals. B da P gansxvavebuli siCqariT moZraoben orive naxazisTvis saerTo x RerZis gaswvriv. grafikze gamosaxul momentSi xBA aris B-s koordinatebi A naxazze. B naxazze P-s koordinatebia xPB. A naxazze koordinatebi xPA = xPB + xBA . momentSi. 4.20 suraTze vxedavT, rom: es formula xPA = xPB + xBA ase ikiTxeba: (4.40) `P-s koordinati A-s mimarT (xPA) tolia P-s koordinats B-s mimarT xPB mimatebuli B-s koordinati A-s mimarT (xBA)~. wakiTxvisas yuradReba miiqcieT indeqsebs. 4.40 formulis drois mimarT gawarmoebiT miviRebT: 107 4.8 erTganzomilebiani fardobiTi moZraoba d d d dt (xPA) = dt (xPB) + dt (xBA). an (vinaidan ν = dx/dt): vPA = vPB + vBA (4.41) es formula ase ikiTxeba: `P-s siCqare A-s mimarT vPA tolia P-s siCqares B-s mimarT vPB mimatebuli B-s siCqare A-s mimarT vBA ~. vBA aris B sistemis siCqare A-s mimarT (vinaidan moZraoba erTi RerZis gaswvrivaa, SegviZlia siCqaris komponentebi am RerZis gaswvriv gamoviyenoT 4.41 formulaSi da ugulebelvyoT veqtoris isrebi). Cven mxolod im sistemebs ganvixilavT, romlebic erTmaneTis mimarT Tanabari siCqariT moZraobs. am SemTxvevaSi es niSnavs, rom barbara (sistema B) aleqsis (sistema A) mimarT moZraobs vBA mudmivi siCqariT. asea Tu ise, P manqanas (moZravi wertili) SeuZlia Secvalos siCqare da mimarTuleba (anu aCqardes). barbarasa da aleqsis mimarT P-s aCqarebis dasakavSireblad, 4.41 formulaSi gavawarmooT dro: d d d dt (vPA) = dt (vPB) + dt (vBA). vinaidan vBA mudmivia, amitom formulis bolo wevri nulis tolia da gveqneba: aPA = aPB . (4.42) sxvagvarad rom vTqvaT: sxvadasxva aTvlis sistemis damkvirveblebi, romlebic erTmaneTis mimarT mudmivi siCqariT moZraoben, miiReben moZravi wertilis erTsa da imave aCqarebas. sakontrolo wertili 7. cxrilSi mocemulia bar- mdgomareoba νBA νPA mdgomareobaSi. ras udris cxrilSi gamotovebuli (a) + 50 + 50 sidideebi da rogor icvleba manZili barbarasa (b) + 30 barasa da P manqanis siCqareebi (km/sT-ebSi) sam da P manqanas Soris? (g) amocanis amoxsnis nimuSi 4.20 suraTze davuSvaT, rom barbaras siCqare aleqsis mimarT mudmivia, vBA = 52 km/ sT, xolo P manqana moZraobs x RerZis uaryofiTi mimarTulebiT. (a) Tu aleqsisTvis P manqanis siCqare vPA = – 78 km/sT, rogori iqneba vPB siCqare barbarasTvis? amoxsna: A aTvlis sistema davukavSiroT aleqss, xolo B aTvlis sistema – barbaras. vinaidan es ori aTvlis sistema erTmaneTis mimarT Tanabrad moZraobs erTi νPB + 40 + 60 – 20 4.10 anu _ 78km/sT = vPB + 52 km/sT maSasadame, vPB = _ 130km/sT (pasuxi) Tu P-s barbaras manqanaze TokiT mivabamdiT, 130 km/sT siCqareze Tokis kvanZi gaixsneboda da ori manqana gancalkevdeboda. (b) Tu P manqana amuxruWebs aleqsis mimarT (maSasadame, miwis mimarT) t = 10 wm-is ganmavlobaSi Tanabari aCqarebiT, rogori iqneba misi aPA aCqareba aleqsis mimarT? RerZis gaswvriv, amitom SegviZlia gamov- amoxsna: aleqsis mimarT P manqanis aCqarebis iyenoT formula 4.41, raTa vPB davukav- gamosaTvlelad unda gamoviyenoT aleqsis vPA = vPB + vBA Tanabaria, amitom viyenebT 2.11 formulas SiroT vPA-sa da vBA-s: mimarT manqanis siCqare. vinaidan aCqareba 108 organzomilebiani da samganzomilebiani moZraoba Tavi 4 (ν = ν0 + at), raTa aCqareba davakavSiroT P-s sawyis da saboloo siCqaresTan. P-s sawyisi siCqare aleqsis mimarT aris vPA = – 78 km/sT, xolo saboloo siCqarea 0. maSasadame, 2.11 manqanis siCqare moZravi barbaras mimarT). formula gvaZlevs: maSasadame, kvlav 2.11 formulis Tanaxmad: aPA = 0 – (–78 km/sT) 1 m/wm v – v0 = 10 wm 3,6 km/sT t = 2,2 m/wm2 viciT P-s sawyisi siCqare barbaras mimarT (νPB = –130 km/sT). P-s saboloo siCqare barbaras mimarT aris –52km/sT (gaCerebuli –52 km/sT – (–130 km/sT) 1 m/wm v – v0 = 10 wm 3,6 km/sT t aPB = (pasuxi) = 2,2 m/wm2 (g) rogori iqneba damuxruWebis dros P (pasuxi) es Sedegi winaswar SegveZlo gvewinas- manqanis aCqareba barbaras mimarT aPB? warmetyvela. vinaidan aleqsi da barbara amoxsna: barbaras mimarT P-s aCqarebis erTmaneTis mimarT Tanabari siCqariT moZ- gamosaTvlelad unda gamoviyenoT barbaras raoben, amitom maT mimarT manqanis aCqa- mimarT manqanis siCqareebi. (a) nawilidan reba erTnairi iqneba. 4.9 organzomilebiani fardobiTi moZraoba axla erTganzomilebiani fardobiTi moZraobidan gadavdivarT organzomilebian (da samganzomilebian) moZraobaze. 4.21 suraTze Cveni ori damkvirvebeli P moZrav wertils uyurebs A da B aTvlis sistemebis aTvlis saTaveebidan. amasobaSi B Tanabari siCqariT vBA moZraobs A-s mimarT (am ori sistemis Sesabamisi RerZebi erTmaneTis paraleluria). 4.21 suraTze moZraobis erTi konkretuli momentia naCvenebi. am momentSi B-s aTvlis saTavis mdebareobis veqtori A-s aTvlis saTavis mimarT aris . amis garda, P wertilis mdebareobis veqtori A-s aTvlis saTavis mimarT aris , xolo B-s aTvlis saTavis mimarT _ . Tu am sam mdebareobis veqtors ise davalagebT, rom erTis saTave meores bolos daemTxves, maSin veqtorebs Soris damyardeba Semdegi kavSiri: = + . (4.43) am formulis drois mimarT gawarmoebiT SesaZlebelia damkvirveblebis mimarT wertilis da P siCqareebis dakavSireba: y y P = + . (4.44) am tolobis gawarmoebiT ki erTmaneTs davukavSirebT wertilis da damkvirveblebis mimarT P aCqarebas. Tumca, vinaidan mudmivia, amitom misi warmoebuli drois mimarT nulis tolia. maSasadame, miviRebT: aTvlis sistema AB aTvlis sistema AA x x sur. 4.21. B sistemas A sistemis mimarT aqvs . B-s mdeTanabari organzomilebiani siCqare bareobis veqtori A-s mimarT aris . P wertilis mdebareobis veqtori A-s mimarT aris , xolo B-s mimarT – . = (4.45) iseve, rogorc erTganzomilebiani moZraobisTvis, aqac arsebobs Semdegi wesi: sxvadasxva aTvlis sistemebis damkvirveblebis mimarT, romlebic erTmaneTis mimarT Tanabari siCqariT moZraoben, moZravi wertilis aCqareba erTnairi iqneba. 109 4.9 organzomilebiani fardobiTi moZraoba amocanis amoxsnis nimuSi 4.22a suraTze TviTmfrinavi aRmosav- 4.1 1 Crd leTisken unda gafrindes, piloti ki mas samxreT-aRmosavleTisken mimarTavs, saidanac Crdilo-aRmosavleTis Zlieri qari uberavs. TviTmfrinavis siCqare qaris mimarT 215 km/sT-is tolia da mimarTulia samxreTiT aRmosavleTis mimarTulebasTan θ kuTxiT. qaris aRm θ 20 o (a) siCqare miwis mimarT aris 65 km/sT da mimarTulia 20° CrdiloaRmosavleTisken. rogori mimarT TviTmfrinavis iqneba miwis y θ siCqaris sidide da θ kuTxe? x amoxsna: es situacia 4.21 suraTis msgavsia. aq P moZravi wertili TviTmfrinavia, A sistema dakavSirebulia miwasTan (davarqvaT mas G), xolo B sistema dakavSirebulia qarTan (davarqvaT mas W). saWiroa (b) sur. 4.22. aRmosavleTisken gasafrenad TviTmfrinavis pilotma qaris mimarTuleba unda gaiTvaliswinos. veqtoruli diagramis ageba, rogorc 4.21 suraTzea naCvenebi. Tumca, amjerad siCqaris sam veqtors gamoviyenebT. an 0 = –(215 km/sT) sin θ + (65 km/sT)(cos 20°). θ-s amoxsna mogvcems: Tavdapirvelad davweroT veqtorebis dama- θ = sin–1 kavSirebeli winadadeba: TviTmfrinavis siCqare miwis mimarT = TviTmfrinavis siCqare qaris mimarT (PG) qaris siCqare + (PW) miwis mimarT (WG) (65 km/sT)(cos 20°) = 16,5° (pasuxi) 215 km/sT aseTive xerxiT x komponentebisTvis vipoviT: vPG,x = vPW,x + vWG,x es kavSiri SeiZleba gamoisaxos 4.22 b suraTis msgavsad, xolo veqtorulad mas = + vinaidan x RerZis paraleluria, amitom vPG,x komponenti vPG sididis tolia. aseTi saxe eqneba: (4.46) CvenTvis ucnobia pirveli veqtoris sidide da meore veqtoris mimarTuleba. ori ucnobis gamo veqtorul kalkulatorze 4.46 formulis amoxsna SeuZlebelia. amis CavsvaT es aRniSvna da kuTxe θ = 16,5°, gve qneba: vPG = (215 km/sT)(cos 16,5°) + (65 km/sT)(sin 20°) = 228 km/sT (pasuxi) nacvlad 4.22 b suraTis koordinatTa sis sakontrolo wertili 8. davuSvaT, temaze veqtorebi komponentebad unda dav piloti TviTmfrinavs aRmosavleTisken SaloT da Semdeg amovxsnaT formula 4.46 abrunebs siCqaris cvlilebis gareSe. TiToeuli RerZisTvis (ixile qveTavi 3.6). gaizrdeba, Semcirdeba Tu igive dar- y komponentebisTvis vipoviT: vPG,y = vPW,y + vWG,y Ceba Semdegi sidideebi: a) vPG,y ; b) vPG,x ; g) vPG? (pasuxis gacema gamoTvlebis ga- reSec SeiZleba). 110 organzomilebiani da samganzomilebiani moZraoba Tavi 4 mimoxilva da Sejameba mdebareobis veqtori. wertilis mde- romelsac myisi aCqareba ewodeba: bareoba koordinatTa sistemis aTvlis saTavis mimarT moicema = mdebareobis veqto- riT. erTeulovan-veqtorul CanawerSi: =x +y +z , d . dt (4.16) erTeulovan-veqtorul CanawerSi: = ax +ay +az , (4.1) (4.17) aq x , y da z veqtoris veqtoruli kompo- sadac ax = dνx /dt, ay = dνy /dt da az = dνz /dt. nentebia, xolo x, y da z – skalaruli kom- Wurvisebri ponentebi (da, aseve, wertilis koordinate- ewodeba 0 moZraoba. sawyisi Wurvisebri siCqariT gasrolili bi). mdebareobis veqtors aRwers sidide da wertilis moZraobas. frenis dros werti- erTi an ori mimarTulebis kuTxe, an misi lis horizontaluri aCqareba nulis tolia, veqtoruli an skalaruli komponentebi. xolo vertikaluri aCqareba Tavisufali gadaadgileba. Tu wertili ise moZraobs, rom misi mdebareobis veqtori 1 icvleba -iT, maSin wertilis D gadaadgileba iqne- 2 ba: D = 2 – 1 . (4.2) gadaadgileba asec SeiZleba Caiweros: D = (x2 – x1) + (y2 – y1) + (z2 – z1) = Dx + Dy + Dz vardnis –g aCqarebaa (zeda mimarTuleba aRebulia dadebiTad). Tu luri x RerZis da vertikaluri y RerZis gaswvriv wertilis moZraobis formulebi Semdegnairi iqneba: x – x0 = (v0 cos θ0)t (4.3) (4.4) tilis gadaadgilebaa Δt drois monakveTSi saS Wurvisebri D (4.8) Δt Tu 4.8 formulaSi Δt nuls uaxlovdeba, masaS Sin saS = aRwevs zRvars, romelsac siCqare an myisi siCqare ewodeba: = d dt (4.10) erTeulovan-veqtorul CanawerSi gveqneba: = vx + vy + vz , sadac νx = dx/dt, νy = dy/dt da νz = dz/dt. (4.11) (4.21) y – y0 = (v0 sin θ0)t – 12 gt2 (4.22) vy = v0 sin θ0 – gt (4.23) vy2 = (v0 sin θ0)2 – 2g(y – y0). siCqare drois am monakveTSi iqneba: gamoisaxeba ν0 sididiTa da θ0 kuTxiT, maSin horizonta- saSualo siCqare da myisi siCqare. Tu weraris D , maSin misi 0 moZraobis dros (4.24) wertilis traeqtoria paraboluria da gamoiTvleba formuliT: y = (tg θ0)x – gx2 2(v0 cos θ0)2 (4.25) roca 4.21 da 4.24 formulebSi x0 da y0 nulis tolia. wertilis R horizontaluri areali aris horizontaluri manZili gasrolis adgilidan im adgilamde, romelzec wertili gasrolis simaRleze brundeba R= myisi v02 sin 2θ0 , g (4.26) siCqare yovelTvis wertilis traeqtoriis Tanabari wriuli moZraoba. rodesac wer- mxebia mocemul momentSi. tili ν Tanabari siCqariT moZraobs r ra- saSualo aCqareba da myisi aCqareba. diusis mqone wrewirze an rkalze, es Tana- Tu wertilis siCqare Δt drois monakveTSi bari wriuli moZraobaa da misi icvleba sididea: 1 -dan -Ti, maSin misi saSualo aC- 2 qareba am periodSi iqneba: saS = 2 – Δt 1 = Δv Δt . a= (4.15) v2 , r aCqarebis (4.34) mimarTulia wrewiris an rkalis centrisken Tu 4.15 formulaSi Δt nuls uaxlovdeba, da mas centriskenuli aCqareba ewodeba. maSin erTi sruli wris Semosawerad daxarjuli saS aRwevs zRvrul mniSvnelobas, 111 sistemis damkvirveblisa da B sistemis dam- dro gamoiTvleba formuliT: 2πr T= v (4.35) T-s brunvis periodi an ubralod periodi ewodeba. fardobiTi moZraoba. rodesac A da B aTvlis sistemebi erTmaneTis mimarT Tanabari siCqariT moZraobs, P wertilis siCqare A kvirveblisTvis erTmaneTisgan gansxvavdeba. am or siCqares Soris Semdegi kavSiria: = + , (4.44) aris B-s siCqare A-s mimarT. orive sadac damkvirveblis mimarT wertilis aCqareba erTnairia: = (4.45) SekiTxvebi 1 4.23 suraTze naCvenebia wertilis i sawyisi da f saboloo mdebareobebi. rogori zedapirs ar ecema. daalage mdgomareobebi iqneba wertilis (a) sawyisi mdebareobis mixedviT. veqtori tori f i Wurvebis saboloo siCqareebis klebadobis da (b) saboloo mdebareobis veq- erTeulovan-veqtorul CanawerSi? (g) D gadaadgilebis x komponenti? y 3m 2m 1m 4m 4m 5m 3m i (a) (b) (g) sur. 4.25. SekiTxva 4. x 5 vapirebT miwidan raketis gasrolas erTerTi mocemuli sawyisi siCqaris veqtoriT: f 3m sur. 4.23 SekiTxva 1. (1) 0 = 20 + 70 , (2) 0 = –20 + 70 , (3) 0 = 20 – 70 , (4) 0 = –20 – 70 . Cvens koordinatTa sistema x miwis donezea, xolo y zeviT izrdeba. a) daalage veqtore- 2 4.24 suraTze naCvenebia lis skunsis bi Wurvis gaSvebis siCqareebis klebadobis saWm- mixedviT; b) daalage veqtorebi maZiebeli moZrao- a i b c frenis drois klebadobis mixedviT. bis traeqtoria i 6 xilis namcx- sawyisi wertili- vars erTaderTi dan. skunss erTi sargebeli Sur- da igive T dro sur. 4.24 SekiTxva 2. sWirdeba traeqtoriaze aRniSnul yovel Si 2 varjiSisas 1 moaqvs. 4.26 su- c wertilebi skunsis saSualo siCqareebis raTze sididis klebadobis mixedviT. bi garkveul = 25 – 4,9 momentSi burTis siCqarea (x RerZi horizontaluria, y mimarTulia zeviT, izomeba m/wm-ebSi). gaiara Tu ara burTma frenis umaRlesi wertili? 4 4.25 suraTze naCvenebia miwidan (erTi da imave simaRlidan) erTnairi sawyisi siCqariTa da kuTxiT nasroli identuri Wurvebis sami mdgomareoba. Wurvebi erTsa da imave naCvene- grafiki asaxavs lili y srola- duliT wertilamde misaRwevad. daalage a, b da 3 Wurvebis 1 gasronamcxvr- A B θ sur. 4.26 SekiTxva 6. is y simaRles da θ kuTxes frenis dros siCqarisa da aCqarebis veqtorebs Soris. (a) grafikze asoebiT aRniSnuli romeli wertili Seesabameba namcxvris miwaze dacemis adgils? (b) grafiki 2 asaxavs srolis imave sawyis siCqares, magram sxva kuTxes. gasrolis adgilidan ufro Sors daecema namcxvari amjerad Tu ufro axlos? 112 7 organzomilebiani da samganzomilebiani moZraoba Tavi 4 350 km/sT lurad siCqariT mfrinavma ketebiT horizonta- TviTmfrinavma saWmlis maragi pa- Camoyara. taluri siCqaris komponentebis mixedviT da (d) sawyisi siCqaris mixedviT. 10 burTi mi- R paketebze haeris zemoqmedeba ugulebel- wis yofilia. rogori iqneba paketebis sawyisi garkveuli (a) yisi vertikaluri da (b) horizonta- donidan isroles. ri suraTze iqneba siCqaris yuTis horizontaluri miwaze dacemamde? a siCqariT luri siCqaris komponentebi? (g) rogokomponenti saw b 4.29 c na Cvenebia θ0 bur sur. 4.29 SekiTxva 10. (d) TviTmfrinavis siCqare rom yofiliyo Tis R 450 km/sT, meti iqneboda vardnis dro, na- cvlileba θ0 gasrolis kuTxis mixedviT. arealis klebi Tu igive? grafikze 8 4.27 suraTze mandarini isroles 1, 2 da 3 fanjrebis gavliT. fanjrebis zomebi dan mciremde daalage (a) burTis frenis identuria da vertikaluradac Tanabrad aRniSnuli simaRleze burTis siCqaris mixedviT. 11 4.30 suraTze maTi avlis drois mixedviT da (b) am momen- naCvenebia oTxi tSi mandarinis saSualo siCqaris mixedviT, traeqtoria (nax- dididan mcireze. evarwreebi an me amis Semdeg mandarini Cauvlis 4, 5 da 6 oTxedi 1 2 wreebi), fanjrebs, romelTa zomebi identuria, ma- romelTac gram romlebic vertikalurad araTanabar matarebeli manZilebzea ganlagebuli. daalage es fan- nabari jrebi (g) maTi Cavlis drois mixedviT da moZraobisas. daa (d) am momentSi mandarinis saSualo siCqa- lage ris mixedviT, dididan mcireze. ebi 3 gadis Ta 4 siCqariT traeqtori dididan mciremde sur. 4.30 SekiTxva 11. matareblis mixedviT. 12 4.31 suraTze P wertili 4 aCqarebis sididis y P Tanabarwri- 5 2 ulad moZraobs. 1 moZrao- 6 vlis sami aT- saTavea. a) θ-s ra mniS- 9 4.28 suraTze naCvenebia miwis doniburTis x koordinatTa sur. 4.27 SekiTxva 8. dartymuli θ bis centri xy sistemis dan didi- mTliani drois mixedviT, (b) maqsimalur aris daSorebuli. daalage fanjrebi (a) 3 wertilebi traeqto- sur. 4.31 SekiTxva 12. vnelobebisTvis iqneba mdebareobis veqtoris ry ver- ria. ugulebelyavi haeris zemoqmedeba da tikaluri komponenti udidesi? b) θ-s ra dididan mciremde daalage traeqtoriebi mniSvnelobebisTvis iqneba wertilis siCqa- (a) frenis drois ris νy vertikaluri komponenti udidesi? g) mixedviT, (b) sawyisi vertikaluri siCqaris komponentebis mixedviT, (g) sawyisi horizon- y θ-s ra mniSvnelobebisTvis iqneba wertilis aCqarebis ay vertikaluri komponenti udidesi? 1 2 3 sur. 4.28 SekiTxva 9. x 13 (a) SesaZlebelia Tu ara aCqareba Tanabari siCqariT moZraobis dros? SesaZlebelia Tu ara rkalis Semowera (b) nulovani aCqarebiT 113 amocanebi SSM amoxsnebi mocemulia Student Solutions Manual-Si. www amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday ILW Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley. com/college/halliday. • – ••• wertilebis raodenoba amocanis sirTules gviCvenebs da (b) aCqarebis Tanabari sididiT? qveTavi 4-2. mdebareoba da gadaadgileba •1 eleqtronis mdebareobis veqtoria = (5 m) – (3 m) – (2 m) . a) ipove -is sidide. b) aage veqtori marjvena koordinatTa sistemaSi. •2 sazamTros koordinatebia: x = –5 m, y = 8 m da z = 0 m. ipove mdebareobis veqtoris a) erTeulovan-veqtoruli Canaweri, b) sidide da g) x RerZis dadebiTi mimarTulebisadmi kuTxe. d) aage veqtori marjvena koordinatTa sistemaSi. Tu sazamTro gadaitanes wertilSi (3 m, 0, 0), rogori iqneba misi gadaadgilebis e) erTeulovan-veqto ruli Canaweri, v) sidide da z) x-is dadebiTi mimarTulebisadmi kuTxe? •3 pozitronis gadaadgileba = 2 – 3 + 6 mTavrdeba = 3 – 4 mdebareobis veqtoriT. rogori iyo pozitronis sawyisi mdebareobis veqtori? D • 5 60 km/sT Tanabari siCqariT moZravi matarebeli 40 wuTis manZilze aRmosavleTisken, Semdegi 20 wuTis manZilze Crdilo-aRmosavleTiT 50°-iT, bolos ki 50 wuTi dasavleTisken moZraobs. rogori iqneba mTeli mgzavrobis manZilze matareblis saSualo siCqaris (a) sidide da (b) kuTxe? SSM •6 eleqtronis formuliT: D mdebareoba = 3t – 4t 2 mocemulia +2 t wameb- Sia, xolo _ metrebSi. (a) rogori iqneba eleqtronis (t) siCqare erTeulovan-veqtorul CanawerSi? rodesac t = 2 wm, rogori iqneba -s (b) erTeulovan-veqtoruli Canaweri, (g) sidide da d) x RerZis dadebiTi mimarTulebisadmi kuTxe? •7 ionis sawyisi mdebareobis veqtoria = 5   – 6 + 2 . 10 wamis Semdeg = –2 + 8 – 2 . rogori iqneba am 10 wamis ganmavlobaSi saS erTeulovan-veqtorul Canaw- erSi? ••8 TviTmfrinavi 45 wuTis ganmavlobaSi 483 km-s aRmosavleTiT mifrinavs A qalaqidan B qalaqisken, Semdeg ki 1,5 sT-is ganmavlobaSi B qalaqidan C qalaqisken mifrinavs 966 km-s samxreTiT. rogori iqneba mTeli frenisTvis TviTmfrinavis gadaadgilebis (a) sidide da (b) mimarTuleba? rogori iqneba saSualo siCqaris (g) sidide da (d) mimarTuleba? (e) rogori iqneba sa- ••4 kedlis saaTis wuTebis maCvene- Sualo manZili? beli isari wveridan RerZamde daaxloebiT 10 sm sigrZisaa. isris wveris gadaadgilebis veqtoris kuTxe da sidide unda ganisazRvros drois sam SualedSi. ra iqneba veqtoris a) sidide da b) kuTxe TxuTmeti wuTidan naxevar saaTamde? rogori iqneba veqtoris g) sidide da d) kuTxe Semdegi naxevari saaTis ganmavlobaSi? rogori iqneba veqtoris e) sidide da v) kuTxe Semdegi erTi saaTis ganmavlobaSi? qveTavi 4.4. saSualo aCqareba da myisi aCqareba qveTavi 4-3. saSualo siCqare da myisi siCqare •9 wertilis moZraobisas mdebareobis (me- trebSi) droze (wamebSi) damokidebulebis funqciaa = + 4t 2 + t . dawere gamosaxulebebi (a) siCqarisTvis da (b) aCqarebisTvis, rogorc drois funqciebi. SSM • 10 garkveul momentSi velosipedisti parkis flagStokidan aRmosavleTiT 40 mis manZilze imyofeba da samxreTisken moZraobs 10 m/wm siCqariT. 30 wamis Semdeg velosipedisti flagStokidan CrdiloeTiT 40 m-is manZilzea da aRmosavleTisken moZ- 114 organzomilebiani da samganzomilebiani moZraoba Tavi 4 raobs 10 m/wm siCqariT. 30-wamian SualedSi desac A wertili y RerZs Cauvlis, B wer- rogori iqneba velosipedistis gadaadg- tili aTvlis saTavidan moZraobas iwyebs = 0,4 m/ ilebis (a) sidide da (b) mimarTuleba? ro- nulovani gori iqneba saSualo siCqaris (g) sidide da wm2 Tanabari aCqarebiT. (d) mimarTuleba? rogori iqneba saSualo dadebiT mimarTulebas Soris ra θ kuTxe aCqarebis (e) sidide da (v) mimarTuleba? ••11 xy sibrtyeze mdebareoba siCqariT da -sa da y RerZis y moZravi wertilis mocemulia = formuliT: (2t – 5t) + (6 – 7t ) , izomeba metrebSi, xolo t – wamebSi. erTeulovan-veqtorul 3 sawyisi 4 CanawerSi, gamoTvale (a) , (b) da (g) A θ , rodesac t = 2 wm. (d) rogori iqneba kuTxe x RerZis dadebiT mimarTulebasa da wertilis B traeqtoriis mxebs Soris, rodesac t = 2 wm? ••12 zomieri qari xy horizontalur tyeze moZrav kenWs = (5 m/wm2) + (7 m/wm2) kenWis siCqarea (4 m/wm) . ras udris kenWis siCqaris (a) sidide da (b) kuTxe, ro desac is gadaadgildeba 12 m-iT x RerZis paralelurad? urika xy horizontalur sibrtyeze moZraobs ax = 4 m/wm da ay = –2 m/wm2 aCqarebis komponentebiT. misi sawyisi siCqaris komponentebia v0x = 8 m/wm da v0y = 12 m/wm. rogori iqneba urikis siCqare erTeulovanveqtorul CanawerSi, rodesac is udides y koordinats miaRwevs? ••14 xy sibrtyeze moZravi siCqare mocemulia wertilis formuliT: xy = (6t – 4t2) + 8 . m/wm-ebSi izomeba, xolo t (>0) – wamebSi. (a) risi tolia aCqareba, rodesac t = 3 wm? b) rodis aris (Tu aris) aCqareba nulis toli? (g) rodis aris (Tu aris) siCqare nulis toli? (d) rodis aris (Tu aris) siCqare 10 m/wm-is toli? •• 15 aTvlis saTavidan wertili moZrao- bas iwyebs = 3 m/wm siCqariT da = (–1 – 0,5 ) m/wm2 Tanabari aCqarebiT. rodesac wertili udides x koordinats miaRwevs, rogori iqneba misi (a) siCqare da (b) mdebareobis veqtori? SSM ILW •••16 4.32 suraTze A wertili y = 30 m wrfis gaswvriv da x RerZis paralelurad ν = 3 m/wm sur. 4.32 amocana 16. sibr- Tanabar aCqarebas aniWebs. rodesac t = 0, ••13 x Tanabari siCqariT moZraobs. ro gamoiwvevs Sejaxebas? qveTavi 4-6. Wurvisebri moZraoba • 17 brtyeli zedapiridan 45 m simaRleze mdebare Tofidan tyvia horizontalurad isroles 250 m/wm siCqariT. (a) ra drois ganmavlobaSi iqneba tyvia haerSi? (b) gasrolis adgilidan horizontalurad ra manZilze daecema tyvia miwaze? (g) ras udris tyviis siCqaris vertikaluri komponenti miwaze dacemis dros? •18 1991 wels tokioSi, mZleosanTa msof lio Cempionatze maik paueli 8,95 m-ze gadaxta da bob bimonis mier damyarebuli da 23 wlis manZilze xeluxlebeli rekordi 5 sm-iT gaaumjobesa. pauelis gadaxtomis siCqare 9,5 m/wm iyo (daaxloebiT sprinteris siCqare), xolo tokioSi g = 9,8 m/wm2. ramdenad mcirea pauelis horizontaluri areali imave siCqariT gasrolili wertilis maqsimalur horizontalur arealze? •19 burTi miwidan haerSi aisroles. 9,1 m-is simaRleze misi siCqarea + 6,1 ) m/wm, rad, xolo = (7,6 mimarTulia horizontalu- – zeviT. (a) ra maqsimalur si- maRles miaRwevs burTi? (b) ra horizontalur manZils gaivlis burTi? risi tolia miwaze dacemis momentSi burTis siCqaris (g) sidide da (d) kuTxe (horizontis mimarT)? • 20 patara burTi 1,2 m simaRlis magidis 115 kidesTan migorda da Camovarda. is magidis kididan 1,52 m horizontalur manZilze wm sawyisi siCqariT da horizontidan 40° kuTxiT. risi tolia srolis adgilidan daeca. (a) ramden xans iyo burTi haerSi? gadaadgilebis (b) ras udrida misi siCqare magididan gad- (b) movardnis dros? = 1,1 wm? gamoTvale •21 isari horizontalurad isroles 10 m/wm rodesac t = 1,8 wm da t = 5 wm. sawyisi siCqariT P wertilisken (samizne). •• 25 0,19 wamis Semdeg isari kidis Q wertils moxvda zustad P-s qvemoT. (a) risi tolia PQ manZili? (b) ra manZilze ascda mizans isari? • 22 4.33 suraTze qva h simaRlis kldis- ken gaisroles 42 m/wm sawyisi siCqariT da horizontisadmi θ0 = 60° kuTxiT. qva A wertilSi daeca gasrolidan 5,5 wamis Semdeg. ipove (a) kldis h simaRle, (b) qvis siCqare dacemamde da (g) qvis maqsimaluri H si- (a) horizontaluri da vertikaluri komponentebi, rodesac t igive komponentebi, myvinTavi wylis zedapiridan 10 m simaRlis platformis kididan 2 m/wm siCqariT gadmoxta. (a) kididan ra horizontalur manZilze iqneba myvinTavi 0,8 wm-is Semdeg? (b) wylis zedapiridan ra vertikalur manZilze iqneba am momentSi myvinTavi? (g) kididan ra horizontalur manZilze Seexeba myvinTavi wyals? SSM WWW ••26 matCis dros CogburTeli burTs 23,6 m/wm siCqariT awvdis. burTis centri raketas horizontalurad scildeba kor- tis zedapiridan 2,37 m-is simaRleze. bade 12 m-is moSorebiTaa da 0,9 m simaRlisaa. H θ0 A h rodesac burTi bades miaRwevs, (a) gadaaxteba Tu ara mas da (b) rogori iqneba manZili burTis centrsa da badis zeda nawils Soris? davuSvaT, burTi imave siCqariT moawodes, magram burTi raketas amjerad sur. 4.33 amocana 22. horizontidan 5° kuTxiT mocilda. rodesac burTi bades miaRwevs, (g) gadaaxteba maRle miwidan. •23 TviTmfrinavis siCqarea 290 km/sT, xolo daxriloba horizontis mimarT _ θ = 30°. piloti TviTmfrinavidan radaris satyuaras agdebs (sur. 4.34). horizontaluri manZili satyuaras gaSvebis adgilidan Tu ara mas da (d) ra manZili iqneba burTis centrsa da badis zeda nawils Soris? •• 27 Tanabari siCqariT moZravi TviTm- frinavi vertikalidan 53° kuTxiT gadaixara da 730 m simaRlidan Wurvi gaisrola. davardnis adgilamde d = 700 m-ia. (a) ram- 5 wamis Semdeg Wurvi miwaze daeca. (a) risi den xans imyofeba satyuara haerSi? (b) ra tolia TviTmfrinavis siCqare? (b) horizontalurad ra manZils gaivlis Wurvi frenis dros? rogori iqneba Wurvis siCqaris (g) θ horizontaluri da (d) vertikaluri komponentebi miwaze dacemis dros? SSM •• 28 fexburTis burTi dartymis Semdeg moZraobs 19,5 m/wm sawyisi siCqariT da miwidan 45°–iani kuTxiT. dartymis wer- d tilidan dartymis mimarTulebiT 55 m-is moSorebiT myofi moTamaSe burTis dasax- sur. 4.34 amocana 23. simaRlidan gauSves satyuara? ILW •24 t = 0 momentSi qva gaisroles 20 m/ vedrad gamorbis. rogori unda iyos misi saSualo siCqare, rom burTs miwaze davardnis dros Seexos? 116 organzomilebiani da samganzomilebiani moZraoba Tavi 4 Wurvis sawyisi siCqare xuTjer me- roles. 1,5 wm-is Semdeg burTi miwaze dae- tia, vidre misi siCqare maqsimalur simaR- ca, Senobidan d = 25 m manZilze da hori- leze. ipove srolis θ0 kuTxe. zontidan θ = 60° kuTxiT. (a) gamoTvale h (miniSneba: erTi gzaa moZraobis ukan `gad- ••29 ••30 burTi kedlisken isroles 25 m/wm siCqariT da horizontidan θ0 = 40° kuTxiT (sur. 4.35). burTis srolis wertilidan kedlamde manZilia d = 22 m. (a) srolis wertilidan ra simaRleze daejaxeba burTi axveva~, rogorc videofirze). risi tolia burTis srolis siCqaris (b) sidide da (g) kuTxe horizontalis mimarT? (d) kuTxe horizontis qveviT aris Tu zeviT? kedels? rogori iqneba kedelze moxvedris •• 34 dros misi siCqaris (b) horizontaluri da siCqaris droze damokidebulebis grafi- golfis burTi miwaze ecema. misi (g) vertikaluri komponentebi? (d) kedelze ki naCvenebia 4.37 suraTze. t = 0 momentSi dajaxebis momentSi gavlili eqneba Tu ara burTi miwaze ecema. (a) horizontalurad burTs traeqtoriis umaRlesi wertili? ra manZils gaivlis burTi miwaze dacemamde? (b) miwidan ra maqsimalur simaRles miaRwevs burTi? 31 29 ν (m/wm) θ0 d 27 25 23 21 sur. 4.35 amocana 30. 19 0 1 2 ••31 SaSxana, romelic tyviebs 460 m/wm siCqariT isvris, 45,7 m-iT daSorebul samiznes daumiznes. Tu samiznis centri SaSxanis simaRlezea, maSin samiznidan ra simaRleze 3 4 t (wm) 5 sur. 4.37 amocana 34. •• 35 4.38 suraTze burTi 10 m/wm siC- unda iyos damiznebuli SaSxanis lula, rom qariT da horizontisadmi 50° kuTxiT is- tyvia zustad centrSi moxvdes? SSM roles. srolis adgili aRmarTis ZirSia, ••32 romlis horizontaluri sigrZea d1 = 6 m, beisbolisti burTs horizontalurad awvdis 161 km/sT siCqariT. momgerie- xolo simaRle d2 = 3,60 m. (a) burTi aRmar- beli misgan 18,3 m manZilzea. a) ra dro Tis Tavze daecema Tu aRmarTze? rodesac dasWirdeba burTs manZilis pirveli naxevris gasavlelad? b) meore naxevris gas- daecema, rogori iqneba srolis wertilidan gadaadgilebis (b) sidide da (g) kuTxe? avlelad? g) rogoria burTis Tavisufali vardnis manZili frenis pirvel naxevarSi? d) meore naxevarSi? e) ratom aris erTmaneTis toli g) da d) sidideebi? ••33 4.36 burTi d2 0 d1 suraTze h simaRlis saxuravis sur. 4.38 amocana 35. marcxena napiridan burTi marcxniv gais- ••36 4.39 suraTze burTi h = 20 h θ m simaRlis saxuravze aisroles. 4 wamSi burTi saxuravze daeca. dacemamde burTsa da saxuravs Soris kuTxe iyo θ = 60°. (a) moZebne burTis d sur. 4.36 amocana 33. mier ganvlili horizontaluri d manZili (ixile 33-e amocanis miniSneba). rogori 117 iqneba burTis sawyisi siCqaris (b) sidide siCqariT unda isrolon. 4.41 suraTze naCvenebia R areali dartymis θ0 kuTxes- da (g) kuTxe (horizontis mimarT)? Tan SedarebiT. θ0-is sidide burTis frenis dros gansazRvravs. tmax iyos frenis maqsi- θ maluri dro. risi tolia burTis umciresi siCqare frenis dros, Tu θ0 isea SerCeuli, h rom frenis dro 0,5 tmax-is tolia? 200 d R (m) sur. 4.39 amocana 36. ••37 momgeriebeli Camowodebul burTs maSin urtyams, rodesac burTis centri miwidan 1,22 m simaRlezea. burTi miwidan 45° kuTxiT mifrinavs. am dartymis Sedegad, burTis horizontaluri areali (dartymis simaRleze dabrunebisas) 107 m-ia. (a) gadalaxavs Tu ara burTi 7,32 m simaRlis kedels, romelic dartymis adgilidan 97,5 m-is moSorebiTaa? (b) rogori iqneba kedlis zeda nawilsa da burTis centrs Soris manZili, rodesac burTi kedels miuaxlovdeba? SSM WWW 100 θ0 sur. 4.41. amocana 40. •••41 burTi 1,52 m/wm siCqariT horizontalurad migoravs kibis napirisken. safexurebis simaRle da sigane 20,3 sm-ia. romel safexurs daecema pirvelad burTi? SSM •••42 gasrolidan 2 wamis Semdeg Wurvis horizontaluri gadaadgilebaa 40 m, xolo vertikaluri – 53 m. risi tolia Wurvis sawyisi siCqaris (a) horizontaluri da (b) vertikaluri rolon x m-iT daSorebuli kedlisken (sur. 4.40a). 4.40b suraTze naCvenebia burTis kedelze moxvedris νy siCqaris y komponenti, rogorc x manZilis funqcia. ras udris dartymis kuTxe? maqsimaluri simaRlis miRwevis momentSi daartyes h = 1 m simaRlidan da Semdeg imave 5 simaRleze daiWires. burTi mifrinavs ked- y x (a) νy (m/wm) •••38 burTi miwis donidan unda is- 0 10 20 x (m) (b) •••39 fexburTels burTis dartyma SeuZlia 25 m/wm sawyisi siCqariT. risi tolia (a) umciresi da (b) udidesi kuTxeebi, romlebiTac karidan 50 m-is daSorebiT mdgomma fexburTelma burTs unda daartyas, rom miwidan 3,44 m-is simaRlis karis CarCoSi goli gaitanos? SSM •••40 (g) miwidan horizontalurad ra manZilze gadaadgilda Wurvi gasrolis wertilidan? •••43 4.42 suraTze beisbolis burTs lis gaswvriv. dartymidan 1 wamis Semdeg -5 sur. 4.40 amocana 38. komponentebi? burTi miwis donidan garkveuli burTi ascdeba kedlis wvers da 4 wamis Semdeg Camoscdeba, rodesac kedlis gaswvriv D = 50 m manZils gaivlis. (a) ra horizontaluri manZili gaiara burTma dartymidan daWeramde? ra iqneba dartymis Semdeg burTis siCqaris (b) sidide da (g) kuTxe (horizontis mimarT)? (d) ra simaRlisaa kedeli? R h sur. 4.42 amocana 43. h 118 organzomilebiani da samganzomilebiani moZraoba Tavi 4 qveTavi 4-7. Tanabarwriuli moZraoba • 44 risi tolia 10 m/wm siCqariT moZravi sprinteris aCqarebis sidide, rodesac is 25 m radiusis mosaxvevs gairbens? •45 15 m radiusis èSmakis borbali~ horizontaluri RerZis garSemo yovel wuTSi 5 bruns akeTebs. risi tolia (a) moZraobis periodi? risi tolia umaRles wertilSi centriskenuli aCqarebis (b) sidide da (g) mimarTuleba? risi tolia yvelaze dabal wertilSi centriskenuli aCqarebis (d) sidide da (e) mimarTuleba? ILW • 46 centriskenuli aCqarebis moyvaruli Tanabarwriulad moZraobs. periodi T = 2 wm da r = 3 m. t1 momentSi misi aCqarebaa = (6 m/wm2) + (–4 m/wm2) ) m/wm2 risi tolia am momentSi a) · • 47 da b) × ? kaci karuselSi vertikaluri RerZis garSemo Tanabari siCqariT brunavs. siCqare aris 3,66 m/wm, xolo centriskenuli aCqareba 1,83 m/wm2. mdebareobis veqtori kacs mimarTulebas emTxveva. risi tolia wriuli traeqtoriis centris a) x da b) y koordinatebi? •• 51 t1 = 2 wm momentSi saaTis isris moZ- raobis sawinaaRmdego mimarTulebiT wriulad moZravi wertilis aCqarebaa (6 m/wm2) + + (4 m/wm2) . wertili Tanabrad moZraobs. t2 = 5 wm momentSi misi aCqarebaa (4 m/wm2) + + (–6 m/wm2) . risi tolia wertilis moZraobis traeqtoriis radiusi? •• 52 wertili Tanabrad moZraobs wreze, romelic horizontalur xy sibrtyeSi mdebareobs. erT momentSi, (4 m, 4 m) wertilSi yofnisas, misi siCqarea –5 m/wm, xolo aCqareba +12,5 m/wm2. risi tolia wriuli traeqtoriis centris a) x da b) y koordinatebi? •••53 biWuna qvas atrialebs 1,5 m radiusis wreze, miwidan 2 m simaRleze. uecrad Zafi wydeba, qva horizontalurad mifrinavs da 10 m manZilis gavlis Semdeg miwaze ecema. risi tolia wriuli moZraobisas brunvis RerZTan aerTebs. a) risi tolia qvis centriskenuli aCqareba? SSM WWW -is sidide? saiT aris mimarTuli , rode- •••54 sac mimarTulia b) aRmosavleTisken da g) samxreTisken? •48 ventilatori wuTSi 1200 bruns asrulebs. ganvixiloT 0,15 m radiusis ventila- kata karuselze zis, romelic Tana- barwriul moZraobas asrulebs. t1 = 2 wm momentSi katis siCqare xy horizontalur koordinatTa sistemaSi aris 1 = (3 m/wm) toris frTis wveri. (a) ra manZils gaivlis + (4 m/wm) . t2 = 5 wm momentSi katis siCqarea = (–3 m/wm) + (–4 m/wm) . risi tolia (a) 2 is erTi brunis dros? rogoria misi (b) katis centriskenuli aCqareba da (b) katis siCqare da (g) aCqareba? (d) risi tolia saSualo aCqareba t2 – t1 drois monakveTSi? brunvis periodi? qveTavi 4-8. erTganzomilebiani fardobiTi moZraoba •• 49 qalis CanTa da safule karuselze Tanabarwriulad moZraobs, Sesabamisad 2 m da 3 m radiusiT. isini centridan erT xa- •55 dasavleTisken 20km/sT-iT moZravi manqanidan kinooperatori pikapze 30 km/sT-iT zze mdebareobs. erT momentSi CanTis aCqa- ufro swrafad morbenal gepards firze rebaa (2 m/wm2) + (4 m/wm2) . amave momentSi da iRebs. uecrad gepardi Cerdeba, aRmosavle- erTeulovan-veqtorul CanawerSi, rogori Tisken brundeba da misi traeqtoriis max- iqneba safulis aCqareba? loblad mdgari jgufis erT-erTi wevris •• 50 mimarT 45 km/sT siCqariT garbis. gepardis wertili Tanabarwriulad moZraobs horizontalur xy koordinatTa sistemaSi. t1 = 4 wm momentSi is imyofeba (5 m, 6 m) adg- siCqaris cvlilebas 2 wami dasWirda. risi tolia operatoris mimarT gepardis aCqa- ilze, aqvs (3 m/wm) siCqare da aCqareba x rebis (a) sidide da (b) mimarTuleba. risi RerZis dadebiTi mimarTulebiT. rodesac tolia jgufis aRelvebuli wevris mimarT t2 = 10 wm, misi siCqarea (–3 m/wm) , xolo aCqarebis mimarTuleba y RerZis dadebiT gepardis aCqarebis (g) sidide da (d) mimarTuleba? 119 • 56 navi mdinaris dinebis sawinaaRmdego mi- imave dros gadis. A xomaldi 24 kvanZs gadis marTulebiT (x RerZis dadebiTi mimarTule- Crdilo-dasavleTisken, xolo B xomaldi ba) mdinaris dinebis mimarT 14 km/sT siC- _ 28 kvanZs 40° samxreT-dasavleTisken. (1 qariT micuravs. wyali miwis mimarT 9 km/sT kvanZi = saaTSi 1 zRvis mili; ixile danar- siCqariT miedineba. risi tolia miwis mimarT Ti D). risi tolia A xomaldis B-s mimarT navis siCqaris (a) sidide da (b) mimarTuleba? siCqaris (a) sidide da (b) mimarTuleba? (g) navze mdgomi bavSvi win da ukan dadis navis ra drois Semdeg daSordebian xomaldebi mimarT 6 km/sT siCqariT. risi tolia miwis erTmaneTs 160 sazRvao miliT? (d) rogori mimarT bavSvis siCqaris (g) sidide da (d) mi- iqneba am dros B-s mdebareobis mimarTule- marTuleba? ba A-s mimarT? SSM ILW •• 57 ••62 saeWvo garegnobis kaci moZravi bilikis erTi bolodan meorisken garbis 2,5 TviTmfrinavma 15 wuTi ifrina 42 km/ sT siCqaris qarSi, romelic aRmosavleTidan wm-is ganmavlobaSi. usafrTxoebis agentis 20°-iT samxreTiT uberavs. amJamad TviTm- gamoCenis Semdeg kaci isev sawyisi werti- frinavi qalaqis Tavzea, sawyisi wertili- lisken mirbis 10 wm-is ganmavlobaSi. risi dan CrdiloeTiT 55 km manZilze. risi to- tolia kacis siCqaris Sefardeba bilikis lia TviTmfrinavis siCqare haeris mimarT? siCqaresTan? ••63 qveTavi 4-9. organzomilebiani far- ragbis moTamaSe burTTan erTad mowi- naaRmdegisken mirbis x RerZis dadebiTi mimarTulebiT. moTamaSes TanagundelisTvis burTis gadawodeba manamde SeuZlia, sanam mindvris (miwis mimarT) siCqariT samxreTisken. wvima qars samxreTisken miaqvs. TiToeuli wvi- dobiTi moZraoba •58 matarebeli wvimaSi moZraobs 30 m/wm mimarT burTis siCqares mis wveTis traeqtoria vertikalidan 70° kuTxiTaa daxrili miwaze mdgomi damkvirveblis mimarT. matarebelze myofi damkvirveblisTvis wvimis wveTebi vertikalurad ecema. gansazRvre miwis mimarT dadebiTi x komponenti ara aqvs. davuSvaT, wvimis wveTebis siCqare. SSM moTamaSe garbis 4 m/wm siCqariT mindvris ••64 4.43 mimarT da burTs Tanagundels awvdis sa- uZravi suraTze mocemul zedxedze P da B jipi moZraobs wrfivad brtyel zeda- -s pirze da Cauvlian uZrav A mesazRvres. sididea 6 m/wm, rogori iqneba pasis umci- mesazRvris mimarT B-s siCqarea 20 m/wm, kuTari Tavis mimarT siCqariT. Tu resi kuTxe wesebis darRvevis gareSe? •59 Tovlis fifqebi miwaze ecema 8 m/wm Tanabari siCqariT. vertikalidan ra kuTxiT Cans fifqebis dacema swor gzaze 50 km/ sT-iT moZravi manqanis mZRolisTvis? SSM ••60 msubuqi qariT TviTmfrinavi 500 km/sT siC- mifrinavs. daniSnulebis adgili zustad CrdiloeTiTaa 800 km-is manZilze, magram piloti xvdeba, rom TviTmfrinavi CrdiloeTidan 20° kuTxiT aRmosavleTisken unda mimarTos, raTa pirdapir daniSnulebis adgilisken gafrindes. 2 saaTis Semdeg TviTmfrinavi daniSnulebis adgils aRwevs. risi tolia qaris siCqaris (a) sidide da (b) mimarTuleba? ••61 A da B xomaldebi portidan erTsa da xolo kuTxe θ2 = 30°. mesazRvris mimarT P- s Tanabari aCqarebaa 0,4 m/wm2, xolo kuTxe θ1 = 60°. garkveul momentSi P-s siCqarea 40 m/wm. risi tolia am momentSi B-s mimarT P-s siCqaris (a) sidide da (b) mimarTuleba? risi tolia B-s mimarT P-s aCqarebis (g) sidide da (d) mimarTuleba? •••65 A xomaldi imyofeba B xomaldidan CrdiloeTiT 4km manZilze da aRmosavleTiT 2,5 km manZilze. A-s siCqarea 22 km/sT samxreTiT, xolo B-s siCqarea 40 km/sT 37° Crdilo-aRmosavleTiT. (a) risi tolia A-s siCqare B-s mimarT erTeulovan-veqtorul CanawerSi, Tu mimarTulia aRmosavle- Tisken? (b) dawere B-s mimarT A-s mdebareobis gamosaxuleba ( -sa da -s saSualebiT), 120 organzomilebiani da samganzomilebiani moZraoba Tavi 4 rogorc t drois funqcia, sadac roca t = 0, mavlobaSi. am momentSi (a) ra manZilze xar xomaldebi imyofebian zemoT aRweril mde- sawyisi wertilidan? (b) moZraobis sawyisi bareobebze. (g) ra droSia xomaldebs So- mimarTulebisadmi ra mimarTulebiT xar? ris daSoreba umciresi? (d) risi tolia 68 sikvdilis farda. liTonis didi aster- umciresi daSoreba? oidi dedamiwaze ecema da miwaSi uzarmaCrd P zar kraters amoTxris. dacemis wertiliaRm qvebi A θ1 dan caSi didi siCqariT da did manZilze amoifrqeva. mocemulia 5 cxrilSi amofrqveuli qvis siCqare da kuTxe hori- θ2 zontis mimarT (aris sxva qvebic, sxva siCqariTa da kuTxiT). davuSvaT, Sen xar x = 20 km B wertilSi, rodesac asteroidi dedamiwas sur. 4.43 amocana 64. ••• 66 200 m siganis mdinare 1,1 m/wm siCqa- ejaxeba t = 0 momentSi da x = 0 wertilSi riT aRmosavleTisken miedineba junglebis (sur. 4.44). (a) rodesac t = 20 wm, rogori gavliT. tyidan patara mindorze gamosul iqneba Seni mimarTulebiT gamosrolili mkvlevars unda mdinaris samxreT napiri- qvebis (A-dan E-mde) x da y koordinatebi? dan kateriT gadavides meore napirze. kat- (b) aRniSne es wertilebi da maTze mrudi eri mdinaris mimarT 4 m/wm Tanabari siC- gaavle, romelSic Sesuli iqneba sxva siC- qariT moZraobs. CrdiloeT napirze aris meore mindori, romelic samxreT napiris mindvris mopirdapire wertilidan 82 kmiT aris daSorebuli dinebis sawiaaRmdego mimarTulebiT. (a) ra mimarTulebiT unda qariTa da kuTxiT amotyorcnili qvebic. grafikma unda mogces warmodgena, ras dainaxav, rodesac qvebi mogiaxlovdeba, da ras xedavdnen dinozavrebi, rodesac uxsovar dros asteroidi dedamiwas Seejaxa. waiyvanos kateri mkvlevarma, raTa pirda- qvebi siCqare (m/wm) kuTxe (gradusi) pir CrdiloeT napiris mindors miadges? A 520 14 (b) ra dro dasWirdeba katers mdinaris B C 630 16 750 18 D 870 20 E 1000 22 gadasakveTad da mindvramde misasvlelad? damatebiTi amocanebi 67 Sen politologebma mogitaces (vinaidan ganucxade, rom politologia namdvili y mecniereba ar aris). miuxedavad imisa, rom Tvalebi akruli gaqvs, SegiZlia gansazRvro maTi manqanis siCqare (Zravis xmauriT), mgzavrobis dro (wamebis daTvliT) da moZraobis mimarTuleba (moxvevebis damaxsovrebiT). Seni varaudiT, wagiyvanes Semdegi kursiT: 50 km/sT-iT 2 wuTis ganmavlobaSi, Seuxvies 90° marjvniv, 20 km/ sT-iT iares 4 wuTis ganmavlobaSi, Seuxvies 90° marjvniv, 20 km/sT-iT iares 60 wamis ganmavlobaSi, Seuxvies 90° marcxniv, 50 km/sT-iT iares 60 wamis ganmavlobaSi, Seuxvies 90° marjvniv, 20 km/sT-iT iares 2 x 0 10 20 sur. 4.44 amocana 68. 69 burTi unda tyorcno 12 m/wm siCqariT da moartya srolis adgilidan h = 5 m simaRleze myof samiznes (sur. 4.45 a). mizanTan miRwevisas burTs horizontaluri siCqare unda hqondes. (a) horizontidan ra θ kuTxiT unda tyorcno burTi? (b) risi wuTis ganmavlobaSi, Seuxvies 90° marcxniv tolia horizontaluri manZili tyorcnis da, bolos, 50 km/sT-iT iares 30 wamis gan- adgilsa da samiznes Soris? (g) risi tolia 121 burTis siCqare samiznis miRwevis dros? wertilisken (t = 5 wT), Semdeg C wertilisken (t = 10 wT), bolos ki D wertilisken mizani (t = 15 wT). dafiqrdi ciyvis saSualo siCqareebze A wertilidan TiToeul wertilamde. am sami saSualo siCqaridan, rogori h iqneba umciresis (a) sidide da (b) kuTxe? θ rogori iqneba udidesis (g) sidide da (d) kuTxe? sur. 4.45 amocana 69. 70 4.46 suraTze talaxis belti wriulad moZraobs 20 sm radiusis borbalze, romelic saaTis isris sawinaaRmdego mimarTulebiT brunavs. borblis brunvis periodia 5 m/wm (borbali 72 4.48 suraTze naCvenebia skunsis moZraobis traeqtoria, i sawyisi wertilidan f saboloo wertilamde. kuTxeebia: θ1 = 30°, θ2 = 50° da θ3 = 80°. manZilebia: d1 = 5 m, d2 = 8 m da d3 = 12 m. risi tolia skunsis mTliani gadaadgilebis (a) sidide da (b) kuTxe? y warmovidginoT saaTis ciberblatis saxiT). rodesac borbalze beltis mdebareoba Seesabameba saaTis θ2 d3 ciferblatze patara isris mdebareobas xuT d1 saaTze, belti mis zedapirs moswydeba h = belti? x i 1,2 m simaRlidan, kedlidan d = 2,5 m daSorebiT. ra simaRleze moxvdeba kedels d2 θ3 f sur. 4.48 amocana 72. 73 wertili Tanabrad moZraobs wrewirze xy koordinatTa sistemis aTvlis saTavis borbali garSemo, saaTis isris moZraobis mimarTule- h talaxi biT, 7 wm periodiT. erT momentSi misi mdebareobis veqtori (aTvlis saTavidan) aris d = (2 m) – (3 m) . risi tolia am momentSi sur. 4.46 amocana 70. 71 4.47 suraTze naCvenebia miwaze moZravi ciyvis traeqtoria: A wertilidan (t = 0) B wertilis siCqare erTeulovan-veqtorul CanawerSi? 74 4.49 suraTze burTi miwidan vertikalurad aisroles v0 = 7 m/wm sawyisi siC- y (m) 50 qariT. amavdroulad, samSeneblo liftis D kabina miwidan zemoT iwyebs moZraobas vc = 3 m/wm sawyisi siCqariT. ra maqsimalur si25 maRles miaRwevs burTi (a) miwis mimarT da (b) kabinis mimarT? ra tempiT icvleba burx (m) 0 25 A 50 C -25 -50 c 0 B sur. 4.47 amocana 71. burTi sur. 4.49 amocana 74. 122 organzomilebiani da samganzomilebiani moZraoba Tavi 4 Tis siCqare (g) miwis mimarT da (d) kabinis romliTac Wurvi mizans moxvdeba? mimarT? 78 mdebareobis veqtori 75 4.50a suraTze ciga Tanabrad moZraobs x-is uaryofiTi mimarTulebiT da vs siCqa- wertilis mdebareobas gansazRvravs, ro riT. am dros cigidan Tovlis gunda is- metrebSi, xolo t – wamebSi. e da f koefici- riT. rodesac gunda miwaze ecema, misi gadaadgileba Δxmiwa izomeba miwis mimarT entebi mudmivebia. 4.52 suraTze naCvenebia wertilis moZra bis θ kuTxe, ro- 4.50 b suraTze Δxmiwa mocemulia, rogorc vs- gorc is funqcia. davuSvaT, gunda daaxloebiT funqcia (θ izo srolis simaRleze daeca. risi tolia (a) meba x RerZis v0x-is da (b) v0y-is sidideebi? cigis mimarT gundis Δxciga gadaadgilebis gazomvac Ses- dadebiTi aZlebelia. davuSvaT, cigis siCqare gun- gamoTvale dis srolis dros ar icvleba. risi tolia koeficienti Δxciga-s, rodesac vs aris (g) 5 m/wm da (d) 15 b) f koeficienti, m/wm? erTeulebis y (a) 0o 10 20 mimar Tulebisadmi). a a) -40o da t (wm) sur. 4.52 amocana 78. ga 79 qals mdore wyalSi niCbis mosma SeuZlia Δ x miwa (m) x drois TvaliswinebiT. gunda ciga 40o obis mimarTule (srolis adgilidan davardnis adgilamde). 40 veqtori izomeba θ = v0x + v0y siCqa0 roles cigis mimarT gorc t drois funqcias. + (et + ft2) = 5t 6,4 km/sT siCqariT, magram amJamad 6,4 km siganis mdinareSi imyofeba, romelic 3,2 km/sT 0 10 20 siCqariT miedineba. davuSvaT, mimarTulia mdinaris erTi napiridan meoresken, xolo – mdinaris dinebis mimarTulebiT. Tu qali -40 v s (m/wm) (b) qalma navi da (b) ra dro dasWirdeba? (g) 76 xy sibrtyeze moZravi wertilis aCqarebaa + 4t , sadac yisi wertilis sapirispiro mimarTulebiT, maSin (a) -sadmi ra kuTxiT unda mimarTos sur. 4.50 amocana 75. = 3t niCbebs usvams da moZraobs wrfivad saw- ra dro dasWirdeba qals, Tu is 3,2 km-iT mdinares dauyveba da Semdeg dabrundeba izomeba m/wm -Si, xolo sawyis wertilSi? (d) ra dro dasWirdeba, t – wamebSi. rodesac t = 0, wertilis mdebare- Tu 3,2 km-s gacuravs mdinarebis dinebis obis veqtoria qare 2 = (20 m) + (40 m) , xolo siC- = (5m/wm) + (2 m/wm) . rodesac t = 4 wm, sawinaaRmdego mimarTulebiT dabrundeba sawyis wertilSi? e) da Semdeg -sadmi ra risi tolia a) wertilis mdebareobis veq- kuTxiT unda mimarTos navi qalma meore tori erTeulovan-veqtorul CanawerSi da napirze umokles droSi gadasasvlelad? b) kuTxe x RerZis dadebiT mimarTulebasa da (v) risi tolia es dro? wertilis moZraobis mimarTulebas Soris? 80 vulkanis amofrqvevis dros krateri- 77 Wurvi R = 20 m moSorebiT miwaze ganlagebul samiznes esroles v0 = 30 m/wm sawyisi dan myari qanis natexebi, e.w. vulkanuri maRali traeqtoria siCqariT, rogorc es 4.51 xuraTzea naCvenebi. risi tolia srolis (a) umciresi udidesi dabali traeqtoria da (b) kuTxe, 0 0 R sur. 4.51 amocana 77. bombebi amoifrqveva. 4.53 suraTze iaponiaSi fuZaiamas mTis ganivkveTia naCvenebi. (a) ra sawyisi siCqariT unda amovardes bombi A gamosasvlelidan horizontisadmi θ = 35o kuTxiT, raTa vulkanis B wertilSi daeces, Tu vertikaluri manZili h = 3,3 km, xolo A-dan B-mde horizontaluri – d = 9,4 km. ha- 123 A miaRwevs r maqsimalur sidides? (d) rogori θ0 iqneba es sidide da srolis adgilidan ra manZilze iqneba qva (e) horizontalurad da (v) vertikalurad? rodesac θ0 = 80°, (z) h rodis miaRwevs r maqsimalur sidides? (T) d B rogori iqneba es sidide da srolis adgilidan ra manZilze iqneba qva (i) horizontalurad da (k) vertikalurad? sur. 4.53 amocana 80. eris zemoqmedeba bombis moZraobaze ug- 84 CrdiloeT naxevarsferos zomier ganede- ulebelyavi. (b) risi toli iqneba bombis bze grZeli gadafrenebis dros aRmosav- frenis dro? (g) gazrdis Tu Seamcirebs leTiT mimarTuli haeris nakadi TviTm- haeris zemoqmedeba (a) kiTxvis pasuxs? frinavis siCqareze gavlenas axdens. Tu 81 A oazisi B oazisidan dasavleTisken 90 km manZilzea. aqlemi A oazisidan gadis da piloti haeris mimarT garkveul siCqares 50 saaTis ganmavlobaSi 75 km gaivlis 37° kuTxiT Crdilo-aRmosavleTiT. Semdegi 35 saaTis ganmavlobaSi aqlemi 65 km-s gaivlis samxreTiT, Semdeg ki 5 saaTs isvenebs. risi tolia aqlemis gadaadilebis (a) sidide da (b) mimarTuleba A-s mimarT? A oazisidan gamosvlidan dasvenebis bolomde risi tolia aqlemis saSualo siCqaris (g) sidide da (d) mimarTuleba? (e) misi saSualo siCqare? aqlemma ukanasknelad wyali A oazisSi dalia da maqsimum 120 saaTi aqvs, raTa B oazisSi dalios. Tu aqlemi B-s droulad miaRwevs, risi toli iqneba dasvenebis Semdeg misi saSualo siCqaris (v) sidide da (z) mimarTuleba? 82 rodesac didi varskvlavi zeaxal varskvlavad anu `supernovad~ gardaiqmneba, ganaviTarebs, dedamiwis mimarT siCqare ufro metia, roca frena haeris dinebis mimarTulebiTaa, da naklebia, roca haeris dinebis sawinaaRmdego mimarTulebiTaa. davuSvaT, TviTmfrinavi 4000 km-iT daSorebul qalaqebs Soris win da ukan mifrinavs. marSrutis pirveli naxevari haeris dinebis mimarTulebiTaa, xolo meore naxevari – sawinaaRmdego mimarTulebiT. kompiuteri pilots urCevs, haeris mimarT 1000 km/sT siCqariT ifrinos, romlisTvisac wina da ukana frenebis xangrZlivobebs Soris gansxvaveba 70 wuTi iqneba. haeris nakadis ra siCqares iTvaliswinebs kompiuteri? 85 4.54 suraTze radiolokaciurma sadgurma aRmoaCina aRmosavleTidan momavali TviTmfrinavi. Tavdapirveli dakvirvebiT, TviTmfrinavi sadguridan d1 = 360 m man- misi birTvi imdenad da ise ikumSeba, rom is Zilze da θ1 = 40° kuTxiT imyofeba. Semdeg 20 km radiusis neitronis varskvlavi xdeba. sadgursa da TviTmfrinavs Soris kuTxe Tu neitronis varskvlavi wamSi erT bruns ∆θ = 123°-iT Seicvala, xolo manZili d2 = akeTebs, maSin (a) risi tolia varskvlavis 490 m gaxda. rogori iqneba am drois ganma- ekvatorze mdebare wertilis siCqare da (b) vlobaSi TviTmfrinavis gadaadgilebis (a) risi tolia wertilis centriskenuli aCqa- sidide da (b) mimarTuleba? rebis sidide? (g) Tu neitronis varskvlavi TviTmfrinavi ufro swrafad brunavs, (a)-s da (b)-s pasuxebi izrdeba, mcirdeba Tu igive rCeba? 83 t = 0 momentSi qva miwidan isroles 16 m/wm sawyisi siCqariT da θ0 kuTxiT. warmoidgine gasrolis adgilidan qvisken d2 das Δθ d1 θ1 mimarTuli mdebareobis veqtori. aage mdebareobis veqtoris r sididis grafiki, Tu (a) θ0 = 40° da (b) θ0 = 80°. rodesac θ0 = 40°, (g) rodis radari sur. 4.54 amocana 85. aRm 124 Tavi 4 organzomilebiani da samganzomilebiani moZraoba 86 ori trasa erTmaneTs kveTs, rogorc es wrewirze 4.55 suraTzea naCvenebi. mocemul momentSi P policiis manqana imyofeba gadakveTis 3×1014 adgilidan dP = 800 m manZilze da moZraobs tolia brunvis periodi? vP = 80 km/sT siCqariT. M manqana gadakveTis adgilidan dM  = 600 m manZilze imyofeba da 90 200 m siganis mdinare aRmosavleTisken moZraobs vM = 60 km/sT siCqariT. (a) risi mimarT 8 m/wm siCqaris mqone navi mdinaris tolia M manqanis siCqare P manqanis mimarT erTeulovan-veqtorul CanawerSi? (b) 4.55 moZravi m/wm2. eleqtronis aCqarebaa (a) risi tolia eleqtronis siCqare, Tu wris radiusia 15 sm? (b) risi miedineba 2 m/wm Tanabari siCqariT. wylis samxreT napiridan miemarTeba 30° kuTxiT Crdilo-dasavleTiT. risi tolia miwis suraTze naCvenebi momentisTvis, risi to- mimarT navis siCqaris (a) sidide da (b) lia kuTxe (a)-Si napovn siCqaresa da ori mimarTuleba? (g) ra dro dasWirdeba navs mZRolis Tvalsawiers Soris? (g) Tu orive mdinaris gadasakveTad? manqana siCqares SeinarCunebs, gadakveTasTan miaxloebisas Seicvleba Tu ara (a) da (b) kiTxvis pasuxebi? 91 amocanis 4.7 b nimuSSi qvemexi Wurvs 686 m horizontalur manZilze isvris. zarbazani zRvis doneze dgas da horizontidan 45°- y iTaa daxrili. ramdeniT gaizrdeba Wurvis frenis horizontaluri manZili, Tu qvemexs 30 m-iT maRla ganvalagebT? M 92 protonis sawyisi mdebareobis veqtoria dM bareobis vM P = –2 + 6 + 2 . veqtori? (b) romeli sibrtyis parale- x luria es veqtori? 93 (a) Tu eleqtroni horizontalurad dP isroles 3×106 m/wm siCqariT, ra manZilze sur. 4.55 amocana 86. daecema is horizontalurad 1 m-is gavlis 87 SaSxana horizontalurad damiznebulia 30 m-iT daSorebul samizneze. tyvia xvdeba veqtori – (a) rogori iqneba protonis gadaadgilebis vP samiznes = 5 – 6 + 2 , xolo momdevno mde- damiznebis adgilidan 1,9 sm-iT qveviT. risi tolia (a) tyviis Semdeg? 94 Wurvis siCqare miwidan maqsimalur simaRleze 10 m/wm-ia. (a) risi tolia Wurvis siCqare maqsimaluri simaRlis miRwevamde frenis dro da (b) misi siCqare SaSxanidan 1 wm-iT adre? (b) risi tolia Wurvis siC- gamosvlis dros? qare 88 swrafi franguli matarebeli 216 km/ sT siCqariT moZraobs. (a) Tu matarebeli am dros wriulad moZraobs, xolo mgzavrebisTvis dasaSvebia 0,05 g-mde aCqareba, risi tolia traeqtoriis mrudis umciresi radiusi? (b) ra siCqariT unda Semouaros matarebelma 1 km radiusis mruds, raTa aCqarebis limiti ar daarRvios? 89 magnituri vels SeuZlia damuxtuli nawilakis davuSvaT, wriuli moZraoba konkretul gamoiwvios. magnitur velSi maqsimaluri simaRlis miRwevidan 1 wm-is Semdeg? Tu maqsimaluri simaRlis wertilia x = 0 da y = 0, xolo siCqaris mimarTuleba x RerZis dadebiT mimarTulebas emTxveva, maSin risi tolia maqsimaluri simaRlis miRwevamde 1 wm-iT adre Wurvis mdebareobis (g) x da d) y koordinati? risi tolia Wurvis (e) x da v) y koordinati maqsimaluri simaRlis miRwevidan 1 wm-is Semdeg? 95 protonis sawyisi siCqarea =4 –2 +3 , xolo 4 wamis Semdeg =–2 –2 +5 (m/wm-ebSi). am oTxi wamis ganmavlobaSi risi tolia 125 (a) protonis saS aCqareba erTeulovan-veqsi- wamis srialis Semdeg kurdRlis a) siCqare da x RerZis dadebiT da b) mdebareoba erTeulovan-veqtorul torul CanawerSi? (b) risi tolia dide da (g) saS-sa Tulebad aiRe aRmosavleTi. risi tolia 3 saS-s mimarTulebas Soris kuTxe? CanawerSi? 96 P wertili Tanabari siCqariT moZrao- 99 wertili moZraobas iwyebs aTvlis saTavi- bs r = 3 m radiusis wreze (sur. 4.56) da erT bruns 20 wamSi asrulebs. wertili O-s Cauvlis t = 0 momentSi. qvemoT mocemuli veqtorebi gamosaxe sidide-kuTxur Canaw- dan t = 0 momentSi 8 m/wm siCqariT da xy sibrtyeze Tanabari (4 + 2 ) m/wm2 aCqarebiT moZraobs. rodesac wertilis x koordinatia 29 m, risi tolia (a) y koordinati da erSi (kuTxe aRebulia x RerZis dadebiTi (b) siCqare? mimarTulebisadmi). ipove O-s mimarT wer- 100 3,5 saaTis ganmavlobaSi sahaero buS- tilis mdebareobis veqtori, Semdeg t mo- ti afrenis wertils Sordeba 21,5 km-iT mentebSi: (a) 5 wm, b) 7,5 wm da (g) 10 wm. (d) CrdiloeTiT, 9,7 km-iT aRmosavleTiT da ipove wertilis gadaadgileba mexuTe wamis 2,88 km-iT vertikalurad zeviT. moZebne (a) dasasrulidan meaTe wamis dasasrulamde burTis saSualo siCqare da (b) kuTxe sa- 5-wamian SualedSi. am SualedisTvis ipove Sualo siCqaresa da horizonts Soris. (e) P wertilis saSualo siCqare da siCqare 101 TviTmfrinavi miwis mimarT aRmosavle- Sualedis (v) dasawyisSi da (z) dasasrulSi. amis Semdeg ipove wertilis aCqareba intervalis (T) dasawyisSi da (i) dasasrulSi. y Tisken mifrinavs 20 km/sT siCqariT samxreTisken mqrolav qarSi. Tu qaris gareSe TviTmfrinavis siCqarea 70 km/sT, risi tolia TviTmfrinavis siCqare miwis mimarT? 102 wriul bilikze morbenali sprinteris r siCqarea 9,2 m/wm, xolo centriskenuli aC- P qarebaa 3,8 m/wm2. risi tolia (a) bilikis radiusi da (b) wriuli moZraobis periodi? x 0 103 burTi horizontalurad isroles 20 m simaRlidan. misi miwaze dacemis siCqare 3-jer metia sawyis siCqareze. risi tolia sur. 4.56 amocana 96. sawyisi siCqare? 97 yinulmWreli gayinuli tbis zedapirze 104 1×109 sm/wm sawyisi horizontaluri qaris mier miniWebuli Tanabari aCqarebiT siCqaris moZraobs. garkveul momentSi yinulmWre- horizontalur damuxtul firfitas Soris mqone eleqtroni liTonis or lis siCqarea (6,3 – 8,42 ) m/wm. sami wamis moZraobs. eleqtroni gaivlis 2 sm horizon- Semdeg qaris Cadgomis gamo yinulmWreli talur manZils firfitebis mimarT 1×1017 Cerdeba. risi tolia yinulmWrelis saSua- sm/wm2 aCqarebiT. risi tolia (a) dro, ro- lo aCqareba am samwamian SualedSi? melic eleqtrons dasWirdeba 2 sm-is gas- 98 aRmosavleTisken 6 m/wm siCqariT morbe- avlelad? (b) risi tolia am droSi ganv- nali kurdReli umniSvnelo xaxunis mqone lili vertikaluri manZili? risi tolia yinulze srialis am firfitebidan gamosvlamde eleqtronis aCqarebas siCqaris (g) horizontaluri da (d) verti- aRmoCndeba. yinulze dros qari kurdRels 1,4 m/wm 2 aniWebs CrdiloeTis mimarTulebiT. airCie kaluri komponentebi? koordinatTa sistema, romlis aTvlis saT- 105 Wurvi 30 m/wm sawyisi siCqariT hori- ave yinulze kurdRlis sawyis mdebareobas zontisadmi 60° kuTxiT isroles. risi to- daemTxveva, xolo x RerZis dadebiT mimar- lia srolidan 2 wamis Semdeg siCqaris (a) 126 organzomilebiani da samganzomilebiani moZraoba Tavi 4 sidide da (b) kuTxe? (g) horizontidan trasis Tavze imyofeba. miwaze myofi dam- maRlaa Tu dablaa es kuTxe? risi to- kvirvebeli pilots gadascems, rom uber- lia srolidan 5 wamis Semdeg siCqaris (d) avs 70 mili/sT siCqaris qari, magram qaris sidide da (e) mimarTuleba? (v) horizonti- mimarTulebas ar eubneba. piloti amCnevs, dan maRlaa Tu dablaa kuTxe? rom, qaris miuxedavad, TviTmfrinavs 1 saa- 106 softbolis momwodebeli burTs mi- Tis ganmavlobaSi trasis Tavze 135 milis widan 3.0 futis simaRlidan isvris. bur- gavla SeuZlia. sxvagvarad rom vTqvaT, mi- Tis mdebareobis stroboskopuli grafiki wis mimarT siCqare qaris drosac igivea. (a) mocemulia 4.57 suraTze. monacemebi asaxu- ra mimarTulebiT uberavs qari? (b) saiTken lia 0,25 wamis intervalSi, xolo burTi aris TviTmfrinavi mimarTuli? gaisroles t = 0 momentSi. (a) risi tolia 110 burTis sawyisi siCqare? (b) risi tolia 2,24 m-is imaRlezea, xolo moednis zomebia burTis siCqare miwis donidan maqsimaluri 9 m×9 m badis TiToeul mxares. Cawodebisas simaRlis miRwevisas? (g) risi tolia maqsi- moTamaSe burTs artyams miwidan 3 m simaR- maluri simaRle? leze da badidan horizontalurad 8 m-is 10 daSorebiT. qalTa frenburTSi Tu burTis bade sawyisi miwidan siCqare y (futi) horizontaluria, maSin (a) ra minimaluri siCqare unda hqondes burTs, rom bades 5 gadaafrindes? (b) ra maqsimaluri siCqare unda hqondes burTs, rom moednis meore naxevris ukana xazs ar gascdes? 0 10 20 30 x (futi) 40 sur. 4.57 amocana 106. vrebis gadasayvanad eskalatori muSaobs. eskalatoris gamoyenebis nacvlad derefans 150 wm-Si fexiT gadis. karli, romelic eskalatorze dgas, imave manZils 70 wamSi gadis. mo eskalatorze dgas da Tan fexiTac midis. ra dro dasWirdeba mos derefnis gasavlelad? dauSvi, rom lari da mo erTi da imave siCqariT dadian. 108 fexburTelma burTs ise daartya, rom burTma haerSi 4,5 wami ifrina B y dinatTa sistemaSi 107 aeroportis grZel derefanSi mgzalari 111 4.58 suraTze naCvenebia xy koor- da 46 m-is moSorebiT daeca. Tu burTi fexburTelis fexs miwidan 150 sm-is simaRleze mowyda, moZravi wertilis wrfivi traeqto- A ria. wertili uZravi mdgomareobidan x sur. 4.58 amocana 111. Cqardeba Δt1 drois SualedSi. aCqareba Tanabaria. A wertilis koordinatebia (4 m, 6 m), xolo B wertilisa – (12 m, 18 m). (a) risi tolia aCqarebis komponentebis Sefardeba ay /ax? (b) risi tolia wertilis koordinatebi, Tu moZraoba grZeldeba kidev erT Δt1 drois SualedSi? 112 sateliti dedamiwis zedapiridan 640 km simaRleze wriul orbitaze moZraobs 98.0- risi toli unda iyos burTis sawyisi siC- wuTiani periodiT. risi tolia satelitis qaris a) sidide da b) kuTxe (horizontis (a) siCqare da (b) centriskenuli aCqareba? mimarT)? 113 xy sibrtyeze moZravi wertilis 109 niu-hemfSiris Statis policia TviT- bareoba mocemulia formuliT: mfrinavs trasaze moZraobis siCqaris kon- [(π/4 rad/wm)t] , sadac trolisTvis iyenebs. davuSvaT, erT-erTi wamebSi. a) gamoTvale wertilis mdebareo- TviTmfrinavis siCqare uqaro amindSi 135 bis x da y komponentebi, rodesac t = 0, mili/sT-ia. Crdilo- 1, 2, 3 da 4 wm. aage xy sibrtyeze wertilis yovelTvis traeqtoria 0 ≤ t ≤ 4 wm SualedSi. (b) ga- eTisken ise es TviTmfrinavi mifrinavs, rom mde- = 2t + 2sin metrebSia, xolo t _ 127 moTvale wertilis siCqaris komponentebi, 10 rodesac t =1, 2 da 3 wm. (a) nawilSi wertilis traeqtoriis grafikze gaavle siCqaris veqtorebi da aCvene, rom siCqare wertilis 5 traeqtoriis mxebia da wertilis moZraobis mimarTuleba aqvs. (g) gamoTvale wertilis aCqarebis komponentebi, rodesac t = 1, 2 da 3 wm. 0 5 114 golfis moTamaSe burTs 43 m/wm sawyisi 10 15 20 sur. 4.59 amocana 119. 25 30 siCqariT da horizontisadmi 30o kuTxiT gaisroles. 4.59 suraTze naCvenebia tyviis urtyams borcvidan. burTi horizontalu- mdebareobebis stroboskopuli grafiki. rad 180 m manZilze mdebare sanaosno arxSi mdebareobebis gansazRvra xdeba 0,5-wamian vardeba. davuSvaT, arxs swori zedapiri intervalSi, xolo tyviis gasrolis droa aqvs. (a) arxidan ra simaRlidan daartyes t = 0. (a) risi tolia tyviis sawyisi siCqare burTi? (b) risi tolia burTis siswrafe erTeulovan-veqtorul CanawerSi? (b) risi arxSi Cavardnis dros? tolia planetaze Tavisufali vardnis aCqa- 115 kosmonavti 5 m radiusis centrifugaSi reba? (g) srolidan ra drois Semdeg daece- brunavs. a) risi tolia kosmonavtis siC- ma tyvia miwaze? (d) Tu analogiuri gasro- qare, Tu centriskenuli aCqarebaa 7 g? b) la dedamiwis zedapiridan ganxorcielda, wuTSi ramdeni brunis gakeTebaa saWiro am ra drois Semdeg Camovardeba tyvia? aCqarebis misaRwevad? g) risi tolia brun- 120 sawyis momentSi (t = 0) deltaplani mi- vis periodi? widan 7,5 m simaRleze imyofeba, moZraobs 116 (a) risi tolia dedamiwis ekvatorze 8 m/wm siCqariT, horizontidan 30° kuTxiT. sxeulis centriskenuli aCqareba, romelic deltaplanis aCqarebaa 1 m/wm2 da mimar- dedamiwis brunviTaa gamowveuli? (b) rogori unda iyos dedamiwis brunvis periodi ekvatorze myofi sxeulisTvis, raTa mas mieniWos 9,8 m/wm2 centriskenuli aCqareba? 117 uWero lifti zeviT moZraobs 10 m/wm Tanabari siCqariT. liftSi myofi biWi iatakidan 2 m simaRlidan burTs zeviT isvris, rodesac lifti miwidan 28 m simaRleze Tulia vertikalurad zeviT. (a) dawere y-is zrdis formula, rogorc t drois funqcia. (b) formulis gamoyenebiT gansazRvre t, rodesac y = 0 (miwis zedapiri). (g) axseni, ratom aqvs (b) kiTxvas ori pasuxi. romelia maTgan is dro, rac deltaplans miwaze dasajdomad dasWirdeba? (d) ra horizontalur manZils gaivlis deltaplani t = 0sa da miwaze dajdomis moments Soris? e) imyofeba. liftis mimarT burTis siCqarea imave sawyisi mdebareobisa da siCqarisT- 20 m/wm. (a) miwidan ra maqsimalur simaRles vis, ra Tanabari moZraoba gamoiwvevs del- miaRwevs burTi? (b) ra dro dasWirdeba taplanis miwaze dajdomas nulovani siCqa- burTs liftis iatakze dasavardnad? riT? pasuxi gaeci erTeulovan veqtorul 118 adamiani 15 m sigrZis gaCerebul es- CanawerSi. kalatorze 90 wamSi adis, moZrav eskala- xolo _ vertikalurad. torze ki uZravad mdgomi imave manZils 60 121 deteqtiur moTxrobaSi sxeuli Seno- wamSi faravs. ra dro dasWirdeba adamians bidan 4,6 m manZilze da Ria fanjridan 24 moZrav eskalatorze fexiT asasvlelad? m-is qvemoT ipoves. (a) davuSvaT, msxverpli damokidebulia Tu ara pasuxi eskalato- fanjridan horizontalurad gadmovarda. ris sigrZeze? risi toli iyo sxeulis sawyisi siCqare? 119 moqmedeba xdeba mzis sistemis Soreul (b) SemTxveviTia Tu ara sikvdili? axseni planetaze. tyvia miwidan 2 m simaRlidan Seni pasuxi. mimarTulia horizontalurad, 128 Tavi 4 organzomilebiani da samganzomilebiani moZraoba 122 beisbolis moTamaSes surs burTi 124 blad? (g) imave grafikze, aRniSne matare- futiT daSorebul pirvel bazas miawvdi- blisa da manqanis mdebareobebis cvlileba nos. misi saukeTeso srolis siCqarea 85 drois mixedviT. mili/sT. (a) Tu moTamaSe burTs miwidan 124 3 futis simaRlidan horizontalurad is- sipediT vris, pirveli bazidan ra manZilze daecema moZraob 30° Crdilo-aRmosavleTiT. burTi miwaze? (b) igive sawyisi simaRlidan 1. gazarde siCqare 0,4 m/wm2 aCqarebiT 6 ra kuTxiT unda isrolos moTamaSem bur- wm-is ganmavlobaSi. Ti, raTa sxva moTamaSem is miwidan 3 fut sawyisi tbaSi wertilidan micurav. wylis velo- Tavdapirvelad, 2. am siCqariT imoZrave 8 wamis ganmavloba- simaRleze daiWiros? (g) rogori iqneba Si. frenis dro am SemTxvevaSi? 3. Semdeg 0,4 m/wm2-iT Seneldi 6 wamis gan- 123 amerikis SeerTebul StatebSi yovelwliurad avtomobilisa da matareblis Sejaxebis 4000 SemTxveva xdeba. umetesoba maSin xdeba, rodesac avtomobilis mZRoli SeniSnavs momaval matarebels da cdilobs, liandagze matareblis mosvlamde gadaiaros (sakmaod suleluri gadawyvetilebaa). ganvixiloTY Semdegi situacia: am 4000-dan erT-erTi mZRoli trasis perpendikularulad momaval matarebels SeniSnavs da mavlobaSi. amis Semdeg miemarTebi dasavleTisken. Semdeg ki: 4. gazarde siCqare 0,4 m/wm2 aCqarebiT 5 wm-is ganmavlobaSi. 5. am siCqariT imoZrave 10 wamis ganmavlobaSi. 6. Semdeg 0,4 m/wm2-iT Seneldi gaCerebamde. fiqrobs, rom matareblis siCqare misi man- sidide-kuTxur CanawerSi, risi tolia sa- qanis siCqaris tolia da 30 m/wm-s udris. Sualo siCqare matarebelic da manqanac gadakveTis adgils raobis ganmavlobaSi? uaxlovdebian (sur. 4.60). manqanis mZRoli 125 garkveul momentSi Citebis sabanaos gadawyvets, rom, Tu manqanas 1,5 mimarT peplis mdebareobis veqtoria m/wm2-iT wertilidan mTeli moZ- = aaCqarebs, maSin pirveli gadakveTs lian- (2 m) + (3 m) + (1 m) . 40 wamis Semdeg pep- dags. elas mdebareobis veqtoria gadawyvetilebis miRebis momentSi manqanisa da matareblis daSoreba gadakve- = (3 m) + (1 m) + (2 m) . gamoTvale (Tu es SesaZlebelia) Tis adgilamde 40 m-ia. manqanis sigrZea 5 m, (a) xolo matareblis sigane _ 3 m. (a) moxdeba CanawerSi, (b) gadaadgilebis sidide, (g) Tu ara Sejaxeba? (b) Tu ara, ra droiT saSualo gaaswrebs manqana matarebels? Tu ki, ra CanawerSi, (d) 40 wamis ganmavlobaSi peplis droa saWiro Sejaxebis Tavidan asacile- frenis saSualo manZili. gadaadgileba siCqare erTeulovan-veqtorul erTeulovan-veqtorul 126 t = 0 momentSi kldidan burTi isroles. burTis sawyisi siCqarea 15 m/wm, xolo kuTxe horizonts qvemoT 20°. rodesac t = 2,3 wm, risi tolia burTis (a) horizontaluri da matarebeli (b) vertikaluri gadaadgileba? 127 beisbolis burTs miwidan daartyes. burTma maqsimalur simaRles dartymidan 3 wamis Semdeg miaRwia. 2,5 wamis Semdeg burTi dartymis adgilidan 97,5 m manZilze manqana aRmarTul kedels gadaafrinda. davuSvaT, miwis zedapiri brtyelia. (a) miwidan ra sur. 4.60 amocana 123. maqsimalur simaRles miaRwia burTma? (b) 129 ra simaRlisaa barieri? (g) barieridan ra manZilze daecema burTi miwas? 128 vivaraudoT, rom kosmosuri sakvlevi raketa 20 g aCqarebas uZlebs. (a) rogori iqneba Semobrunebis minimaluri radiusi aseTi raketisTvis, Tu is sinaTlis siCqaris erTi meaTediT moZraobs? (b) aseTi siCqariT moZraobisas ra dro dasWirdeba raketas 90°-iT SemobrunebisTvis? 129 manqana 12 m/wm Tanabari siCqariT wri- sur. 4.61 amocana 132. ulad moZraobs. garkveul momentSi manqanis aCqarebaa 3 m/wm2 aRmosavleTisken. rogori iqneba am momentSi manqanis daSoreba Caagdos? horizontaluri manZilebia d1 = 1 futi da d2 = 14 futi, xolo simaRleebia da mimarTuleba wrewiris centridan, Tu h1 = 7 futi da h2 = 10 futi. manqana moZraobs (a) saaTis isris moZrao- 133 vertmfreni brtyeli zedapiridan 9,5 m- bis mimarTulebiT da (b) saaTis isris moZ- is simaRleze wrfivad mifrinavs 6,2 m/wm raobis sawinaaRmdego mimarTulebiT? siCqariT. iqidan agdeben pakets vertmfre- 130 golfis moTamaSe burTebs 20 m daSorebis vertikalur kedels esvris da cdilobs kedelze daxatul 30 sm diametris nis mimarT 12 m/wm sawyisi siCqariT da misi moZraobis sawinaaRmdego mimarTulebiT. (a) moZebne paketis sawyisi siCqare dedamiwis wiTel rgolSi moaxvedros. samiznis cen- mimarT. (b) rogori iqneba horizontaluri tri miwidan daaxloebiT 1.2 m simaRlezea. manZili vertmfrensa da pakets Soris, rode- 15 m/wm sac paketi miwaze daecema? (g) rogori kuTxe erTi cdisas, burTi moZraobs sawyisi siCqariT, horizontis mimarT 35° iqneba paketis siCqaris veqtorsa da dedami- kuTxiT da kedels rgolis centrze gama- was Soris, rodesac paketi miwaze daecema? val vertikalur xazze xvdeba. (a) ra dro 134 kinofilmebSi msaxiobebi haerSi var- dasWirdeba burTs kedelTan misaRwevad? debian afeTqebis, motocikliT Sejaxebis, (b) moxvdeba Tu ara burTi wiTel rgols tornados da bevri sxva ramis gamo. sinamd- (gansazRvre rgolis centramde manZili)? vileSi, msaxiobebs haerSi agdebs sahaero (g) risi tolia burTis siCqare kedelze tramplini, romelic Sedgeba liTonis daf- moxvedrisas? (d) gaiara Tu ara burTma isa da SekumSuli haeris pnevaturi cilin- traeqtoriis umaRlesi wertili kedelze drebisgan. rodesac msaxiobi dafas Seexeba, moxvedramde? eleqtroCamrTveli irTveba, haeri awveba 131 Wurvis areali ara mxolod v0-sa da θ0-zea damokidebuli, aramed Tavisufali vardnis aCqarebis g sididezec, romelic mxars, romelic dafas zeviT aagdebs. gebSi moZebne msaxiobis saSualo aCqareba adgilm- Semdeg situaciebSi: a) aCqareba grZeldeba debareobasTan erTad icvleba. 1936 wels 0,25 wamis ganmavlobaSi, msaxiobi isroles berlinis olimpiur TamaSebze, sadac g = 9,8128 m/wm2, jesi ouensma sigrZeze xtomis msoflio rekordi daamyara – 8,09 m v0-is da θ-is imave sidideebis pirobebSi, ramdenad gansxvavebuli iqneboda ouensis rekordi 1956 wels, melburnSi, sadac g=9,7999 m/wm2? 132 ra sawyisi siCqariT unda tyorcnos 4.61 suraTze kalaTburTelma burTi horizontisadmi θ0 = 55o kuTxiT, raTa kalaTSi patara rkalze horizontidan 30° kuTxiT, xolo misi arealia 4 m. b) aCqareba grZeldeba 0,29 wamis ganmavlobaSi, msaxiobi isroles did rkalze horizontidan 45° kuTxiT, xolod misi arealia 12 m. 135 bostonis stadionze beisbolis burTi `saxlidan~ 0,762 m simaRlidan isroles. burTis sawyisi siCqarea 33,53 m/wm da is horizonts zemoT 55° kuTxiTaa mimarTuli. 130 Tavi 4 organzomilebiani da samganzomilebiani moZraoba dartymidan 5 wamis Semdeg burTi unda ga- SaSxanidan. tyviam manqanis gverdebSi gai- dascdes 11,28m simaRlis Soreul kedels. ara da, rogorc manqanidan moCans, Sesvlis (a) gamoTvale manZili `saxlsa~ da kedels da gasvlis xvrelebi zustad erTmaneTis Soris. (b) vertikaluri manZili, romliTac mopirdapirea. liandagis mimarT ra mimar- burTi kedels gadaaxteba. `saxlis~ mimarT TulebiT isroles tyvia? davuSvaT, manqa- burTis (g) horizontaluri da (d) verti- naSi Sevardnis Semdeg tyviam mimarTuleba kaluri gadaadgilebebi kedelze gadaxto- ar Seicvala da misi siCqare Semcirda 20%- mamde 0,5 wamiT adre. iT. davuSvaT, ν1 = 85 km/sT da ν2 = 650 m/wm. 136 xis satvirTo vagoni swor liandagze (ratom ar aris saWiro satvirTo vagonis moZraobs ν1 siCqariT. am dros snaiperi mas siganis codna?) tyvias esvris (ν2 siCqariT) didi simZlavris 5 Zala da moZraoba aviamzididan afrenisas mZlavri ZravebiT aRWurvil reaqtiul TviTmfrinavs aviamzidis gembanze damontaJebuli sasroli meqanizmi win ubiZgebs. amis Sedegad miRebuli didi aGqarebis gamo TviTmfrinavi afrenisTvis saWiro siGqares mokle droSi aRwevs. samagierod, gembanidan afrenis dros aseTi aGqareba pilots ubiZgebs, TviTmfrinavis cxviri mkveTrad qveviT daxaros. pilotebs am impulsis ugulebelyofas aswavlian, magram zogjer TviTmfrinavi mainc pirdapir okeaneSi vardeba. romeli fizikuri movlenebi iwvevs am saSiSi impulsis gaCenas? pasuxs am TavSi SeityobT 5.1. ra aris fizika? rogorc ukve vnaxeT, fizikis erTi nawili moZraobasa da am dros sxeulis aCqarebas (siCqaris cvlilebas) Seiswavlis. fizika Seiswavlis agreTve sxeulis aCqarebis gamomwvev mizezsac. es mizezi aris Zala, romelic, uxeSad rom vTqvaT, sxeuls ubiZgebs. vambobT, rom Zala sxeulze moqmedebs da amiT siCqares ucvlis. magaliTad, sportuli avtomobilis aCqarebisas mis ukana saburavze amaCqarebeli Zala moqmedebs. rodesac sportuli TamaSisas erTi gundis Tavdamsxmeli meore gundis mcvels miwaze anarcxebs, am dros Tavdamsxmeli mcvelze moqmedebs ZaliT, romelic mcvels ukan mimarTul aCqarebas aniWebs. rodesac avtomanqana satelefono boZs ejaxeba, Zala, romliTac boZi manqanaze moqmedebs, mis gaCerebas iwvevs. samecniero, sainJinro, iuridiuli da samedicino Jurnalebi savsea statiebiT sxeulebze, maT Soris adamianebze moqmedi Zalebis Sesaxeb. 5.2. niutonis meqanika Zalasa da mis mier gamowveul aCqarebas Soris kavSiri pirvelad isaak niutonma (1642-1727) aRmoaCina. fizikis im nawils, romelic am kavSirs Seiswavlis, niutonis meqanika ewodeba. Cven yuradRebas moZraobis sam ZiriTad kanonze gamavaxvilebT. niutonis meqanika yovelTvis ar gamodgeba. Tu urTierTmoqmedi sxeulebis siCqare Zalian maRalia, maSin niutonis meqanikas ainStainis fardobiTobis Teoria enacvleba, romelic nebismieri siCqarisTvis gamosadegia. Tu urTierTmoqmedi sxeulebi atomuri struqturisaa (magaliTad, eleqtronebi atomSi), maSin niutonis meqanikas kvanturi meqanika cvlis. Tanamedrove fizikosebi niutonis meqanikas am ori yovlismomcveli Teoriis kerZo SemTxvevad ganixilaven. 131 132 Tavi 5 Zala da moZraoba Tumca, es sakmaod mniSvnelovani SemTxvevaa, vinaidan is bevr rames miesadageba, Zalian mcire (TiTqmis atomuri struqturis) sxeulebiT dawyebuli da astronomiuli sxeulebiT (galaqtikebiTa da galaqtikaTa jgufebiT) damTavrebuli. 5.3. niutonis pirveli kanoni vidre niutoni meqanikis kanonebs Camoayalibebda, miiCnevdnen, rom sxeuls Tanabari siCqariT moZraobis SenarCunebisTvis garkveuli zemoqmedeba, `Zala~ sWirdeba. aseve fiqrobdnen, rom uZravi sxeuli Tavis `bunebriv mdgomareobaSi~ imyofeboda. egonaT, rom sxeulis Tanabari siCqariT moZraobisTvis saWiroa mudmivi biZgi, sxva SemTxvevaSi ki is moZraobas `bunebrivad~ Sewyvets. es mosazrebebi sakmaod racionaluri iyo. Tu hokeis Saibas xis zedapirze gaacureb, is Seneldeba da Semdeg gaCerdeba. Tu gsurs Saibam Tanabari siCqariT imoZraos, amisTvis mas periodulad unda ubiZgo. hokeis Saiba ufro Sors wava, Tu mas sacigurao moednis yinulis zedapirze gavacurebT. SegiZlia warmoidgino ufro grZeli da xaxunisgan Tavisufali zedapiri, romelzec Saiba Znelad Seneldeba da Zalian Sors gasrialdeba. am dakvirvebebis Semdeg SeiZleba davaskvnaT, rom sxeuli Tanabari siCqariT imoZravebs, Tu masze Zala ar moqmedebs. am daskvnas mivyavarT niutonis pirvel kanonamde: niutonis pirveli kanoni: Tu sxeulze Zala ar moqmedebs, maSin misi siCqare ar Seicvleba, e.i. sxeuli ver aCqardeba. sxvagvarad rom vTqvaT, roca sxeuli uZravia, is uZravad rCeba, xolo roca moZraobs, moZraobas imave siCqariT ganagrZobs (imave sididiT da imave mimarTulebiT). 5.4. Zala axla Zalis erTeuli unda ganvsazRvroT. ukve viciT, rom Zalas sxeulis aCqarebis gamowveva SeuZlia. maSasadame, Zalis erTeuls ganvsazRvravT aCqarebiT, romelsac Zala 1 kg masis sxeuls aniWebs (sur. 1.4.). sxeuls horizontalur, xaxunisgan Tavisufal magidaze movaTavsebT da marjvnidan ise vubiZgebT, rom 1 m/wm2 aCqareba mivaniWoT. amis Semdeg vambobT, rom 1 kg masis sxeulze   sur. 5.1. F Zala 1 kg masis sxeuls a aCqarebas aniWebs. moqmedi Zala 1 niutoni (n) sididisaa. SegviZlia amave sxeulze 2 n ZaliT vimoqmedoT, anu ise vubiZgoT, rom 2 m/wm2 aCqareba mivaniWoT, da a.S. maSasadame, Tu standartuli sxeuli 1 kg masisaa da a sididis aCqareba aqvs, maSin viciT, rom masze F Zala moqmedebs da am Zalis sidide (niutonebSi) aCqarebis sididis tolia (m/wm2). Zala mis mier miniWebuli aCqarebiT izomeba. Tumca, aCqareba veqtoruli sididea, e.i. sidide da mimarTuleba aqvs. aris Tu ara Zala veqtoruli sidide? Zalas mimarTulebas iolad mivaniWebT (ubralod aCqarebis mimarTulebas mivusadagebT), magram es sakmarisi ar aris. eqsperimentiT unda dadgindes, rom Zala veqtoruli sididea. aseTi eqsperimenti ukve Catarda: Zala namdvilad 5.4. Zala 133 veqtoruli sididea, mas aqvs sidide da mimarTuleba da maTi Sekreba xdeba me-3 TavSi moyvanili veqtoruli Sekrebis wesebiT. rodesac sxeulze ori an meti Zala moqmedebs, veqtoruli SekrebiT jamuri Zalis anu Zalebis tolqmedis povna SegviZlia. jamuri Zalis sididisa da mimarTulebis mqone erTi Zala sxeulze imave zemoqmedebas axdens, rasac cal-calke moqmedi es Zalebi erTad. am movlenas Zalebis zeddebis principi ewodeba. samyaro sakmaod ucnauri iqneboda, Tu Sen da Seni megobari 1 kg masis sxeuls erTi mimarTulebiT ubiZgebdiT, TiToeuli 1 n ZaliT, da jamuri Zala rogorRac 14 n gaxdeboda.  am wignSi Zala veqtoruli simboloTia ( F ) warmodgenili, xolo tolqmedi _ simboloTi. iseve, rogorc sxva veqtorebs, Zalisa da tolqmedis veqtorebsac aqvs komponentebi koordinatTa RerZebis gaswvriv. rodesac Zalebi mxolod erTi RerZis gaswvriv moqmedebs, isini erTkomponentiani Zalebia. am SemTxvevaSi Zalis simbolos isari SegviZlia movxsnaT da RerZebis gaswvriv Zalis mimarTulebis saCveneblad `+~ da `–~ niSnebi gamoviyenoT. 5.3 qveTavSi mocemuli formulirebis nacvlad, niutonis pirveli kanoni tolqmedi Zalis gamoyenebiT ase SeiZleba Camoyalibdes: niutonis pirveli kanoni: Tu sxeulze tolqmedi Zala ar moqmedebs ( = 0), maSin misi siCqare ar Seicvleba, e.i. sxeuli ver aCqardeba. sxeulze ramdenime Zala SeiZleba moqmedebdes, magram Tu tolqmedi Zala nulis tolia, sxeuls aCqareba ar eqneba. aTvlis inerciuli sistemebi niutonis pirveli kanoni aTvlis yvela sistemas ar miesadageba, magram yovelTvis SegviZlia iseTi aTvlis sistemebis moZebna, romelSic es kanoni samarTliani iqneba (iseve, rogorc niutonis meqanikis danarCeni nawili). aseT sistemebs aTvlis inerciuli sistemebi an, ubralod, inerciuli sistemebi ewodeba. aTvlis inerciuli sistemaSi niutonis yvela kanoni samarTliania. magaliTad, dedamiwa aTvlis inerciuli sistemaa, Tu dedamiwis brunvas ugulebelvyofT. es daSveba maSin gamodgeba, rodesac hokeis Saibas yinulis mokle zolze vasrialebT. am SemTxvevaSi Saiba niutonis kanonebs emorCileba. magram, davuSvaT, rom Saiba CrdiloeT polusidan did manZilze gavacureT (sur. 5.2a). Tu Saibis moZraobas kosmosSi mdebare uZravi sistemidan davakvirdebiT, maSin Saiba samxreTisken wrfivad moZraobs, vinaidan CrdiloeT polusis garSemo dedamiwis brunva Saibis qveS yinuls ubralod asrialebs. Tu Saibas miwidan davakvirdebiT, maSin misi traeqtoria wrfivi ar iqneba. imis gamo, rom dedamiwis siCqare, romelic aRmosavleTiskenaa mimarTuli, miT metia, rac ufro samxreTiTaa Saiba, miwaze uZravad myof damkvirvebels eCveneba, sur. 5.2. a) kosmosis uZravi wer rom Siaba dasavleTisken moZraobs (sur. 5.2b). Saibis gadaxra tilidan danaxuli, CrdiloeT polusidan mosriale Saibis moZra niutonis kanoniT ki ar aris gamowveuli, aramed imiT, rom obis traeqtoria. dedamiwa brunavs Saibas moZravi aTvlis sistemidan vakvirdebiT. am SemTxvevaSi aRmosavleTisken. b) dedamiwidan danaxuli Saibis moZraobis tra dedamiwa aTvlis arainerciuli sistemaa. eqtoria. 134 Tavi 5 Zala da moZraoba am wignSi vuSvebT, rom dedamiwa aTvlis inerciuli sistemaa da Zalebs da aCqarebebs swored am sistemis mimarT vzomavT. Tu gazomvebi Catarda, magaliTad, dedamiwis mimarT aCqarebuli liftis kabinaSi, maSin gazomvebi Catarebula aTvlis arainerciul sistemaSi da Sedegebic sakmaod ucnauri iqneba. amis magaliTs amocanis amoxsnis 5.8 nimuSSi vnaxavT. sakontrolo wertili 1. eqvsidan romel suraTzea naCvenebi, ise, rom miviRoT mesame veqtori, romelic (a) (b) (g)   F1 da F2 veqtorebis Sekreba sworad tolqmedi iqneba? (d) (e) (v) 5.5. masa yoveldRiuri yofa gviCvenebs, rom mocemuli Zala sxvadasxva sxeuls sxvadasxva sididis aCqarebas aniWebs. beisbolisa da keglis burTebi iatakze daawyve da orives erTi ZaliT daartyi. Sedegi winaswar ici: beisbolis burTi keglis burTze did aCqarebas SeiZens SeiZens. aCqarebebs Soris gansxvaveba masebs Soris gansxvavebiTaa gamowveuli. ra aris masa? SegviZlia ramdenime eqsperimentiT vaCvenoT, Tu rogor unda gavzomoT masa aTvlis inerciul sistemaSi. pirvel eqsperimentSi Zala m0 = 1 kg masis sxeuls movdoT. davuSvaT, sxeuli 1 m/wm2-iT aCqarda. am SemTxvevaSi SegviZlia vTqvaT, rom sxeulze 1 n Zala moqmedebs. Semdeg igive Zala ucnobi masis meore X sxeuls movdoT. davuSvaT, es X sxeuli 0,25 m/wm2-iT aCqarda. ukve viciT, rom naklebi masis mqone beisbolis burTi ufro did aCqarebas iZens, vidre meti masis mqone keglis burTi, rodesac maTze erTi da igive Zala moqmedebs. aqedan gamomdinare, mivdivarT Semdeg daskvnamde: ori sxeulis masebis Sefardeba maTi aCqarebebis Sebrunebuli Sefardebis tolia, rodesac am sxeulebze erTi da igive Zalaa modebuli. X sxeulisa da 1 kg masis sxeulisTvis gveqneba: mx-is amoxsna mogvcems: mx a0 m0 = ax . a 1.0 m/w2 mx = m0 a0 = (1 kg) = 4 kg. 0.25 m/w2 x Cveni daskvna mxolod im SemTxvevaSia swori, Tu is Zalis sididis Secvlis SemTxvevaSic samarTliania. magaliTad, 1 kg masis sxeuls 8 n Zala movdeT da miviReT 8 m/wm2 aCqareba. igive Zala X sxeuls movdeT da miviReT 2 m/wm2 aCqareba. Cveni daskvnis Tanaxmad: a 8.0 m/w2 mx = m0 a0 = (1 kg) = 4 kg. 2.0 m/w2 x es pasuxi pirveli eqsperimentis Sedegs ar ewinaaRmdegeba. rodesac bevri eqsperimenti erTsa da imave Sedegs gvaZlevs, es imas niSnavs, rom Cveni daskvna saimedo da samarTliania nebismieri sxeulis masis dasadgenad. Cveni eqsperimentebi gviCvenebs, rom masa sxeulis Sinagani maxasiaTebelia anu isaa, rac sxeuls bunebrivad aqvs. igive eqsperimentebi aCvenebs, rom masa skalaruli sididea. Tumca kvlav upasuxod rCeba kiTxva: `ra aris masa?~ vinaidan sityva `masa~ yoveldRiur saubarSi xSirad gamoiyeneba, albaT is ukve intuiciurad gvesmis. ra aris masa _ sxeulis zoma, wona Tu simkvrive? arc erTi, Tumca sxeulis es maxasiaTebelebi masaSi xSirad erevaT. mxolod imis Tqma SegviZlia, rom sxeulis masa aris maxasiaTebeli, romelic sxeulze moqmed Zalas am moqmedebis Sedegad miRebul aCqarebasTan akavSirebs. masas sxva gansazRvreba ara aqvs. masis fizikurad SegrZneba mxolod sxeulis aCqarebisas SeiZleba, magaliTad, rodesac beisbolis an keglis burTs urtyam. 5.6. niutonis meore kanoni 135 5.6. niutonis meore kanoni aqamde ganxiluli yvela gansazRvreba, eqsperimenti da dakvirveba SeiZleba Sejamdes erTi winadadebiT: niutonis meore kanoni: sxeulze moqmedi Zalebis tolqmedi sxeulis masisa da misi aCqarebis namravlis tolia. formulis saxiT gveqneba:   Fnet = ma (niutonis meore kanoni) (5.1) formula martivia, magram is frTxilad unda gamoviyenoT. pirvel rigSi unda ganisazRvros sxeuli, romlisTvisac viyenebT am formulas. Semdeg unda davrwmundeT, rom jamur veqtorSi mxolod am da ara romelime sxva sxeulze moqmedi Zalebi Sedis. magaliTad, ragbiSi burTisTvis brZolisas Senze moqmedi Zalebis tolqmedi Seni sxeulis mier miRebuli biZgebi da dartymebia da ara _ sxva moTamaSeze Sens mier miyenebuli dartymebi. sxva veqtoruli formulebis msgavsad, 5.1 formulac sami komponenturi formulis eqvivalenturia. xyz koordinatTa sistemis TiToeuli RerZisTvis gveqneba: Ft,x= max , Ft,y= may da Ft,z = maz (5.2) TiToeuli formula tolqmedi Zalis komponentebs Sesabamisi RerZebis gaswvriv mimarTul aCqarebasTan akavSirebs. magaliTad, pirveli formula gvauwyebs, rom yvela Zalis komponentebis jami x RerZis gaswvriv sxeulis aCqarebis x komponents ax iwvevs, xolo y da z mimarTulebebiT aCqareba ara gvaqvs. sxvagvarad rom vTqvaT, ax aCqarebis komponenti sxeulze moqmedi Zalebis x komponentebis jamiTaa gamowveuli. zogadad:L mocemuli RerZis gaswvriv aCqarebis komponenti Zalebis aseTive komponentebis jamiTaa gamowveuli da ara _ sxva komponentebiT. 5.1 formulis Tanaxmad, Tu sxeulze moqmedi Zalebis tolqmedi nulis tolia, maSin sxeulis aCqareba  a =0. roca sxeuli uZrav mdgomareobaSia, uZravadve rCeba; roca moZraobs, moZraobas Tanabari siCqariT ganagrZobs. aseT SemTxvevaSi sxeulze moqmedi Zalebi erTmaneTs awonasworebs da am mdgomareobas wonasworuli ewodeba. aseve amboben, rom Zalebi erTmaneTs abaTilebs, magram termini `gabaTileba~ ar aris zusti, vinaidan es ar niSnavs, rom Zalebi aRar arsebobs. yvela Zala sxeulze kvlav moqmedebs. cxrili 5.1 erTeulebi niutonis meore kanonSi (formulebi 5.1 da 5.2) sistema Zala masa aCqareba SI niutoni (n) kilogrami (kg) m/wm2 CGSa dini grami (g) sm/wm2 britanulib girvanqa (lb) slagi futi/wm2 a 1 dini = 1 g⋅sm/wm2 b 1 girvanqa = 1 slagi⋅futi/wm2 136 Zala da moZraoba Tavi 5 SI erTeulebSi, 5.1 formulis Tanaxmad: 1 n = (1 kg)(1 m/wm2) = 1 kg⋅m/wm2 (5.3) Zalis zogierTi erTeuli sxva erTeulTa sistemebSi mocemulia 5.1 cxrilsa da D danarTSi. niutonis meore kanonis daxmarebiT amocanebis amosaxsnelad xSirad Tavisufali sxeulis grafiks vxazavT. am grafikze mxolod is sxeulia gamosaxuli, romelze moqmed Zalebsac vkribavT. zogjer maswavleblebi Tavad sxeulis daxatvas arCeven, magram adgilis ekonomiis mizniT am TavebSi sxeuls wertilis saxiT gamovsaxavT. sxeulze moqmedi TiToeuli Zala veqtoriT gamoisaxeba, romlis saTave am sxeulzea. viyenebT, agreTve, koordinatTa sistemas, xolo sxeulis aCqareba zogjer veqtoris isriTaa naCvenebi. sistema erTi an meti sxeulisgan Sedgeba da nebismier Zalas, romelic am sxeulebze sistemis garedan moqmedebs, gare Zala ewodeba. Tu sistemis Semadgeneli sxeulebi erTmaneTTan mWidrodaa dakavSirebuli, maSin sistema SegviZlia ganvixiloT, rogorc erTi mTliani sxeuli da masze moqmedi tolqmedi yvela gare Zalis jamis toli iqneba (Sida Zalebs anu sistemis sxeulebs Soris moqmed Zalebs ugulebelvyofT). magaliTad, erTmaneTTan dakavSirebuli lokomotivi da vagoni erT sistemas qmnis. roca lokomotivs buqsiriT ewevian, buqsirze modebuli Zala mTel sistemaze moqmedebs. iseve, rogorc erTi sxeulisTvis, SegviZlia sistemaze moqmedi tolqmedi sistemis aCqarebas davukavSiroT. es xdeba niutonis meore kanonis Tanaxmad _  gare Zalac  Fnet = ma , sadac m mTliani sistemis masaa. sakontrolo wertili 2. suraTze naCvenebia xaxunisgan Tavisufal zedapirze  mdebare sxeulze horizontaluri moqmedi Zalac ori horizontaluri moqmedebs, risi tolia Zala. Tu sxeulze  F3 -is F3 mesame sidide da mimarTuleba, rodesac sxeuli a) uZravia da b) marcxniv moZraobs 5 m/wm Tanabari siCqariT? 5.1 amocanis amoxsnis nimuSi 5.3a-5.3g suraTebze xaxunisgan Tavisufal yinulze x RerZis gaswvriv erT ganzomilebaSi moZrav Saibaze erTi an ori Zala moqmedebs. Saibis masaa m = 0,2 kg.   F1 da F2 Zalebi RerZis gaswvrivaa mimarTuli  da maTi sidideebia amoxsna: TiToeul mdgomareobaSi, niutonis meore kanonis   F(net = ma ) saSualebiT aCqareba Saibaze moqmed SegviZlia  a Zalebis tolqmeds davukavSiroT. Tumca, vinaidan moZraoba mxolod x RerZis gaswvriv xdeba, SegviZlia yoveli mdgomareoba gavamartivoT da niutonis meore kanoni mxolod x komponentebisTvis davweroT: Ft,x=max (g) (d) (e) (v) sur. 5.3. a)-g) am sam mdgomareobaSi Zalebi x RerZis gaswvriv moZrav Saibaze moqmedeben. d)-v) Tavisufali sxeulis grafikebi. 5.3d suraTisTvis, sadac mxolod erTi horizontaluri Zala moqmedebs, formula 5.4 mogvcems: F1 = max CvenTvis cnobili monacemebis CasmiT miviRebT: (5.4) 5.3d-5.3v suraTebze mocemulia Tavisufali sxeulis grafiki sam mdgomareobaSi. Saiba aRniSnulia wertiliT. (b) Saiba F1 = 4 n da F2 = 2 n. F3 Zala θ = 30° kuTxiTaa mimarTuli da misi sididea F3=1 n. risi tolia Saibis aCqareba TiToeul mdgomareobaSi? (a) ax = F1 4n = = 20 m/w2 m 0.2 kg (pasuxi) dadebiTi pasuxi imas niSnavs, rom aCqarebas x RerZis dadebiTi mimarTuleba aqvs. 137 5.6. niutonis meore kanoni F3,x–F2 F cosθ–F2 = 3,x m m (1 n)(cos 30ο) - 2 n = = –5,7 m/w2 (pasuxi) 0.2 kg 5.3e suraTze Saibaze ori horizontaluri  Zala moqmedebs. F1 modebulia x-is dadebiTi  mimarTulebiT, xolo F2 _ uaryofiTi ax = mimarTulebiT. am SemTxvevaSi formula 5.4 mogvcems: uaryofiTi mimarTulebis aCqarebas aniWebs. F 1 –F 2 =ma x CvenTvis cnobili maSasadame, tolqmedi Zala Saibas x RerZis monacemebis CasmiT miviRebT sakontrolo wertili suraTze F 1 –F 2 4n–2n = = 10 m/w2 (pasuxi) m 0.2 kg maSasadame, tolqmedi Zala Saibas x RerZis gamosaxulia dadebiTi mimarTulebiT aCqarebs.  5.3v suraTze, F3 Saibis aCqarebis gaswvriv aniWebs. daalage mdgomareobebi a) blokze ar aCqarebis klebadobis mixedviT. ax = aris mimarTuli. F3,x mxolod aCqarebis gaswvriv  komponentia mimarTuli ( F3 zevidan 3. danaxuli oTxi mdgomareoba, sadac ori Zala xaxunisgan Tavisufal zedapirze sxeuls aCqarebas moqmedi tolqmedi Zalis sididisa da b) blokis Zala organzomilebiania, Tumca moZraoba mxolod erT ganzomilebaSi xdeba). maSasadame, formula 5.4 daiwereba Semdegi saxiT: F3,x – F2 = max suraTidan vxedavT, rom F 3,x = F 3 cos θ. aCqarebis amoxsna da F3,x-is Casma mogvcems: amocanis amoxsnis nimuSi 5.2 5.4a suraTze 2 kg namcxvris qila xaxunis gan Tavisufal horizontalur zedapirze  3 m/wm2-iT Cqardeba a veqtoris mimarTulebiT. aCqareba gamowveulia sami horizontaluri Zalis moqmedebiT, romelTagan mxolod oria   F1 -is sididea10 n, xolo F2 -isa – 20 n. risi tolia mesame F3 Zala erTeulovannaCvenebi: veqtorul da sidide-kuTxur CanawerSi? amoxsna: qilaze moqmedi tolqmedi  sami Zalis jamis tolia da a aCqarebasTan meore kanoniTaa dakavSirebuli niutonis  F(net = ma ). maSasadame,     F1 + F2 + F3 = ma , rac mogvcems:     F3 = ma - F1 - F2 sur. 5.4. a) samidan ori Zala, romlebic namcxvris   qilaze moqmedeben da mas a aCqarebas aniWeben. F3 ar    Cans. b) ma , - F1 da - F2 veqtorebis dalageba F3 -is mosaZebnad. (5.5) am amocanaSi yvelaferi or ganzomilebaSi Seikribos, rogorc es 5.4b suraTzea naCvenebi. xdeba, maSasadame, 5.5 formulis marjvena Sekreba SeiZleba veqtoruli kalkulatoris mxares saSualebiT moxdes, vinaidan samive veqtoris veqtoruli sidideebis ricxviTi mniSvnelobebis CasmiT    nacvlad  F3 -s ver vipovniT. amitom ma , – F1 ( F1 -is Sebrunebuli)   da – F2 ( F2 -is Sebrunebuli) veqtorulad unda sidide da kuTxe viciT. asea Tu ise, am SemTxvevaSi 5.5 formulis marjvena mxares komponentebis saSualebiT gamovTvliT, jer x 138 Tavi 5 Zala da moZraoba RerZis gaswvriv, Semdeg ki y RerZis gaswvriv. x RerZis gaswvriv gvaqvs: F3,x = max – F1,x – F2,x maSasadame, erTeulovan-veqtorul Cana werSi gveqneba: = m(a cos50°)–F1cos(–150°)–F2cos90°. cnobili monacemebis CasmiT vipoviT: F3,x=(2 kg)(3 m/wm2)cos50°–(10 n)cos(–150°) –(20 n)cos90° = =12,5 n. aseTive meTodiT vpoulobT y komponen tsac: F 3,y= may – F 1,y – F 2,y = m(asin50°)–F 1sin(–150°)–F2sin90°= = (2kg)(3 m/wm2)sin50°–(10n)sin(–150°) – (20n)sin90° = = –10,4 n. amocanis amoxsnis nimuSi (pasuxi)  F3 -is sididisa da kuTxis gasagebad SegviZlia veqtoruli kalkulatori gamo viyenoT. sididisa da kuTxis (x RerZis dadebiTi mimarTulebisadmi) dadgena 3.6 formulis saSualebiTac SeiZleba: da (pasuxi) 5.3 bagiris qaCvis SejibrSi aleqsi, beti da Carlzi avtomobilis saburavs 5.5a suraTze aleqsi Car naCvenebi kuTxeebiT eqaCebian. saburavi, sami li Zalis moqmedebis miuxedavad,  uZravia. aleqsi saburavs 220 n sididis FA ZaliT  eqaCeba, FC ZaliT. ferso xolo Carlzi _ 170 n sididis  FC -s mimarTuleba naCvenebi ar aris. risi  F tolia betis B Zalis sidide? amoxsna: vinaidan sami aCqarebas ar aniWebs, amitom Zala beti saburavs  a =0 (anu Zalebi (b) (a) erTmaneTs awonasworebs). SegviZlia niutonis   meore kanonis daxmarebiTF(net = ma ) es aCqareba saburavze moqmedi Zalebis tolqmeds sur. 5.5 a) sami adamiani eqaCeba saburavs. b) Tavisufali sxeulis grafiki saburavisTvis. davukavSiroT. niutonis meore kanoni ase daiwereba: an 5.5b suraTze naCvenebia (5.6) Tavisufali saSualebiT. vinaidan  FB mimarTulia y RerZis gaswriv, virCevT am RerZs da vwerT: sxeulis grafiki, romelzec koordinatTa F By = –FAy – F Cy sistemis aTvlis saTave saburavs davamTxvieT,  komponentebis kuTxeebis gansazRvriT da FA -sTvis 133° (=180°–47°) kuTxis gamoyenebiT  xolo Fc -s kuTxe f asoTi aRvniSneT.  viciT  gvinda FB -s sididis gageba.  FA -s sidide da kuTxe, agreTve Fc -s mxolod sidide. maSasadame, 5.6 formulis orive mxares ucnobebi gvaqvs da veqtorul kalkulatorze mas ver amovxsniT. amis nacvlad, formula 5.6 unda gadavweroT x an y RerZis komponentebis miviRebT: F Bsin(–90°)=–FAsin133°–FCsinf monacemebis Casma mogvcems: –F B=–(220 n )(sin133°)–(170 n )sinf magram φ kuTxe jer kidev ar viciT. (5.7) 139 5.7 zogierTi konkretuli Zala φ kuTxis povnas SevZlebT, Tu 5.6 formulas x RerZisTvis gadavwerT, rogorc: F Bx = –FAx – F Cx 0 = – (220 n)(cos133°) – (170 n) cosφ xolo φ Semdeg gveqneba: FBcos(–90°) = –FAcos133° – FCcosφ rac mogvcems: am pasuxis 5.7 formulaSi Casma mogvcems: FB = 241 n (pasuxi) amocanis amoxsnis taqtika taqtika 1: ganzomilebebi da veqtorebi bevri studenti ver acnobierebs imas, rom moqmedeba erT ganzomilebaSi ar xdeba, da es Secdoma wignze muSaobis procesSi Tan sdevs. Zalebis tolqmedis misaRebad am Zalebis sidideebis ubralod mimateba an gamokleba ar SeiZleba, Tu isini erTi da imave RerZis gaswvriv araa mimarTuli. sxva SemTxvevaSi unda gamoiyenoT veqtoruli veqtoruli kalkulatoriT an Sekreba, RerZebis gaswvriv komponentebis povniT, rogorc es amocanis amoxsnis 5.2 nimuSSia gakeTebuli. taqtika 2: ZalebTan dakavSirebuli amocanebis wakiTxva amocanis piroba ramdenjerme waikiTxe, raTa kargad gaiazro mocemuli mdgomareoba, monacemebi da isic, Tu ras moiTxovs Sengan amocana. Tu ukve ici, ras Seexeba amocana, magram ver xvdebi, ra unda gaakeTo, maSin amocana gverdze gadade da saswavlo teqsti gadaikiTxe. Tu niutonis meore kanoni kargad ver gaige, es Tavi kidev waikiTxe. Seiswavle amocanis amoxsnis nimuSebi da daimaxsovre, rom fizikis amocanebis amoxsnas varjiSi sWirdeba (iseve, rogorc manqanis SekeTebas an kompiuteruli sqemebis daproeqtebas). Sen xom amocanis amoxsnis unari dabadebidan ar mogyolia. taqtika 3: ori tipis naxazi amoxsnisas SeiZleba ori naxazi dagWirdes. pirveli _ rodesac Zalebs mdgomareobis xazav, uxeSi eskizia. TiToeuli Zalis veqtoris saTave sxeulis zedapirs an centrs daamTxvie. meore naxazi Tavisufali sxeulis grafikia, sadac sxeuli wertilis an martivi figuris saxiTaa warmodgenili. TiToeuli Zalis veqtoris saTave am wertils an figuras daamTxvie. taqtika 4: romelia Seni sistema? Tu niutonis meore kanons iyeneb, unda icode, romel sxeuls an sistemas miusadageb. amocanis amoxsnis 5.1 nimuSSi es sxeuli hokeis Saibaa (da ara yinuli), amocanis amoxsnis 5.2 nimuSSi _ namcxvris qila, amocanis amoxsnis 5.3 nimuSSi ki _ saburavi (da ara adamianebi). taqtika 5: RerZebi gonivrulad airCie amocanis amoxsnis 5.3 nimuSSi uamravi Sroma davzogeT, roca koordinatTa RerZebi ise ganvalageT, rom y RerZi erT-erT Zalis  mimarTulebas daemTxva (y RerZi FB Zalas). 5.7 zogierTi konkretuli Zala mizidulobis Zala anu gravitacia  sxeulze moqmedi Fg mizidulobis Zala anu gravitacia is Zalaa, romelic meore sxeuliskenaa mimarTuli. wina TavebSi ar gagvirkvevia am Zalis buneba da ganvixilavdiT iseT situaciebs, sadac meore sxeuli dedamiwa iyo. Tumca, rodesac vlaparakobT sxeulze moqmed  Fg mizidulobis Zalaze, vgulisxmobT Zalas, romelic sxeuls dedamiwis centris mimarTulebiT ubiZgebs anu mimarTulia qveviT. momavalSi davuSvebT, rom dedamiwa aTvlis inerciuli sistemaa. roca m masis sxeuli Tavisufali vardnis g aCqarebiT Tavisuflad ecema, haeris zemoqmedebas ugulebelvyofT da sxeulze moqmed erTaderT Zalad  Fg mizidulobis Zalas miviCnevT. 140 Tavi 5 Zala da moZraoba qveviT mimarTuli mizidulobis Zala da aCqareba SegviZlia erTmaneTs niutonis meore kanonis   ( F = ma ) saSualebiT davukavSiroT. vertikalur y RerZs sxeulis moZraobis traeqtoriis gaswvriv gavavlebT da zeda mimarTulebas dadebiTad miviCnevT. am RerZisTvis niutonis meore kanoni Semdeg saxes miiRebs: Ft ,y = may . Cvens SemTxvevaSi gveqneba: –Fg = m(–g) Fg = mg an (5.8) anu mizidulobis Zala mg namravlis tolia. imave sididis mqone mizidulobis Zala sxeulze maSinac moqmedebs, rodesac is Tavisuflad ki ar vardeba, aramed, magaliTad, magidaze devs an magidaze moZraobs (mizidulobis Zalis gasaqrobad dedamiwa unda gauCinardes). veqtorul formaSi niutonis meore kanoni mizidulobis ZalisTvis ase SeiZleba Caiweros: sadac , (5.9) ˆj aris zeda mimarTulebis mimaniSnebeli erTeulovani veqtori y RerZisTvis, xolo g _ Tavisufali vardnis aCqareba (veqtoruli saxiT), romelic qveviTaa mimarTuli. wona sxeulis wona W im tolqmedi Zalis sididea, romelic sxeulis Tavisufali vardnis SesaCerebladaa saWiro. magaliTad, miwaze dgomis dros burTis xeliT dasaWerad unda warmoqmna zeviT mimarTuli Zala, romelic sxeulze moqmed mizidulobis Zalas gaawonasworebs. davuSvaT, mizidulobis Zalaa 2 n, maSin zeviT mimarTuli Zala 2 n unda iyos, da burTis W wonac 2 niutoni iqneba. am SemTxvevaSi vambobT, rom burTi 2 niutons iwonis. 3 n wonis burTis gaCerebas ufro didi Zala – 3 n dasWirdeba. amis mizezi ufro didi (3 n) mizidulobis Zalaa. am SemTxvevaSi vambobT, rom meore burTi pirvelze mZimea. axla ki modiT situacia ganvazogadoT. ganvixiloT sxeuli, romlis  a aCqareba dedamiwis mimarT nulis tolia. dedamiwas miviCnevT aTvlis inerciul sistemad. sxeulze moqmedebs ori Zala: qveviT mimarTuli  Fg mizidulobis sur. 5.6. tolmxariani saswori. rodesac saswori gawonasworebulia, asawon mizidulobis sxeulze moqmedi gL Zala (marcxena TefSze) da miTiTebul sxeulebze (marjvena TefSze) moqmedi mizidulobis Zala tolia. mTliani gR amgvarad, asawoni sxeulis mL masa da miTiTebuli sxeulebis mTliani mR masa tolia. Zala da misi gamawonasworebeli W sididis Zala. vertikaluri y RerZisTvis, romlis zeda mimarTuleba dadebiTadaa miCneuli, niutonis meore kanoni Semdegi saxiT SeiZleba daiweros: F t ,y =ma y Cvens magaliTSi es iqneba: anu W=F g W–F g =m·0 (wona, rodesac dedamiwa aTvlis inerciuli sistemaa) (5.10) (5.11) es toloba gvauwyebs, rom: sxeulis W wona sxeulze moqmedi Fg mizidulobis Zalis sididis tolia. 5.8 formulaSi Fg-s nacvlad W-is CasmiT miviRebT: W=mg (wona) es formula sxeulis wonas masasTan akavSirebs. (5.12) 5.7 zogierTi konkretuli Zala 141 sxeulis awonva misi wonis gazomvas niSnavs. sxeuls vdebT Tanabarmxriani sasworis erT-erT TefSze (sur. 5.6), xolo cnobili masis mqone sawonebs meoreze vawyobT, sanam wonasworoba ar damyardeba (sanam orive mxares moqmedi mizidulobis Zala erTmaneTis toli ar gaxdeba). am SemTxvevaSi sasworis TefSebis masebi tolia da vigebT sxeulis masas. Tu wonasworobis adgilze g-s mniSvneloba viciT, sxeulis wonas 5.12 formulis saSualebiT vipoviT. sxeulis awonva SesaZlebelia zambariani sasworiTac (sur. 5.7). sxeuli zambaras gaWimavs da masis an wonis 5.7. zambariani skala (dagra erTeulebiT dagraduirebuli skalis gaswvriv maCvenebels duirebuli zambara). misi Cveneba TefSze mdebare sxeulis wonis proporciulia. Tu miTiTebulia wonis erTeuli, skala wonas uCvenebs. Tu, piriqiT, miTiTebulia masis erTeuli, maSin skala sxeulis wonas uCvenebs im SemTxvevaSi, Tu g-s mniSvneloba skalis gamoyenebis adgilmdebareobaSi igivea, rac skalis dayofis adgilmdebareobaSi. gadaadgilebs (aSS-Si sasworebis umetesoba am principiT sur. muSaobs da dagraduirebulia Zalis girvanqebSi). roca saswors masis erTeulebi aqvs, Cveneba mxolod maSin iqneba swori, Tu g-s mniSvneloba sasworis damzadebis adgilsa da awonvis adgilze erTnairia. sxeulis awonva maSin xdeba, rodesac sxeuli vertikalurad dedamiwis mimarT ar Cqardeba. magaliTad, awonva SesaZlebelia saabazanoSi an swraf matarebelSi. magram, Tu aCqarebul eskalatorze awonvas gadawyvet, maSin Cveneba Seni namdvilis wonisgan gansxvavebuli iqneba. aseT gazomvas moCvenebiTi ewodeba. yuradRebiT: sxeulis wona masa ar aris. wona Zalis sididea da masasTan 2.15 formuliTaa dakavSirebuli. Tu sxeuls gadaitan wertilSi, sadac g gansxvavebulia, maSin misi masa (Sinagani maxasiaTebeli) ucvleli darCeba, wona ki Seicvleba. magaliTad, 7,2 kg masis keglis burTis wona dedamiwaze 71 niutonia, mTvareze ki _ mxolod 12 n. masa dedamiwazec da mTvarezec erTnairia, magram mTvareze Tavisufali vardnis aCqarebaa 1,7 m/wm2. drekadobis Zala leibze dgomisas dedamiwa qveviT geqaCeba, Sen ki mainc uZrav mdgomareobaSi xar. amis mizezi leibia, romelic Sens fexebs qvemoT deformirdeba da maRla gibiZgebs. aseve, roca iatakze dgaxar, is deformideba (SeikumSeba an CaiRuneba) da zeviT gibiZgebs. erTi SexedviT myari betonis iatakic ki aseve iqceva (Tu is dedamiwaze myarad ar aris damagrebuli, bevri adamianis dgomisas is SeiZleba Catydes kidec).  leibidan an iatakidan Senze moqmed Zalas FN drekadobis Zala ewodeba. iatakidan Senze moqmedi drekadobis Zala iatakis perpendikularulia. rodesac sxeuli zedapirs awveba, zedapiri (erTi SexedviT  myaric ki) deformirdeba da sxeulze FN drekadobis ZaliT sur. 5.8. a) magidaze dadebuli uZravi sxeuli magidis perpendikularuli  FN drekadobis Zalis moqmedebas ganicdis. b) Tavisufali sxeulis grafiki am sxeulisTvis. moqmedebs, romelic zedapiris perpendikularulia. 5.8a suraTze amis magaliTia naCvenebi. m masis sxeuli magidas awveba da mis deformacias  Fg FN drekadobis ZaliT moqmedebs. 5.8b   F suraTze naCvenebia Tavisufali sxeulis grafiki. sxeulze vertikalurad moqmedebs g da FN mizidulobis Zalis gavleniT axdens. magida sxeulze Zalebi. maSasadame, am SemTxvevaSi niutonis meore kanoni zeviT mimarTuli y RerZis pirobebSi (Ft,y=may) Semdegi saxiT daiwereba: F N– F g= m a y 142 Tavi 5 Zala da moZraoba 5.8 formulidan Fg-s nacvlad CavsvaT mg, miviRebT: F N –mg=ma y maSin drekadobis Zala iqneba: F N =mg+ma y =m(g+a y ) (5.13) magidis an sxeulis nebismieri vertikaluri ay aCqarebis SemTxvevaSi (magaliTad, moZrav liftSi), Tu magida da sxeuli dedamiwis mimarT aCqarebas ar ganicdis, maSin ay=0 da formula 5.13 gvaZlevs: sakontrolo wertili 4. 5.8 suraTze FN=mg (5.14)  FN drekadobis Zala mg-ze metia, naklebia Tu tolia, rodesac sxeuli da magida liftiT zeviT moZraobs a) Tanabari siCqariT da b) zrdadi siCqariT? xaxuni Tu sxeuls SevecdebiT, zedapirze sxeulsa da gavasrialebT zedapirs Soris an gasrialebas winaaRmdegoba gaCndeba (am winaaRmdegobas ufro dawvrilebiT Semdeg TavSi ganvixilavT). es winaaRmdegoba erTi  f ZaliTaa warmodgenili da mas xaxunis Zala an, ubralod, xaxuni ewodeba. es Zala sur. 5.9. f xaxunis Zala zedapirze mimarTulia zedapiris gaswvriv moZraobis sawinaaRmdego sxeulis srials ewinaaRmdegeba. mimarTulebiT (sur. 5.9). situaciis gasamartiveblad xaxuns zogjer ugulebelyofen (da amboben, rom zedapiri xaxunisgan Tavisufalia). daWimuloba rodesac Toki (Zafi, kabeli  an sxva msgavsi sagani) mimagrebulia sxeulze da Zlierad iWimeba, amiT sxeulze moqmedebs T ZaliT, romelic mimarTulia sxeulidan gareT da Tokis gaswvriv (sur. 5.10a). am Zalas xSirad daWimulobis Zalas uwodeben, vinaidan Toki daWimulia. Tokis daWimuloba sxeulze moqmedi T Zalis sididea. magaliTad, Tu Tokidan sxeulze moqmedi Zalaa T=50 n, maSin Tokis daWimulobac 50 n yofila. vuSvebT, rom Tokis masa nulovania (anu misi masa sxeulis masasTan SedarebiT umniSvneloa), Tavad Toki ki araelastiuria. am SemTxvevaSi Toki ubralod or sxeuls akavSirebs. is orive sxeulze erTi da imave T ZaliT moqmedebs, im SemTxvevaSic ki, rodesac sxeulebi da Toki (a) (b) (g)  sur. 5.10. a) Toki daWimulia. Tu misi masa umniSvneloa, Toki sxeulsa da xelze T ZaliT moqmedebs, im SemTxvevaSic ki, rodesac Toki uwono da xaxunisgan Tavisufal blokzea Semoxveuli, rogorc b) da g) SemTxvevebSi. 143 5.7 zogierTi konkretuli Zala aCqarebiT moZraobs an Toki umniSvnelo masis, xaxunisgan Tavisufal blokzea (suraTebi 5.10b da g) Semoxveuli. aseT bloks sxeulebTan SedarebiT umniSvnelo masa da xaxuni aqvs. Tu Toki blokze naxevradaa Semoxveuli (rogorc 5.10g suraTze), maSin Tokidan blokze moqmedi Zalis tolqmedia 2T. sakontrolo wertili 5. 5.10g suraTze Tokze dakidebuli sxeulis wonaa 75 n. amaze metia, naklebia Tu tolia T, rodesac sxeuli zeviT moZraobs a) Tanabari siCqariT, b) mzardi siCqariT da g) klebadi siCqariT? amocanis amoxsnis taqtika taqtika 6: drekadobis Zala oRond misi mimarTuleba ucvleli unda formula 5.14 sxeulze moqmedi drekadobis ZalisTvis  mxolod rodesac maSinaa gamosadegi, FN mimarTulia zeviT da sxeulis vertikaluri aCqareba nulis  tolia. FN -is darCes. magaliTad, 5.8a suraTze SesaZlebelia  FN ise gadavaadgiloT, rom misi saTave sxeulsa da magidas Soris sazRvars daemTxves.  FN -is arasworad gamosaxvis albaToba sxva yvelaze naklebia, Tu mis saTaves sxeulis mimarTulebebis da vertikaluri aCqarebis kideze an Tavad sxeulSi (rogorc naCvenebia) arsebobis dros ar viyenebT. mis nacvlad movaTavsebT. kidev ufro ukeTesia Tavisufali maSasadame, niutonis am meore formulas kanonisgan FN -is axali formula unda gamoviyvanoT.  SegviZlia suraTze FN gadavaadgiloT, amocanis amoxsnis nimuSi sxeulis grafikis ageba (sur. 5.8b), rodesac  FN -is saTave sxeulis amsaxvel wertils an gamosaxulebas emTxveva. 5.4 afrenis iluzia. adamianis mier vertikalis damajerebelia, amitom pilots eCveneba, rom SegrZneba vizualur niSnebsa da Sida yurSi TviTmfrinavi gembanidan mkveTrad maRla ganlagebul vestibularul sistemas efuZ aweuli cxviriT frindeba. specialuri wvrTnis neba. es sistema siTxeSi moTavsebuli pa gareSe piloti Seecdeba TviTmfrinavis cxviri tara bususebisgan Sedgeba. rodesac Tavi daxaros da pirdapir okeaneSi aRmoCndeba.  sworad giWiravs, bususebis mimarTuleba Fg davuSvaT, piloti Tanabar horizontalur mizidulobis Zalas emTxveva da sistema tvins aCqarebas ganicdis da afrenis 85 m/wm siCqares atyobinebs, rom Tavi sworad giWiravs. rodesac Tavs raime φ kuTxiT ukan gadaxri, bususebi 90 m manZilze aRwevs. risi tolia pilotisTvis moCvenebiTi daxris φ kuTxe? gadaixreba da sistema tvins gadaxris Taobaze amoxsna: saWiroa im wrfis φ daxrilobis acnobebs. es bususebi aseve gadaixreba, Tu gageba, romelzec mdebareobs hor Zalis moqmedebiT horizontalurad aCqardebi. j j veqtori. aris pilotze moqmedi vertikaluri  Fg am SemTxvevaSi tvinisTvis gagzavnili mcdari mizidulobis Zalisa da zurgidan moqmedi signali ambobs, rom Tavi ukanaa gadaxrili da horizontaluri veqtorTa jamis j= g + hor mimarTulebas povna hor SesaZlebelia, Zalebis jami. φ kuTxis Tu veqtorebs 5.11b emTxveva (sur. 5.11a). miuxedavad amisa, am mcdar suraTis mixedviT davalagebT da Semdeg signals goneba ugulebelyofs, Tu Tvalebi davwerT: xedavs, rom Tavi gadaxrili ar aris, rogorc, tgφ magaliTad, manqaniT aCqarebis SemTxvevaSi. aviamzidis gembanidan swrafad afrenili piloti vizualur niSnebs RamiT ver xedavs. gadaxris iluzia sakmaod Zlieri da an φ (5.15) 144 Zala da moZraoba Tavi 5 niutonis meore kanonis saSualebiT Sesa Zlebelia Fhor modebuli Zalisa da pilotis ax aCqarebis erTmaneTTan dakavSireba: Fhor=max,  a moCvenebiTi gadaxra φ sadac m pilotis masaa. vinaidan aCqareba hor Tanabaria, amitom ax-is sapovnelad SegviZlia x x φ 2.1 cxrilis formulebi gamoviyenoT. cnobilia sawyisi siCqare (ν0=0), saboloo g siCqare j g j (νx= 85m/wm) da gadaadgileba (x–x 0 =90 m), Sesaba hor 2 misad, viyenebT 2.16 formulas (ν2= ν 0 +2a(x–x0)) da vwerT: (85 m/wm)2 = 02 + 2ax(90 m) a x = 40,1 m/wm2. anu niutonis meore kanonis Tanaxmad, Fhor= m(40 sur. 5.11. a) marjvniv mimarTuli hor Zala pilotze afrenis drosaa modebuli. pilots eCveneba, rom Tavi wyvetili zolis gaswvriv gadaixara. b) veqtorebis jami j (= g + hor) vertikalisadmi φ kuTxiTaa dax rili. m/wm2). CavsvaT es Sedegi da Fg=m(9,8 m/wm2) 5.15 formulaSi, miviRebT: φ (pasuxi) gadaxris iluzia eqmneba anu eCveneba, rom TviTmfrinavis cxviri 76°-iT aiwia. am TviT iluziam pilots SeiZleba afrenis Semdeg mfrinavis aCqarebisas pilots 76°-iT ukan TviTmfrinavis cxviris 76°-iT daxra aiZulos. maSasadame, aviamzidis gembanze 5.8. niutonis mesame kanoni amboben, ori sxeuli urTierTqmedebs, rodesac es sxeulebi erTmaneTs ubiZgebs an eqaCeba, anu TiToeuli sxeuli meoreze ZaliT moqmedebs. magaliTad, roca B wigns C yuTs miayudeb (sur. 5.12a), wigni da yuTi urTierTqmedeben. yuTi wignze wigni yuTze _ BC horizontaluri ZaliT moqmedebs, xolo horizontaluri ZaliT. Zalebis wyvili naCvenebia 5.12b suraTze. CB niutonis mesame kanoni: ori sxeuli urTierTqmedebs ZalebiT, romlebic sididiT erTmaneTis tolia, xolo mimarTulebiT _ sawinaaRmdego. wignisa da yuTiTvis es kanoni skalarulad ase SeiZleba B wigni Caiweros: FBC = FCB C yuTi (Tanabari sidideebia) xolo veqtorulad: =– BC CB (Tanabari sidideebi d a sawinaaRmdego mimarTulebebi) “minus” niSani gvamcnobs, rom am veqtorebs urTierTsawinaaRmdego mimarTuleba aqvT. or sxeulze moqmed Zalebs mesame kanonis ZalTa wyvili SeiZleba vuwodoT. rodesac sur. 5.12. a) B wigni miyudebulia C yuTze. b) BC (yuTidan wignze) ori sxeuli erTmaneTze moqmedebs, am dros aucileblad da arsebobs mesame kanonis ZalTa wyvili. wignica da yuTic 5.12a Tanabari sidide da sawinaaRmdego suraTze uZravia, magram mesame kanoni maTi moZraobisa da mimarTuleba aqvs. CB (wignidan yuTze) Zalebs aCqarebis SemTxvevaSic imoqmedebda. kidev erTi magaliTis saxiT mesame kanonis ZalTa wyvilebi 5.13a suraTze naCvenebi nesvisTvis ganvixiloT, romelic miwaze dadgmul magidaze Zevs. nesvi magidaze da dedamiwaze moqmedebs (amjerad mocemuli gvaqvs sami sxeuli, romelTa urTierTqmedeba unda SeviswavloT). 5.9. niutonis kanonebis gamoyeneba Tavdapirvelad yuradReba gavamaxviloT nesvze moqmed Zalebze (sur. 5.13b). CT nesvi C aris magidis mxridan nesvze moqmedi drekadobis Zala, xolo CE 145 _ nesvze moqmedi dedamiwis magida T mizidulobis Zala. aris Tu ara es Zalebi mesame kanonis ZalTa wyvili? ara, vinaidan isini or sxeulze ki ara, aramed erTze – nesvze moqmedebs. mesame kanonis ZalTa wyvilis sapovnelad yuradReba unda dedamiwa E (a) gamavaxviloT ara nesvze, aramed nesvsa da sxva sxeuls Soris urTierTqmedebaze. nesvi-dedamiwis urTierTqmedebisas (sur. 5.13g) dedamiwa nesvze mizidulobis ZaliT moqmedebs, CE xolo nesvi dedamiwaze _ magidis perpendikularuli Zala EC mizidulobis ZaliT. aris Tu ara es Zalebi mesame kanoni ZalTa wyvili? aris, vinaidan isini or gravitaciuli Zala (b) sxeuls Soris moqmedebs. maSasadame, niutonis mesame kanonis Tanaxmad: nesvi CE =– EC (dedamiwa-nesvis urTierTqmedeba) magida-nesvis urTierTqmedebaSi magida nesvze kadobis ZaliT moqmedebs, xolo nesvi magidaze _ CT dre TC dre kadobis ZaliT (sur. 5.13d). es Zalebic mesame kanonis ZalTa dedamiwa wyvilia da, maSasadame:L CT =– TC (nesvi-magidis urTierTqmedeba) (g) sakontrolo wertili 6. davuSvaT, 5.13 suraTze naCvenebi nesvi da magida moTavsebulia liftis kabinaSi, romelic zeda mimarTulebiT Cqardeba. a) gaizrdeba, Semcirdeba Tu igive darCeba TC -s da CT -s sidideebi? b) aris Tu ara es ori Zala kvlav Tanabari sididisa da sawinaaRmdego mimarTulebis? g) gaizrdeba, Semcirdeba Tu igive darCeba CE -s da EC -s sidideebi? d) aris Tu ara es ori Zala kvlav Tanabari sididisa da sawinaaRmdego mimarTulebis? (d) sur. 5.13. a) nesvi miwaze dadgmuli magidaze devs. b) nesvze moqmedi Zalebia CT da CE . g) mesame kanonis ZalTa wyvili nesvisa da dedamiwis urTierTqmedebisas. d) mesame ka nonis ZalTa wyvili nesvisa da ma gidis urTierTqmedebisas. 5.9. niutonis kanonebis gamoyeneba am Tavis danarCeni nawili amocanis nimuSebisgan Sedgeba. dawvrilebiT unda Seiswavlo isini da daakvirde ara mxolod pasuxebs, aramed amocanis amoxsnis procedurebsac. gansakuTrebiT mniSvnelovania mdgomareobis gadatana Tavisufali sxeulis grafikze ise, rom niutonis kanonebis gamoyeneba SeiZlebodes. amocanis amoxsnis pirveli nimuSi ukiduresad detalurad aris gadmocemuli da kiTxva-pasuxis meTods iyenebs. 146 Zala da moZraoba Tavi 5 amocanis amoxsnis nimuSi 5.5 S sxeuli (mosriale sxeuli). is xaxunisgan 2. Toki H Camokidebul sxeuls igive T sididis ZaliT zeviT eqaCeba. es Zala H sxeulis Tavisufal horizontalur zedapirze mo Tavisufal vardnas ewinaaRmdegeba. Zraobs da meore H sxeulTan (Camokidebuli 3. dedamiwa S mosriale sxeuls qveviT eqaCeba  Fg S mizidulobis ZaliT, romlis sididea Mg. 5.14 suraTze naCvenebia M=3,3 kg masis sxeuli) xaxunisgan Tavisufal m=2,1 kg masis blokze Semoxveuli TokiTaa dakavSirebuli. Tokisa da blokis masebi sxeulebis masebTan SedarebiT umniSvneloa. H Camokidebuli sxe uli S mosriale sxeulis marjvniv aCqarebis dros qveviT Cadis. gamoTvale a) S sxeulis aCqareba, b) H sxeulis aCqareba da g) Tokis 4. dedamiwa H Camokidebul sxeuls qveviT eqa  Ceba Fg H mizidulobis ZaliT, romlis sididea mg.  5. magida S sxeulze FN drekadobis ZaliT moqmedebs. arsebobs daWimuloba. kidev erTi mniSvnelovani faqtori. Cven davuSviT, rom Toki ar iWimeba S mosriale sxeuli anu Tu drois garkveul monakveTSi H sxeuli 1 mm-iT qveviT Cadis, drois imave monakveTSi S sxeuli 1 mm-iT marjvniv gadaadgileba. es niSnavs, rom sxeulebi erTdroulad moZraobs xaxunisgan Tavisufali zedapiri da Tanabari a aCqareba aqvT. H Camoki debuli sxeuli SekiTxva: rogori klasifikacia mivaniWo amocanas? unda mimaniSnos Tu ara man fizikis romelime kanonze? sur. 5.14. M masis S sxeuli blokze Semoxveuli TokiT SekiTxva: raSi mdgomareobs amocanis arsi? mocemuli gaqvs ori sxeuli – mosriale da Camokidebuli, magram maT garda ganvixilavT romelic orive aCqarebebs. maSasadame, is unda migvaniSnebdes  m masis H sxeulTan aris dakavSirebuli. dedamiwasac, diax. es amocana Seexeba Zalebs, masebsa da sxeulze moqmedebs (dedamiwis gareSe saerTod araferi moxdeboda). rogorc 5.15 suraTzea naCvenebi, sxeulebze 5 Zala moqmedebs: 1. Toki S mosriale sxeuls T sididis ZaliT marjvniv ubiZgebs. niutonis meore kanonze, Fnet  = ma . SekiTxva: Tu am amocanaSi niutonis meore kanons gamoviyeneb, romel sxeuls unda mivusadago? Cveni yuradReba mipyrobilia or sxeulze – mosriale da Camokidebul sxeulebze. miuxedavad imisa, rom arcerTi ar aris wertili, mainc SegviZlia isini aseTebad ganvixiloT, vinaidan erTnairad moZraobs. maTi yvela nawili niutonis meore kanoni TiToeuli sxeulisTvis calke unda gamoviyenoT. sxeuli S SekiTxva: blokze ras ityviT? bloks wertilis saxiT ver warmovadgenT, vinaidan misi sxvadasxva nawilebi sxvada sxvanairad sxeuli H moZraobs. brunvis ganxilvis dros blokze ufro detalurad visaubrebT. amjerad blokis moqmedebas gamovricxavT da vuSvebT, rom misi masa sxeulebis masebTan SedarebiT umniSvneloa. blokis erTaderTi sur. 5.15. 5.14 suraTis or sxeulze moqmedi Zalebi. funqcia Tokis mimarTulebis Secvlaa. 147 5.9. niutonis kanonebis gamoyeneba  SekiTxva: keTili. rogor gamoviyeno Fnet formula mosriale sxeulisaTvis?  = ma M masis S sxeuli wertilis saxiT warmo vadginoT da masze moqmedi yvela Zala davxatoT, rogorc es 5.16a suraTzea naCvenebi. es iqneba Tavisufali sxeulis grafiki. Semdeg gavavloT RerZebi. x RerZi aviRoT magidis paralelurad, sxeulis moZraobis mimarTulebiT. y RerZis gaswvriv aCqareba gvaqvs, amitom viyenebT 5.16 formulis y nawils (Ft,y=May) da vwerT: T–F gH=ma y FgH-is nacvlad SegviZlia CavsvaT mg, xolo ay-is nacvlad –a (a uaryofiTia, vinaidan H Cqardeba y RerZis uaryofiTi mimarTulebiT). miviRebT: T–mg=–ma (5.18) axla gaiTvaliswine, rom 5.17 da 5.18 SekiTxva: gmadlob, magram jer ar giTqvams,  = ma mosriale sxeu formulebi orucnobian (T da a) gantolebaTa lisTvis. mxolod is axseni, Tu rogor unda gamovaklebT,amiT T-s gamovricxvaT. a-sTvis  rogor gamoviyenoFnet avago Tavisufali sxeulis grafiki.   marTalia. gamosaxulebaFnet = ma veqto rulia, amitom misi Cawera sami komponentis sistemas Seadgens. Tu am formulebs erTmaneTs amoxsna ki mogvcems: (5.19) saxiT SeiZleba: Ft,x=Max, Ft,y=May, Ft,z=Maz , (5.16) sadac Ft,x , Ft,y da Ft,z tolqmedi Zalis komponentebia sami RerZis gaswvriv. axla TiToeuli komponentis formula Sesabamisi mimarTulebisTvis gamoviyenoT. vinaidan S Camokidebuli sxeuli H mosriale sxeuli S sxeuli vertikalurad ar Cqardeba, amitom Ft,y=May gardaiqmneba ase: F N–F gS= 0 anu F N = F gS maSasadame, y mimarTulebiT drekadobis Zalis sidide mizidulobis Zalis sididis sur. 5.16. a) Tavisufali sxeulis grafiki, 5.14 suraTis S sxeulisTvis. b) Tavisufali sxeulis grafiki 5.14 suraTis H sxeulisTvis. tolia. am gverdis perpendikularuli z mimarTu lebiT Zala ar moqmedebs. x mimarTulebiT gvaqvs Zalis mxolod erTi komponenti – T. amitom, Ft,x=Max gardaiqmneba ase: T=Ma (5.17) es formula Seicavs or ucnobs – T-sa da a-s. ase rom, misi amoxsna jerjerobiT SeuZlebelia. miuxedavad amisa, gavixsenoT, rom Camokidebul sxeulze araferi ar gviTqvams. esec marTalia. rogor gamoviyeno  SekiTxva:  Fnet = ma formula Camokidebuli sxeulisa Tvis? iseve, rogorc S sxeulisTvis. avagoT Tavisufali sxeulis grafiki H sxeulisTvis, es 5.16b suraTzea naCvenebi. Semdeg  rogorc  Fnet = ma davSaloT komponentebad. amjerad 5.17 formulaSi am Sedegis CasmiT miviRebT: (5.20) ricxvebis Casmis Semdeg gveqneba: 2,1 kg = 3,3 kg + 2,1 kg (9,8 m/w2) = 3,8 m/w2 (pasuxi) da (3,3kg)(2,1kg) = 3,3 kg + 2,1 kg (9,8 m/w2) = 13 n (pasuxi) SekiTxva: axla ki amocana amoixsna, sworia? kargi SekiTxvaa. amocana amoxsnili ar aris, vidre Sedegebis sisworeSi ar davrwmundebiT. Tavdapirvelad darwmundi, rom 5.19 formulas tolobis orive Sexede. mxaris erTeulebi erTnairia da a aCqareba g-ze naklebia. es yovelTvis ase unda iyos, vinaidan 148 Tavi 5 Zala da moZraoba Camokidebuli sxeuli Tavisuflad ar vardeba. Camokidebul sxeulze moqmed mizidulobis kidebuli sxeuli zeviT aCqardeboda. pasuxebis Semowmeba aseve SeiZleba, Tu gansakuTrebul SemTxvevebs ganvixilavT, rodesac pasuxi winaswar viciT. martivi magaliTia g = 0-is Casma, TiTqos eqsperimenti kosmosSi mimdinareobdes. am SemTxvevaSi viciT, rom sxeulebi ar unda moZraobdes, Tokis boloebze Zalebi ar unda iyos modebuli da Tokis daWimulobac ar unda arsebobdes. miiReba Tu ara aseTi Sedegi Cveni formulebiT? diax, miiReba. Tu 5.19 da 5.20 formulebSi Casvam g = 0, miiReb a = 0 da T = 0. SegiZlia mosinjo sxva SemTxvevebic, Zalaze. magaliTad M = 0 da m→∞. Toki mas zeviT eqaCeba. axla 5.20 formulas Sexede, romelic SeiZleba Semdegi saxiTac daiweros: (5.21) am gamosaxulebaSi kargad Cans, rom erTeu lebis ganzomileba sworia, vinaidan T-sa da mg-s Zalis erTeulebi aqvs. 5.21 formulaSi agreTve Cans, rom Tokis daWimuloba yovelTvis mg-ze naklebia da, maSasadame, yovelTvis naklebia es damakmayofilebeli daskvnaa, vinaidan, T rom mg-ze meti yofiliyo, Camo amocanis amoxsnis nimuSi 5.6 5.17a suraTze M=15 kg masis B samSeneblo amoxsna: modiT, blokidan daviwyoT, vinaidan bloki mk masis K kvanZze TokiT kidia. Tavis is mxolod erT Tokzea Camokidebuli. 5.17b mxriv, es Toki ori sxva TokiT Werzea mibmuli. suraTze, Tokebis gamosaxulia blokze moqmedi Zalebi: masa umniSvneloa, iseve, rogorc maTze moqmedi mizidulobis Zala blokze moqmed ZalasTan SedarebiT. risi tolia Tokebis daWimuloba? Tavisufali sxeulis grafikze  Fg (Mg sididis) mizidulobis Zala da Tokidan blokze moqmedi T3 Zala. niutonis meore   kanonis saSualebiT F(net = ma ) SesaZlebelia am Zalebis aCqarebasTan dakavSireba. vinaidan orive Zala vertikaluria, amitom virCevT kanonis vertikaluri komponentis variants (Ft,y=may) da vwerT: T 3 –F g =Ma y . Fg-s nacvlad Mg-s CasmiT da ay aCqarebis nacvlad 0-is CasmiT miviRebT: T 3 –Mg=M·0=0 es niSnavs, rom blokze moqmedi ori Zala erTmaneTs awonasworebs. M-isa (15kg) da g-s Casma mogvcems: Semdeg T3=147 n ganvixiloT (pasuxi) 5.17g-ze naCvenebi Tavisufali sxeulis grafiki K kvanZisTvis, sadac kvanZze moqmedi mizidulobis Zala ugulebelyofilia. SegviZlia kvanZze moqmedi niutonis meore kanonis saSualebiT sami Zala  F(net = ma )kvanZis aCqarebas davukavSiroT da sur. 5.17. a) M masis bloki sami TokiT K kvanZzea Camokidebuli. b) Tavisufali sxeulis grafiki blo kisTvis. g) Tavisufali sxeulis grafiki kvanZisTvis. davweroT:     T1 + T2 + T3 = mk a k .  ak aCqarebis nacvlad 0-is Casma mogvcems:    , (5.22) T 1 + T2 + T3 = 0 149 5.9. niutonis kanonebis gamoyeneba rac niSnavs, rom kvanZze moqmedi sami Zala  erTmaneTs awonasworebs. T3 -is sidide da   kuTxe cnobilia, T1 -isa da T2 -is ki mxolod kuTxeebi viciT. ori ucnobis gamo veqtoruli kalkulatoris saSualebiT 5.22 formulis  152° aris T1 -sa da x-is dadebiT mimarTulebis Soris kuTxe). 5.22 formula y RerZisTvis Semdegi saxiT gadaiwereba: T 1y+T 2y+T 3y=0 amoxsna am ori veqtorisTvis SeuZlebelia. x da y komponentebis saSualebiT gadavweroT. x amis nacvlad, 5.22 formula an T1sin28°+T2sin47°–T3=0 T3-is adre miRebuli Sedegis Casma mogvcems: T1sin28°+T2sin47°–147 n = 0 RerZisTvis gveqneba: (5.24) formulebis 5.23 da 5.24 cal-calke amoxsna T1x+T2x+T3x=0 SeuZlebelia, vinaidan maTSi ori ucnobia, arsebuli monacemebis Casma mogvcems: –T 1 cos28°+T 2 cos47°+0=0 (5.23) (gamosaxulebis pirveli wevrisTvis alterna tivac gvaqvs. –T1cos28°-is nacvlad SegviZlia misi eqvivalenturi T1cos152° davweroT, sadac amocanis amoxsnis nimuSi magram es orucnobiani gantolebaTa sistemaa da misi amoxsna SeiZleba (CasmiT, formulebis mimatebiT an gamoklebiT, formulebis amom xsneli kalkulatoris saSualebiT). miviRebT: T1=104 n da T2=134 n (pasuxi) 5.7 5.8 suraTze Tokze damagrebuli 15 kg masis bloki θ=27° kuTxiT daxril xaxunisgan Tavi sufal zedapirze uZravadaa gaCerebuli.  T Zala da zedapiridan blokze moqmedi FN a) risi tolia Tokidan blokze moqmedi drekadobis Zala? amoxsna: es ori Zala da Tokze moqmedi mizidulobis  Fg Zala 5.18b suraTze naCveneb Tavisufali sxeulis grafikzea mocemuli. blokze mxolod es sami Zala moqmedebs. niu  tonis meore kanonis saSualebiT F(net  = ma ) Zalebs blokis aCqarebas vukavSirebT da vwerT:     T + F N + Fg = ma .  a aCqarebis nacvlad CavsvaT 0, miviRebT:    (5.25) T + F N + Fg = 0 formula gvamcnobs, rom Zalebi erTmaneTs awonasworeben. 5.25 formulaSi mocemuli gvaqvs ori ucnobi veqtori, amitom veqtoruli kalkulatoris saSualebiT misi amoxsna SeuZlebelia. maSa sur. 5.18. a) m masis A bloki Tokze uZravadaa mibmuli. b) Tavisufali sxeulis grafiki blokisTvis.  g) Fg -is x da y komponentebi. y komponenti mimarTulia zedapiris Suagulisken, xolo x komponenti – zeda piris gaswvriv, magram misi daxazva samkuTxedis gverdis saxiTac SeiZleba (orive gza naCvenebia). sadame, formula komponentebis saxiT unda gadavweroT. viyenebT koordinatTa sistemas, romlis x RerZi zedapiris paraleluria, rogorc es 5.18b suraTzea naCvenebi. am   SemTxvevaSi FN da T Zalebi RerZebs emTxveva, amitom maTi koordinatebis povna  advilia. Fg Zalis komponentebis sapovnelad, gaviTvaliswinoT, rom zedapiris daxris  F θ kuTxe g -sa da y RerZs Soris kuTxecaa (sur. 5.18g). maSin Fgx komponenti aris –Fgsinθ, romelic tolia –mgsinθ. Fgy komponentisTvis gveqneba –mgcosθ. 150 Zala da moZraoba Tavi 5 5.25 formula x komponentisaTvis Caiwereba aCqarebasTan dakavSireba x komponentebisTvis ase: niutonis meore kanoniT SeiZleba (Ft,x=max). T+0-mgsinθ=0, miviRebT: Fgx = ma – mg sin θ = ma, saidanac gamomdinareobs: T=mgsinθ =(15kg)(9,8m/wm2)(sin27°) =67 n anu rac mogvcems: (pasuxi) y RerZisTvis 5.25 formula mogvcems: (pasuxi) b) Toki gadavWeriT. aCqardeba Tu ara bloki zedapirze (5.26) a = – (9,8 m/wm2) (sin27°) = – 4,4 m/wm2 FN=mgcosθ =(15kg)(9,8m/wm2)(cos27°) =131 n ≈130 n daxril a = –g sin θ CvenTvis cnobili monacemebis Casma mogvcems: 0+F N –mgcosθ=0 anu Camocurebisas? Tu aCqardeba, risi tolia aCqareba?  amoxsna: Tokis gadaWriT blokze moqmedi T Zala gaqreba. y RerZis gaswvriv drekadobis Zala da Fgy komponenti kvlav erTmaneTs awonasworebs. asea Tu ise, x RerZis gaswvriv blokze mxolod Fgx komponenti moqmedebs. vinaidan es komponenti mimarTulia zedapiris gaswvriv (x RerZis mimarTulebiT), is zeda pirze blokis aCqarebas gamoiwvevs. Fgx-is a amocanis amoxsnis nimuSi (pasuxi) a aCqarebis sidide Tavisufali vardnis aCqarebaze naklebia, vinaidan a aCqarebas  mxolod Fg komponenti (zedapiris gaswvriv mimarTuli) iwvevs. sakontrolo wertili 7. suraTze  F horizontaluri Zala daxril zedapirze moTavsebul blokzea modebuli. a) zedapirisadmi perpendikularuli  F -is komponenti Fcosθ-is tolia Tu Fsinθ-isa? b)  F -is arseboba zedapiridan blokze moqmed dreka dobis Zalas zrdis Tu amci rebs? 5.8 5.19a suraTze m=72,2 kg masis mgzavri liftSi sasworze dgas. ganvixilavT sasworis Cvenebas liftis uZraobisas da zeviT an qveviT moZraobis dros. a) gamoTvale sasworis Cveneba kabinis nebismieri mimarTulebiT moZraobisas. amoxsna: mgzavri sasworis  Cveneba sasworidan mgzavrze moqmedi FN drekadobis Zalis toli iqneba. mgzavrze moqmedi erTaderTi sxva Zala mizidulobis  Fg Zalaa, rogorc es (a) (b) 5.19b suraTze mocemul Tavisufali sxeulis grafikzea naCvenebi. mgzavrze moqmedi Zalebi niutonis meore    kanonis saSualebiTF(net = ma ) mis a aCqarebas SegviZlia davukavSiroT. magram gavixsenoT, sur. 5.19. a) platformis saswori masze mdgomi mgzavris wonas an moCvenebiT wonas aCvenebs. b) Tavisufali sxeulis grafiki mgzavrisTvis. grafikze naCvenebia sasworidan mgzavrze moqmedi   drekadobis Zala FN da mizidulobis Zala Fg . rom am kanonis gamoyeneba mxolod aTvlis inerciuli sistemisTvis SeiZleba. Tu kabina Cqardeba, maSasadame, is aRar aris aTvlis inerciuli sistema. ase rom, aTvlis inerciul sistemad virCevT dedamiwas da mgzavris aCqarebasac mis mimarT vzomavT. vinaidan mgzavrze moqmedi orive Zala da aCqareba mimarTulia vertikalurad, y RerZis gaswvriv, rogorc es 5.19b suraTzea naCvenebi, amitom SegviZlia gamoviyenoT niutonis 5.9. niutonis kanonebis gamoyeneba 151 meore kanoni y komponentisTvis (Ft,y=may) da Cveneba davweroT: moCvenebiT wonas zomavs, vinaidan gazomva F N = F g + ma (5.27) formula gvauwyebs, rom sasworis Cveneba anu FN – vertikalur aCqarebazea damokidebuli. Fg-s nacvlad mg-s Casma mogvcems: FN = m(g + a) wonaze metia. saswori aTvlis arainerciul sistemaSi xdeba. qveda F N – F g = ma anu mgzavris (pasuxi) (5.28) nebismieri a aCqarebisTvis. b) risi tolia sasworis Cveneba, Tu kabina uZravia an 0,5 m/wm Tanabari siCqariT moZraobs zeviT? amoxsna: nebismieri Tanabari siCqarisTvis (maT Soris, nulisTvisac) mgzavris a aCqareba nulis tolia. 5.28 formulaSi am da sxva mocemuli sidideebis CasmiT miviRebT: FN=(72,2 kg)(9,8 m/wm2+0)=708 n (pasuxi) mimarTulebiT aCqarebisas (zeviT mimarTuli siCqaris Semcirebisas an qveviT mimarTuli siCqaris zrdisas) sasworis Cveneba mgzavris wonaze naklebia. d) g) SemTxvevaSi zeda mimarTulebiT aCqa rebisas risi tolia mgzavrze moqmedi tol qmedi Zala Ft da risi tolia liftis kabinis gazomili ap,kab aCqareba? aris Tu ara  sistemaSi  Fnet = map,kab? amoxsna: mgzavrze moqmedi Fg mizidulobis Zalis sidide mgzavris moZraobaze an kabinaze ar aris damokidebuli. ase rom, b) nawilis Tanaxmad, Fg=708 n. g) nawilis Tanaxmad, zeviT mimarTuli aCqarebisas mgzavrze moqmedi drekadobis Zala FN sasworis Cvenebis (939n) es aris mgzavris wona, romelic mgzavrze tolia. maSasadame, mgzavrze moqmedi Zalebis moqmedi Fg mizidulobis Zalis tolia. tolqmedi zeviT mimarTuli aCqarebisas aris: Ft= FN–Fg = 939 n – 708 n = 231 n g) risi tolia sasworis Cveneba, Tu kabina (pasuxi) 3,2 m/wm aCqarebiT moZraobs zeviT da 3,2 m/wm - magram mgzavris ap,kab aCqareba liftis aTvlis iT aCqarebiT moZraobs qveviT? sistemis mimarT nulis tolia. maSasadame, 2 amoxsna: 2 a=3,2 m/wm -isTvis formula 5.28 2 gvaZlevs: FN = (72,2 kg)(9,8 m/wm2+3,2 m/wm2)=939 n (pasuxi) a = –3,2 m/wm2-isTvis formula 5.28 gvaZlevs: FN = (72,2 kg)(9,8 m/wm2–3,2 m/wm2) = 477 n (pasuxi) aCqarebuli liftis arainerciul sistemaSi Ft ar aris map,kab-is toli da niutonis meore kanonis gamoyeneba ar SeiZleba. sakontrolo wertili 8. amocanis amoxsnis am nimuSSi rogori iqneba sasworis zeda mimarTulebiT aCqarebisas (rodesac Cveneba, Tu liftis bagiri gawydeba da kabinis zeviT mimarTuli siCqare izrdeba an kabina Tavisuflad vardnas daiwyebs? qveviT mimarTuli siCqare mcirdeba) sasworis amocanis amoxsnis nimuSi 5.20a suraTze 20 n sididis mudmivi horizontaluri Zala mA = 4 kg masis A blokze moqmedebs. A bloki mB = 6 kg masis B bloks awveba. blokebi x RerZis gaswvriv, xaxunisgan Tavisufal zedapirze moZraoben. zuli Secdomis Semcvel amoxsnas, Semdeg amoxsnas. da, bolos, x (a) A (b)  FAB x  B FBA x (g) Tanabari horizontaluri Zala moqmedebs. A bloki B bloks awveba. amoxsna: Tavdapirvelad ganvixilavT serio amoxsnas B A sur. 5.20. a) A blokze a) risi tolia blokebis aCqareba? uperspeqtivo 5.9 swor B blokidan ori horizontaluri Zala, b) A blokze  da F AB moqmedebs. A blokidan mxolod erTi horizontaluri g) B blokze  Zala, F BA moqmedebs. 152 Tavi 5 Zala da moZraoba Zala pirdapir A seriozuli Secdoma: kidev erTxel x RerZisTvis vwerT: F hor = (m A + m B)a blokze moqmedebs, amitom niutonis meore sworad SevusabameT mA + mB mTliani kanonis saSualebiT am Zalas A blokis a axla aCqarebas vukavSirebT. moZraoba x RerZis masis gaswvriv xdeba, amitom viyenebT kanons x monacemebis Casma mogvcems: komponentebisTvis (F t,x=ma x) da vwerT: a= F hor = m Aa magram es formula arasworia, vinaidan A blokze moqmedi erTaderTi Zala ar aris.  masze B blokidan F AB Zalac moqmedebs (sur. 5.20b). uperspeqtivo amoxsna: axla  F AB Zalac CavTva loT da kvlav x RerZisTvis davweroT:  F AB-s mimarTulebis aRsa niSnavad viyenebT). vinaidan FAB meore ucnobia, amitom a-s gansazRvra SeuZlebelia. swori amoxsna: a-s amoxsna da cnobili Fhor 20 n = = 2 m/w2 (pasuxi) mA + mB 4 kg + 6 kg maSasadame, sistemisa da TiToeuli blokis aCqarebas x RerZis dadebiTi mimarTuleba aqvs da misi sididea 2 m/wm2. b) risi tolia A blokidan B blokze moqmedi F BA horizontaluri Zala (sur. 5.20g)? amoxsna: B blokze moqmedi Zalebis tolqmedis Fhor – FAB = mAa („minus“ niSans sistemas. -is mimarTulebis gamo es ori bloki mWidrod dakavSirebul sistemas qmnis. sistemaze moqmedi tolqmedi Zalis aCqarebasTan dakavSireba niutonis meore kanonis saSualebiTaa SesaZlebeli. maSasadame, aCqarebasTan dakavSireba niutonis meore kanonis saSualebiT SegviZlia. kanons x komponentebisTvis vwerT: FBA = mBa, arsebuli monacemebis Casma mogvcems: FBA = (6 kg)(2 m/wm2) = 12 n (pasuxi)  maSasadame, F BA Zala x RerZis dadebiTi mimar Tuleba aqvs da misi sididea 12 n. mimoxilva da Sejameba niutonis meqanika. sxeulis siCqare SeiZleba samarTliania, Seicvalos ewodeba. (sxeuls SeuZlia aCqardes), rodesac masze sxva sxeulebidan erTi an meti Zala moqmedebs. niutonis meqanika aCqarebebsa da Zalebs erTmaneTTan akavSirebs. Zala. Zala veqtoruli misi romelsac is 1 kg masis sxeuls mianiWebs. im Zalis sidide, romelic aseT sxeuls 1 m/wm2 aCqarebas aniWebs, aris 1 n. Zalis mimarTuleba mis mier miniWebuli aCqarebis mimarTulebas Zalebze moqmedeba veqtoruli algebris Tanaxmad xdeba. sxeulze moqmedi Zalebis tolqmedi sxeulze moqmedi yvela Zalis veqtoruli jamia. niutonis pirveli kanoni. Tu sxeulze an wrfivad, Tanabari siCqariT moZraobs. sistemas, inerciuli romelSic arainerciul sistema sistemebSi niutonis kanonebi ar moqmedebs. masa. sxeulis masa sxeulis Sinagani maxasia gamomwvev ZalasTan akavSirebs. masa skalaruli sididea. niutonis meore kanoni. m masis sxeulze moqmedi   tolqmedi FnetZala = ma aCqarebas Semdeg nairad ukavSirdeba:   Fnet = ma am formulis komponenturi versiebia: Ft, x=max, Ft, y=may da Ft, z=maz (5.1) (5.2) SI sistemis erTeulebSi, niutonis meore tolqmedi Zala ar moqmedebs, sxeuli uZravia aTvlis aTvlis inerciuli Tebelia da sxeulis aCqarebas am aCqarebis sididea. ricxobrivi sidide ganisazRvreba aCqarebiT, emTxveva. aTvlis kanonis Tanaxmad, 1 n = 1 kg · 1 m/wm2 Tavisufali sxeulis grafiki aris gamar sistemebi. aTvlis tivebuli gamosaxuleba, romelzec mxolod niutonis kanonebi erTi sxeulia ganxiluli. es sxeuli wertilis 5.9. niutonis kanonebis gamoyeneba an marTkuTxedis saxiTaa warmodgenili. aseT drekadobis grafikze mocemulia sxeulze moqmedi gare perpendikularulia. Zalebi, xolo koordinatTa sistemis RerZebi iseTi mimarTulebiTaa gavlebuli, rom Zala zedapiris  f xaxunis Zala sxeulze maSin moqmedebs, rodesac is zedapirze srialebs an srials amocanis amoxsna gaamartivos. cdilobs. zogierTi konkretuli Zala. sxeulze moqme paraleluria  di Fg mizidulobis Zala is Zalaa, romliTac yovelTvis 153 es Zala da yovelTvis zedapiris mimarTulia sawinaaRmdegod. moZraobis xaxunisgan Tavisufal mas sxva sxeuli miizidavs. am wignSi ganxilul zedapirze xaxunis Zala umniSvneloa. umetes situaciebSi es sxva sxeuli aris rodesac Toki daWimulia, misi TiToeuli dedamiwa an raime sxva astronomiuli sxeuli. bolo dedamiwis mimarTulia SemTxvevaSi mizidulobis Zala sxeuls miizidavs. Tokis gaswvriv es mizidva da sxeulis mimarTulia qveviT, miwisken, romelic aTvlis Tokze mibmis wertilSi iwyeba. masis armqone inerciul sistemad miiCneva. am daSvebiT, Tokis (umniSvnelo masis Tokis) TiToeul  Fg -s sididea: Fg=mg (5.8) sadac m sxeulis masaa, xolo g _ Tavisufali vardnis aCqareba. sxeulis W wona zeviT mimarTuli im Zalis sididea, romelic sxeulze moqmed mizidulobis Zalas awonasworebs. sxeulis wonasa da masas Soris Semdegi kavSiria: W=mg awveba, am (5.12) sxeulze masis armqone, xaxunisgan Tavisufal blokze (romelsac umniSvnelo masa aqvs da umniSvnelo xaxuni axasiaTebs). niutonis mesame kanoni. Tu C sxeulidan B F BC Zala sxeulze moqmedebs, maSin arsebobs  B-dan C sxeulze moqmedi  FN drekadobis ZaliT zedapiri, romelsac sxeuli boloSi am Zalas erTi da igive T sidide aqvs, im SemTxvevaSic ki, Tu Toki Semoxveulia F CB Zalac. es Zalebi sididiT tolia, xolo mimarTulebiT   F BC = – F CB sawinaaRmdego: moqmedebs. SekiTxvebi 1. 5.21 suraTze naCvenebia oTxi mdgomareoba, Rvre, 5.22 suraT rodesac ze naCveneb Tavi Zalebi xaxunisgan Tavisufal zedapirze mdebare blokze moqmedeben. Tu sufali Zalis sidideebi sworadaa arCeuli, maSin grafikze romel mdgomareobaSi iqneba SesaZlebeli li veqtori war a) blokis uZravad yofna da b) blokis moZraoba moadgens Tanabari siCqariT? sxeulis rome sauke Tesod a) F1 -s da  b) F2 -s? risi tolia tolqmedi Zalis komponenti sur. 5.22. SekiTxva 2. g) x RerZis gaswvriv da d) y RerZis gaswvriv? romeli meoTxedisken mianiSnebs e) tolqmedi Zalis veqtori da v) namcxvris aCqarebis veqtori? 3. sur. 5.21. SekiTxva 1.  2. ori horizontaluri Zala: F1 =(3n) î –(4n) ˆj  5.23  suraTze F1 da F2 Zalebi  da F2 =–(1n) î –(2n) ˆj , xaxunisgan Tavisufal moqmedebs magidaze moTavsebul namcxvarze moqmedebs. xunisgan kalkulatoris gamoyenebis gareSe gansaz fal ze, yuT romelic xa Tavisu zedapirze sur. 5.23. SekiTxva 3. 154 Zala da moZraoba Tavi 5 mudmivi siCqariT misrialebs.  F1 -is sidi dis cvlilebis gareSe unda SevamciroT θ saukeTesod 5.24 suraTze da me-5 SekiTxvaSi mocemul mdgomareobebs? kuTxe. yuTis Tanabari siCqariT srialis  SesanarCuneblad, F2 -is sidide unda gavzar doT, SevamciroT Tu SevinarCunoT?  4. t=0 momentSi mudmivi sididis F Zala kosmosSi x RerZis dadebiTi mimarTulebiT moZrav qvaze moqmedebas iwyebs. qva am RerZis gaswvriv moZraobas ganagrZobs. a) rodesac t>0, romeli x(t) funqcia SeiZleba asaxavdes qvis mdebareobas: 1) x=4t–3, 2) x=–4t 2 +6t–3, 3) x=4t 2 +6t–3 ? b) romeli funqciisTvisaa  F qvis moZraobis sawyisi mimarTulebis sawinaaRmdegod mimarTuli? 5. 5.24 suraTze sxeulis grafiki naCvenebia oTx Tavisufali mdgomareobaSi, rodesac xaxunisgan Tavisufal zedapirze myof sxeulze ramdenime Zala moqmedebs.  romel mdgomareobaSi eqneba sxeulis a sur. 5.25. SekiTxva 6. 7. 5.26 suraTze oTx mdgomareobaSi naCvenebia erTi da igive yuTi, romelzec horizontaluri aCqarebas a) x komponenti da b) y komponenti? Zalebi moqmedebs. daalage mdgomareobebi g) yuTis aCqarebis klebadobis mixedviT. TiToeul mdgomarobaSi aCvene  a -s mimarTuleba (amisTvis cotaodeni gamoTvlebi dagWirdeba). sur. 5.26. SekiTxva 7. 8. 5.27 suraTze naCvenebia oTxi blokisgan Semdgari “matarebeli”, romelsac xaxunisgan  F Zala miaTrevs. ra  mTliani masa Cqardeba marjvniv a) F ZaliT, Tavisufal zedapirze b) me-3 TokiT da g) TokiT 1? d) daalage blokebi aCqarebis klebadobis mixedviT. e) daalage Tokebi daWimulobis klebadobis mixedviT (es SekiTxva 42-e da 45-e amocanebisTvis mogamzadebs). sur. 5.24. SekiTxva 5. 6. 5.25 suraTze naCvenebia siCqaris νx(t) komponentis sami grafiki da siCqaris νy(t) komponentis sami grafiki. masStabebi daculi ar aris. romeli grafiki Seesabameba sur. 5.27. SekiTxva 8. 9. zedapirze mdebare m masis blokze verti kaluri  F Zala moqmedebs. ra mouva zeda 155 piridan blokze moqmed  FN drekadobis Zalas,  Tu F Zalis sidide izrdeba da F mimarTulia  F 21 ZaliT da g) meoredan mesameze moqmedi F 32 ZaliT? d) daalage a) qveviT da b) zeviT? blokebi SekiTxva 43-e amocanisTvis mogamzadebs). zedapirze mdebare 12. planeri haerSi x RerZis gaswvriv moZra moqmedi F sididis Zalis marTuleba. mimar klebadobis mixedviT.  F , F 21 da F 32 Zalebi sididis klebadobis mixedviT, dididan mcireze (es naCvenebia daxril oTxi SesaZlo mi aCqarebis   e) daalage 10. 5.28 suraTze blokze meoreze moqmedi obs. 5.30 suraTze naCvenebia planeris siCqaris νx komponenti, rogorc t drois funqcia, rodesac planerze Fx komponentis mqone Zala sur. 5.28. SekiTxva 10. Tulebebi horizontaluri an vertikaluria moqmedebs. TiToeuli drois SualedisTvis (a da b variantebSi Zala blokis asawevad gansazRvre, dadebiTia Fx, uaryofiTi Tu nu sakmarisi ar zedapiridan aris). blokze daalage variantebi moqmedi drekadobis lis tolia. agreTve gansazRvre, Fx mudmivia, izrdeba Tu mcirdeba. Zalis klebadobis mixedviT. 11. 5.29 suraTze naCvenebia xaxunis gan Tavisufal ze dapirze mdebare sami bloki, rom lebzec  F hori sur. 5.29. SekiTxva 11. zontaluri Zala  moqmedebs. ra masa aCqardeba marjvniv a) sur. 5.30. SekiTxva 12. F ZaliT, b) pirveli blokidan amocanebi SSM www amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday ILW Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley. com/college/halliday. • – ••• da b) y komponenti? g) risi tolia Zalebis amoxsnebi mocemulia Student Solutions Manual-Si. wertilebis raodenoba amocanis sirTules tolqmedi erTeulovan-veqtorul CanawerSi? •3. 2 kg masis xorcis saWrel ficarze ori horizontaluri Zala moqmedebs. ficari gluv  magidaze, xy sibrtyeze Zevs. erTi Zalaa F1 = =(3n) î +(4n) ˆj . moZebne ficris aCqareba erTeu gviCvenebs lovan-veqtorul CanawerSi, rodesac meore   Zalaa a) F2 =(–3n) î +(–4n) ˆj , b) F2 =(–3n) î +(4n) ˆj  da g) F2 =(3n) î +(–4n) ˆj . qveTavi 5.6. niutonis meore kanoni •1. 3 kg masis sxeulze ori horizontaluri Zala moqmedebs. erTi Zalis sididea 9 n, xolo •4. sami kosmonavti, romelTac zurgze Zra vebi aqvT damagrebuli, 120 kg masis aste mimarTuleba _ aRmosavleTiT. meore Zalis roids kosmosuri xo sididea 8 n, mimarTuleba _ 62° Crdilo- maldisken ubiZgebs. dasavleTisken. risi tolia sxeulis aCqareba? 5.31 suraTze naCve •2. 1 kg masis sxeulis aCqarebaa 2m/wm da mimarTulia x RerZis dadebiTi mimar Tulebisadmi 20°kuTxiT. risi tolia sxeulze moqmedi Zalebis tolqmedis a) x komponenti nebia asteroidze moq 2 medi Zalebi, F1=32 n, F2=55 n, F3=41 n. θ1=30° da θ3=60°. risi tolia sur. 5.31. amocana 4. 156 Zala da moZraoba Tavi 5 asteroidis aCqareba a) erTeulovan veqtorul CanawerSi? risi tolia aCqarebis b) sidide da g) x RerZis dadebiTi mimarTulebisadmi kuTxe? ••5. 2 kg masis yuTze ori Zala moqmedebs, magram 5.32 suraTze mxolod erTia naCve nebi. F1 = 20 n-sTvis a=12 m/wm2 da θ = 30°. moZebne meore Zalis a) sur. 5.32. amocana 5. erTeulovan- veqtoruli Canaweri, b) sidide da g) x RerZis dadebiTi mimarTulebisadmi kuTxe. SSM ••6. 2 kg masis sxeulze sami Zala moqmedebs,  romlebic mas a =–(8m/wm2) î +(6m/wm2) ˆj aniWebs.   F2 =(–12n) î +(8n) ˆj . da F1 =(30n) î +(16n) ˆj moZebne mesame Zala. sur. 5.34. amocana 9. meore Zexvi gvaqvs da sistema uZravia. risi tolia sasworis Cveneba? SSM. ••7. ori Zalis moqmedebis Sedegad wertili  Tanabari ν =(3m/wm) î –(4m/wm) ˆj siCqariT mo  Zraobs. erT-erTi Zalaa F1 = (2 n ) î +(–6 n ) ˆj . risi tolia meore Zala? •10. 3 n wonis bloki horizontalur zeda pirze uZrav mdgomareobaSia. blokze 1 n ze viT mimarTuli Zala moqmedebs, romelic gamowveulia blokze vertikalurad mibmuli •••8. 4 kg masis disko TokiT. risi tolia blokidan horizontalur yinulze misrialebs. zedapirze   masze F1 da F2 ho rizontaluri Zale bi yinu moqmedebs. lze gavlebulia xy koordinatTa sistema.  F1 -is moqmedi Zalis a) sidide da b) mimarTuleba? •11. 5.35 suraTze oTxi disko sur. 5.33. amocana 8. mimarTuleba emTxveva x RerZis dadebiT mimarTulebas,  xolo misi sididea 7 n. F2 Zalis sididea 9 n. 5.33 suraTze naCvenebia diskos siCqaris νx komponenti, rogorc t drois funqcia. risi   tolia kuTxe F1 da F2 Zalebis mimarTulebebs Soris? qveTavi 5.7. zogierTi konkretuli Zala •9. a) 11 kg masis Zexvi TokiT zambarian sasworzea Camokidebuli, TviT saswori ki Werze meore TokiT kidia (sur. 5.34a). risi tolia sasworis Cveneba wonis erTeulebSi? b) 5.34b suraTze Zexvze da blokze Semoxveulia Toki. sasworis meore mxare kedelze TokiTaa mimagrebuli. risi tolia sasworis Cveneba? g) 5.34g suraTze kedelis nacvlad 11 kg masis Tokze kidia. Tokis zeda, yvelaze grZeli nawili xaxu nisgan Tavisufal blokzea Semoxveuli da kedelze 98 n Za liT moqmedebs. mok le nawilebis daWi T1=58,8n, sur. 5.35. amocana 11. T2=49n da T3=9,8n. risi tolia a) A diskos masa, b) B diskos masa, g) C diskos masa da d) D diskos masa? mulobebia ••12. zogierT mwers Txeli totis qveda zedapirze SeuZlia siaruli. davuSvaT, m masis mqone aseTi mweri horizontalur totzea Camokidebuli θ=40° kuTxiT (sur. 5.36). fexi misi eqvsive Tanabradaa sur. 5.36. amocana 12. 157 horizontaluria. a) risi tolia TiToeuli •17. raketa TanabaraCqarebuli moZraobisas 1,8 wm-is ganmavlobaSi 1600 km/sT siCqares wvivis Zvlis daWimulobis fardoba mweris aRwevs. risi tolia am dros tolqmedi Zala? wonasTan? b) Tu mweri fexebs gaasworebs, SSM. daWimuli da sxeulTan axlos fexis nawilebi Zvlebis daWimuloba gaizrdeba, Semcirdeba •18. daWimulobas, romlis drosac ankesis Tu igive darCeba? Zua wydeba, ankesis `Zalas~ uwodeben. ra qveTavi 5.9. niutonis kanonebis gamoyeneba •13. davubrundeT 5.18 suraTs. blokis masa iyos 8,5 kg, xolo θ kuTxe 30°. moZebne a) Tokis minimaluri `Zala~ sWirdeba ankess, romelmac 85 n wonis oraguli 11 sm-is manZilze unda SeaCeros, Tu Tevzis sawyisi siCqarea 2,8 m/wm? CaTvale, rom Seneleba xdeba Tanabrad. •19. `mzis iaxta~ kosmosuri xomaldia didi daWimuloba da b) blokze moqmedi drekadobis ialqniT, romelsac mzis sinaTle “awveba”. Zala. g) Tu Toks gadavWriT, moZebne blokis miuxedavad imisa, rom aseTi “miwola” Zalian aCqarebis sidide. SSM WWW. mcirea, is sakmarisia kosmosuri xomaldis •14. 45 kg masis qali gayinuli tbis zedapirze nela aRmosavleTisken misrialebs da dasavleTisken mimarTulebiT. davuSvaT, kosmosuri xomal asamoZraveblad mzis sawinaaRmdego mosriale 90 kg masis kacs ejaxeba. Sejaxebis dis masaa 900 kg, xolo mzis “miwola” _ 20 n. dros kaci qalze dasavleTisken mimarTuli, a) risi tolia aCqarebis sidide? Tu xomaldi maqsimum 180 n ZaliT moqmedebs. risi tolia moZraobas uZravi mdgomareobidan iwyebs, qalidan kacze moqmedi maqsimaluri Zalis maSin b) ra manZils gaivlis 1 dReSi da g) ra a) sidide da b) mimarTuleba? risi tolia siswrafiT imoZravebs? qalis maqsimaluri aCqarebis g) sidide da d) •20. 53 km/sT siCqariT moZravi manqana xidis mimarTuleba? risi tolia kacis maqsimaluri aCqarebis e) sidide da v) mimarTuleba? moajirs daejaxa. manqanis mgzavrma gzis mimarT 65 sm-iT win waiwia da Semdeg gaxsnilma  •15. horizontaluri Fa Zala 2kg masis yuTs damcavma baliSma SeaCera. ra sididis Zala xaxunisgan Tavisufal zedapirze ubiZgebs. (davuSvaT, zedapirze gavlebulia xy koordinatTa sistema. 5.37 suraTze mocemulia yuTis x da mudmivi) moqmedebs mgzavris sxeulis zeda nawilze, romlis masaa 41 kg? •21. 712 n wonis mexanZre vertikalur boZze 3 m/wm2 aCqarebiT Camocurda. risi tolia y komponentebis t droze damokidebulebis  grafiki. risi tolia Fa -s a) sidide da boZidan mexanZreze moqmedi vertikaluri b) mimarTuleba? Zalis a) sidide da b) mimarTuleba (zeviTaa es Zala mimarTuli Tu qveviT)? ILW. •22. 1,3×104 n wonis manqana 40 km/sT siCqariT moZraobs, uecrad muxruWdeba da 15 m-is gavlis Semdeg Cerdeba. davuSvaT, manqanis SemaCerebeli Zala mudmivia. moZebne a) am sur. 5.37. amocana 15. •16. 5.38 suraTze 2 kg masis yuTze moqmedi ori horizontaluri Zalidan mxolod erTia (F1=20n) naCvenebi. yuTi x RerZis gaswvriv moZraobs. ipove meore Zala erTeulovanveqtorul CanawerSi, Tu yuTis ax aCqarebaa: a) 10 m/wm2, b) 20 m/wm2, saWiro dro. Tu sawyisi siCqare orjer metia da manqanaze damuxruWebisas igive Zala moqmedebs, maSin ra koeficientebze unda gamravldes g) SeCerebis manZili da d) SeCerebis dro? •23. 1,2×107 m/wm siCqaris eleqtroni hori zontalurad moZraobs, masze ki am dros g) 0, d) –10 m/wm2 da e) –20 m/wm2. Zalis sidide da b) siCqaris cvlilebisTvis 4,5×10-16 n vertikalurad mimarTuli mudmivi sur. 5.38. amocana 16. 158 Zala da moZraoba Tavi 5 Zala moqmedebs. eleqtronis masaa 9,11×10-31 kg. horizontaluri ZaliT. risi tolia a) cigis vertikalurad ra manZiliT Seicvlis kurss aCqareba da b) gogonas aCqareba? g) gogonas eleqtroni 30 mm-is horizontalurad gada sawyisi mdebareobidan ra manZilze Sexvdebian adgilebis Semdeg? SSM. gogona da ciga erTmaneTs? •24. 5.39 suraTze m=100 kg masis xis ••28. 40 kg masis moTxilamure xaxunisgan Tavisufal, horizontisadmi 10° kuTxiT yuTs daxril ferdobze eSveba. davuSvaT, moTxi xaxunisgan T a v i s u f a l lamure daxril zedapirze mimarTulebiT.  F (θ=30°) sur. 5.39. amocana 24. horizontaluri ZaliT awvebian.  risi tolia a) F Zalis sidide da b) zedapiridan moZraobs x RerZis moTxilamureze uaryofiTi moqmedebs Fx komponentis mqone qaris Zala. risi tolia Fx, Tu moTxilamuris siCqare a) mudmivia, b) izrdeba 1 m/wm2 aCqarebiT da b) izrdeba 2 m/wm2 aCqarebiT? kldidan Camokida lianaze, romelic xidan ••29. bloks xaxunisgan Tavisufal daxril zedapirze ν0=3,5 m/wm sawyisi siCqariT zeviT ubiZgeben. zedapiris daxris kuTxea θ=32°. eSveba vertikalisadmi 22° kuTxiT. davuSvaT, a) ra manZils gaivlis bloki zedapirze? x RerZi horizontaluria, xolo y RerZi _ b) ra dro dasWirdeba am manZilis gasavlelad? vertikaluri. tarzanis lianaze dakidebis g) rogori iqneba misi siCqare sawyis wertilSi momentSi lianis daWimulobaa 760n. risi dabrunebisas? SSM WWW tolia a) lianidan tarzanze moqmedi Zala ••30. 5 kg masis erTeulovan-veqtorul CanawerSi da risi yuTs tolia tolqmedi Zalis (b) erTeulovan- Tavisufal, hori veqtoruli Canaweri, g) sidide da d) kuTxe zontisadmi θ yuTze moqmedi Zalis sidide? •25. 820 n wonis tarzani 20 m simaRlis x RerZis dadebiTi mimarTulebisadmi. risi xaxunisgan ku TxiT daxril ze tolia tarzanis aCqarebis e) sidide da dapirze v) kuTxe? ubiZgeben. ••26. 5.40 suraTze suraTze naCvenebia yuTis siCqaris νx kom zeviT 5.41 sur. 5.41. amocana 30. naCvenebia xaxunis ponenti, rogorc t drois funqcia, roca x gan Tavisufal ma RerZi zustad daxrili sibrtyis gaswvrivaa gidaze mimarTuli. risi tolia zedapiridan yuTze dadebuli moqmedi drekadobis Zala? 0,025kg masis li moni da masze mo qmedi samidan ori sur. 5.40. amocana 26. horizontaluri  Zala. F1 Zalis sididea 6n, xolo kuTxe θ1=30°.  F2 Zalis sididea 7n, xolo kuTxe θ2=30°. risi tolia mesame Zala erTeulovan-veqtorul CanawerSi, Tu limoni a) uZravia, b) moZraobs  mudmivi siCqariT ν =(13 î –14 ˆj )m/wm da g) mo  Zraobs cvladi siCqariT ν =(13t î –14t ˆj ) m/wm, ••31. 3×10-4 kg masis sfero Tokzea Camoki debuli. msubuqi horizontaluri qari sferos mimarTulebis ubiZgebs da Toks vertikalisadmi 37° kuTxiT gadaxris. moZebne a) biZgis Zala da b) Tokis daWimuloba. ILW. ••32. 50 kg masis moTxilamure sabuqsiro Toks eWideba da horizontisadmi 8° kuT xiT daxril, xaxunisgan Tavisufali saTxi sadac t droa? lamuro ferdobze adis. risi tolia To ••27. 40kg masis gogona da 8,4 kg ciga gayinuli tbis zedapirze arian, erTmaneTisgan 15 m a) moTxilamuris siCqare mudmivia da ν = 2 manZilze da erTmaneTTan umniSvnelo masis Toki akavSirebT. gogona Tokze moqmedebs 5,2n kidan moTxilamureze moqmedi F Zala, Tu m/wm, b) ν=2 m/wm da siCqare izrdeba 0,1 m/wm2 aCqarebiT? 159 ••33. liftisa da misi tvirTis saerTo masaa 1600 kg. gamoTvale liftis bagiris moqmedi daWimuloba, rodesac qvemoT moZravi lifti, xeze moqmedi Zala. romlis sawyisi siCqarea 12 m/wm, 42 m-is mudmivi Semdeg gamoTvale aCqarebiT gavlis Semdeg Cerdeba. e) ••34. lampa 2,4 m/wm2 SenelebiT zeviT moZrav adamianisgan bolo liftSi vertikalur Tokzea Camokidebuli.  F Zala da v) Ti a) Tu Tokis daWimulobaa 89 n, risi tolia lampis masa? b) risi toli iqneba Tokis daWi muloba, Tu lifti zeviT imoZravebs 2,4 m/wm2 aCqarebiT? ••35. 27,8 kn wonis liftis kabina zeviT adis. risi tolia liftis bagiris daWimuloba, Tu liftis siCqare a) izrdeba 1,22 m/wm2-iT da b) mcirdeba 1,22 m/wm2iT. ••36. lifts bagiri maRla eweva. liftisa da erTi mgzavris saerTo masaa 2000 kg. mgzavrs moneta gauvarda da kabinis mimarT 8 m/wm2 aCqarebiT iatakze daeca. risi tolia bagiris Zala da d) mexuTedan meoT- jaWvis amwevi rgolze moqmedi Toeuli rgolis amaCqarebeli sur. 5.43. amocana 39. Za lebis tolqmedi. SSM. ••40. Zvelad cxenebi arxSi gemebs 5.44 suraTze naCvenebi xerxiT eqaCebodnen. davuSvaT, cxeni Toks eqaCeba 7900 n ZaliT gemis moZraobis mimarTulebisadmi (x RerZis dadebiTi mi θ = 18° marTulebisadmi) kuTxiT. gemis masaa 9500 kg, xolo aCqareba _ 0,12 m/wm2. risi tolia wylidan gemze moqmedi Zalis a) sidide da b) mimarTuleba (x RerZis dadebiTi mimarTulebisadmi)? daWimuloba? ••37. gvaqvs Toki, romelic 387 n-ze meti daWi mulobisas wydeba. am TokiT 449 n wonis tvirTi 6,1 m simaRlidan miwaze unda Camoitano. a) tvirTis rogori aCqareba gamoiwvevs Tokis gawyvetas? b) ra aCqarebiT da ra siCqariT daecema tvirTi miwaze? TokiT miaTreven. Tokze F=12 n Zala θ=25° kuTxiT moqmedebs. a) risi tolia blokis bagiriTaa aCqareba? gadabmuli da ma Ti amoZraveba kabinis zeda nela ba F izrdeba. risi tolia Zala iatakidan kabinis masaa 1700 blokis srulad awevis Sem B kabinisa – 1300 kg. A kabinis iatakze 12kg masis b) Zalis sidide nel- A giriT SeiZleba. A kg, ••41. 5.45 suraTze m=5 kg masis bloks xaxunis gan Tavisufal horizontalur zedapirze ••38. 5.42 suraTze A da B liftebi mokle sur. 5.44. amocana 40. sur. 5.45. amocana 41. deg? g) risi tolia sur. 5.42. amocana 38. yuTi devs. kabinebis damakavSirebuli bagiris blokis aCqareba misi iatakidan srulad awevis momentSi? daWimulobaa 1,91×104 n. risi tolia kabinis ••42. 5.46 suraTze TokiT gadabmuli oTxi iatakidan yuTze moqmedi drekadobis Zala? pingvinia naCvenebi, romelTac ••39. 5.43 suraTze 0,1 kg masis xuTi rgolisgan Semdgar jaWvs a=2,5 m/wm2 aCqarebiT vertika lurad zeviT sweven. gamoTvale a) meore rgolidan pirvelze moqmedi Zala, b) mesamedan meoreze moqmedi Zala, g) meoTxedan mesameze sur. 5.46. amocana 42. xaxunisgan 160 Zala da moZraoba Tavi 5 Tavisufal yinulze adamiani miaTrevs. sami Semoxveuli. yu pingvinis masa da ori Tokis daWimuloba Tebis Semdegia: m1=12kg, m3=15kg, m4=20kg, T2=111n da mA=30kg, mB=40kg da mC=10 kg. ro ••43. desac T4=222 n. gamoTvale meore pingvinis masa m2. ori bloki xaxunisgan Tavi sufal rogorc 5.47 su sistema sur. 5.50. amocana 46. amoZravdeba, a) risi tolia B da C yuTebis damakavSirebeli Tokis daWimuloba? b) ra magidaze gverdigverd dgas. masebia manZilze gadaadgileba A yuTi 0,25 wm-Si (im sur. 5.47. amocana 43. daSvebiT, rom is bloks ar miaRwevs)? raTze vxedavT, did blokze horizontaluri ••47. 5.51 suraTze Zala moqmedebs. a) roca m1=2,3 kg, m2=1,2 kg da F=3,2n, gamoTvale or bloks Soris moqmedi Zalis sidide. b) daamtkice, rom Tu imave F naCvenebia xaxuni sididis Zala patara blokze sawinaaRmdego sis blokze Semo mimarTulebiT moqmedebs, maSin or bloks xveuli sgan Tavisufal, umniSvnelo ma TokiT Soris 2,1 n Zalaa modebuli, romlis sidide d a k a v S i r e b u l i gasxvavdeba a) nawilSi gamoTvlili sididisgan. ori yuTi. am mo g) axseni am gansxvavebis mizezi. SSM ILW WWW wyobilobas etvu sur. 5.51. amocana 47.  ••44. 5.48a suraTze A blokze Fa mudmivi horizontaluri Zala moqmedebs. A bloki B bloks 20 n ZaliT horizontalurad marjvniv  awveba. 5.48b suraTze igive Fa Zala B blokze moqmedebs. amjerad A bloki B bloks 10 n – m2=2,8 kg. risi tolia a) yuTebis aCqareba da ZaliT miwaze Camodis. Toki Semoxveulia xaxunisgan horizontalurad marjvniv awveba. dis manqana ewo deba. erTi yuTis masaa m1=1,3 kg, meore yuTisa b) Tokis daWimuloba? ••48. 85 kg masis kaci 10 m simaRlidan TokiT blokebis saerTo masaa 12 kg. risi tolia Tavisufal blokze da mis meore boloze 65kg a) 5.48a suraTze maTi aCqarebebi da b) Fa Zalis masis qviSis tomaraa dakidebuli. ra siCqariT sidide? daecema kaci miwaze?  ••49. 10 kg masis maimuni umniSv nelo masis mqone Tokze sur. 5.48. amocana 44. Toki Semoxveulia ••45. 5.49 suraTze sam gadabmul bloks xaxu nisgan Tavisufal zedapirze T3=65n ZaliT horizontalurad marjvniv micocavs. miaTreven. m1=12 kg, m2=24 kg da m3=31 kg. gamoTvale a) sistemis aCqareba, b) T1 daWimuloba da g) T2 daWimuloba. xaxunisgan Tavi sufal totze da mibmulia miwaze mdgar, 15 kg masis yuTze (sur. 5.52). a) ra minimaluri aCqareba unda hqondes maimuns, sur. 5.52. amocana 49. raTa yuTi miwidan sur. 5.49. amocana 45. ••46. 5.50 suraTze sami yuTi TokebiTaa awios? Tu yuTis awevis Semdeg maimuni co cvas Sewyvets da Toks Camoekideba, risi toli iqneba maimunis aCqarebis b) sidide da gadabmuli. erT-erTi Toki xaxunisgan Tavi g) mimarTuleba? d) risi toli iqneba Tokis sufal da umniSvnelo masis mqone blokzea daWimuloba? SSM 161 10 kg. roca yuTebs xels gavuSvebT, risi toli ••50. 5.53 suraTze naCvenebia iqneba marjvena Tokis daWimuloba? skam Camokide •••53. M masis sahaero burTi qveviT mimarTuli a aCqarebiT vertikalurad qveviT eSveba. bulia umniSvnelo ra masis zedmeti tvirTi unda gadmoagdon masis Tok burTidan, raTa man zeviT mimarTuli a aCqareba Se SeiZinos? davuSvaT, rom haeridan moqmedi xaxu zeviT mimarTuli Zala masis Semcirebisas ar ze mjdomare kaci. skami ze, mqone romelic moxveulia nisgan Tavisufal blokze, xolo misi bolo meore kacs xelSi ravs. icvleba. SSM ILW. •••54. 5.56 suraTze naCvenebia m2 = 1 kg masis uWi kacisa sur. 5.53. amocana 50. da skamis saerTo masaa 95 kg. ra ZaliT unda imoqmedos kacma Tokze, raTa aiwios a) Tanabari siCqariT da b) zeviT mimarTuli 1,3 m/wm2 aCqarebiT? (miniSneba: Tavisufali sxeulis grafiki namdvilad dagexmarebaT). Tu Tokis marjvena mxare miwamde eSveba da kacis mewyvile Toks qaCavs, ra ZaliT unda imoqmedos mewvilem Tokze, raTa kaci awios yuTi xaxunisgan Tavisufal, θ = 30° kuTxiT daxril zedapirze. es yuTi umniSvnelo masis TokiT dakavSirebulia m1 = 3 kg masis meore yuTTan, romelic horizontalur xaxunisgan zedapirze Tavisufal Zevs. bloki xaxunisgan Tavisufalia da umniSvnelo masa  aqvs. a) Tu horizontaluri F Zalis sididea 2,3 n, maSin risi tolia damakavSirebeli Tokis daWimuloba? b) ra maqsimaluri sididisa  SeiZleba iyos F Tokis gawydomis gareSe? g) Tanabari siCqariT da d) maRla mimarTuli 1,3 m/wm2 aCqarebiT? risi tolia blokis sistemidan Werze moqmedi Zala e) a) nawilSi, v) b) nawilSi, z) g) nawilSi da T) d) nawilSi? m1 = 3,7 ••51. kg masis yuTii xaxunisgan Ta sur. 5.56. amocana 54. θ = 30° visufal, kuTxiT daxril zedapirze TokiT dakavSirebulia m2 sur. 5.54. amocana 51. •••55. 5.57 suraTze naCvenebia x RerZis gaswvriv moZravi 3 kg masis yinulis blokze moqmedi Zalis Fx komponenti, rogorc t drois funqcia. t = 0 momentSi bloki RerZis dadebiTi mimarTulebiT 3 m/wm siCqariT moZraobs. risi = 2,3 kg masis vertikalurad dakidul yuTze. tolia blokis a) siCqare da b) moZraobis Toki Semoxveulia umniSvnelo masis mqone, mimarTuleba, rodesac t = 11 wm? xaxunisgan Tavisufal blokze (sur. 5.54). risi tolia a) TiToeuli yuTis aCqareba, b) dakiduli yuTis aCqarebis mimarTuleba da g) Tokis daWimuloba? ••52. 5.55 suraTze naCvenebia xaxunisgan Tavisufal blokebze dakavSirebuli Semoxveuli TokiT sa mi yuTi. B yuTi xa xunisgan magidaze Tavisufal Zevs. •••56. yu mA=6 kg, mB=8 kg da mC= Tebis sur. 5.57. amocana 55. 5.58 suraTze naCvenebia sabagiro gzis monakveTi. TiToeuli vagonisa da misi masebia sur. 5.55. amocana 52. mgzavrebis dasaSvebi saerTo masaa 2800kg. bagirze moZrav vagonze moqmedebs Zala meore 162 Zala da moZraoba Tavi 5 pasuxi kabelidan, romelic Zvel sistemas efuZneboda, sadac vagonis koeficienti 1,77 gamoiyeneboda ara litrebis Tavzea mimagrebuli. kilogramebSi, aramed litrebis girvanqebSi davuSvaT, TiToeuli kabelebi gadasayvanad. g) sinamdvileSi ramdeni kilo daWimulia da θ=35° grami sawvavi iyo TviTmfrinavSi? (garda kuTxiTaa mocemuli daxri 1,77-isa, koeficientebi oTxi li. rogori iqneba niSnis sizustiT gansxvaveba kabe leSi ram-deni damatebiTi sawvavi iyo saWi mo ro? e) monrealidan gafrenisas saWiro sa daWimu wvavis ra procentuli raodenoba hqonda lis mezobeli nakveTebis lobebs Soris, TiToeul TviTmfrinavs? Tu vagons maqsimaluri daamrgvale) d) sinamdvi sur. 5.58. amocana 56. dasa edmontonis gzaze, 7,9 km simaRleze, TviT mfrinavs sawvavi gauTavda da vardna daiwyo. Svebi masa aqvs da 0,81 m/wm aCqarebiT zeviT miuxedavad moZraobs? imdeni moaxerxa, rom TviTmfrinavi Tanabrad 2 sawvavis arqonisa, pilotma eSveboda. vinaidan uaxloesi moqmedi aero damatebiTi amocanebi 57. erTeulebis porti ugulebelyofa. fizikis mTeli kursis ganmavlobaSi maswavlebeli erTeulebTan yuradRebiT mopyrobas iT xovs. Tumca, zogierT daudevar students sjera, rom erTeulebi yovelTvis wesrigSia. imedia, realuri cxovrebidan aRebuli Semdegi magaliTi am aseT midgomaze uars gaTqmevinebs. 1983 wlis 23 ivliss aviakompania “ear kanadis” 143-e reisis TviTmfrinavi monrealidan edmontonamde xangrZlivi frenisTvis emza deboda, rodesac ekipaJma aeroportis mom metismetad Soris iyo, pilotma TviTmfrinavi Zveli aeroportisken mimarTa. saubedurod, am aeroportis asafreni biliki manqanebis trasad iyo gadakeTebuli da mis garSemo rkinis jebiri iyo Semovlebuli. trasaze daSvebisas TviTmfrinavis wina Sasi motyda da TviTmfrinavi cxviriT trasaze daeca. xaxunma TviTmfrinavi Seanela da jebirTan axlos gaaCera. manqanebis mZRolebi da mgzavrebi SeZrwunebulebi uyurebdnen am sanaxaobas. TviTmfrinavis yvela mgzavri cocxali gadarCa. am ambis morali aseTia: yuradRebiT moepyari erTeulebs. frenisTvis saWiro iyo 22300 kg sawvavi. 58. sxeulze 20 n da 35 n ori Zala erTmaneTisgan 80° kuTxiT gansxvavebuli mimarTulebiT moqmedebs. amis Sedegad miRebuli aCqarebaa 20 maT m/wm2. risi tolia sxeulis masa? saxure personals hkiTxa, ramdeni sawvavi iyo TviTmfrinavSi. sawvavis ekipaJma raodenoba icoda, rom kilogramebSi icodnen, vinaidan kanada metrul sistemaze 59. x RerZis gaswvriv moZrav 2 kg masis iyo gadasuli. manamde sawvavi girvanqebSi wertilze am RerZis gaswvriv mimarTuli izomeboda. aeroportis personali sawvavis cvladi Zala moqmedebs. wertilis mdebareoba raodenobas litrebSi zomavda da amitom mocemulia formuliT x = 3 m + (4 m/wm)t + ct2 – upasuxa, 7682 litrio. TviTmfrinavSi sawvavis arsebuli raodenobisa da SesaZlo dasamatebeli raodenobis gasagebad ekipaJma aeroports hkiTxa, ra koeficienti iyo saWiro litrebis kilogramebSi gadasayvanad. (2m/wm3)t3, sadac x metrebSia, xolo t _ wamebSi. c koeficienti mudmivaa. rodesac t=3 wm, wertilze x RerZis uaryofiTi mimarTulebiT 36 n sididis Zala moqmedebs. risi tolia c? 60. M masis A bloks xaxunisgan Tavisufal pasuxi iyo 1,77 (1,77 kg _ 1 litri). a) ekipaJis horizontalur zedapirze m masis TokiT varaudiT, ramdeni kilogrami sawvavi iyo miaTreven, rogorc TviTmfrinavSi? b) ramdeni litri sawvavis es damateba moiTxova ekipaJma? naCvenebi. saubedurod, aeropoertis TanamSromlebis boloze 5.59 suraTzea  F Tokis hori sur. 5.59. amocana 60. 163 zontaluri Zala moqmedebs. a) daamtkice, d = 1×10–14 m diametris birTvma daatyveva. rom Toki odnav mainc Caizniqeba. Semdeg neitronze moqmedi Zala Tanabaria. gamoTvale dauSvi, rom Cazneqa umniSvneloa da moZebne b) am Zalis sidide. neitronis masaa 1,67×10-27 kg. Tokisa da blokis aCqareba, g) Tokidan blokze 65. 0,2 kg masis Saiba 2 m/wm siCqariT xaxunisgan moqmedi Zala da d) Tokis daWimuloba mis Sua wertilSi. Tavisufal hokeis moednis zedapirze aRmo savleTisken misrialebs. risi tolia im tol 61. 5.60 suraTze naCvenebia 12 kg sa qmedi Zalis a) sidide da b) mimarTuleba, buravi,romelsac intervalSi Saiba dasavleTisken wavides 5 m/ sami, wm siCqariT? risi tolia tolqmedis g) sidide horizonta lurad gaWimuli, TokiT eqaCebian. erT romelmac Saibaze unda imoqmedos, raTa 0,4 wm Tokze mocemulia. sur. 5.60. amocana 61. modebuli danarCen (F1=50n) Zala Tokebze modebulma Zalebma saburavze ise unda imoqmedos, rom mas, sul cota, a aCqareba mianiWos. risi tolia a, Tu a) F2=30 n, F3=20 n; b) F2=30 n, F3=10 n; g) F2=F3=20 n?  62. ucvleli ν = (2 m/wm) î – (7 m/wm) ˆj siCqariT da d) mimarTuleba, Tu Saiba am intervalSi aRmosavleTisken wava 5 m/wm siCqariT? 66. 5.62-e suraTze 4 kg masis A bloki da 6kg masis B bloki umniSvnelo masis ZafiTaa  gadabmuli. A blokze moqmedebs FA =(12n) î  Zala, xolo B blokze _ FB =(24 n) î Zala. risi tolia Zafis daWimuloba? moZrav wertilze sami Zala moqmedebs. ori   Zalaa F1 = (2 n) î + (3 n) ˆj + (–2 n) k̂ da F2 = (–5 n) î + (8 n) ˆj + (–2 n) k̂ . risi tolia mesame Zala? 63. muSa yuTs TokiT qarxnis iatakze miaTrevs (sur. 5.61). muSa Tokze F = 450 n ZaliT moqmedebs. Toki horizontalisadmi sur. 5.62. amocana 66. 67. 5.63 suraTze naCvenebia xaxunisgan Tavisufal, θ1=30° daxril zedapirze mdebare yuTi (m1=3 kg). es yuTi umniSvnelo masis TokiT θ2=60° daxril zedapirze mdebare meore yuTTanaa mibmuli (m2=2 kg). bloki xaxunisgan θ = 38° kuTxiTaa daxrili, xolo iataki yuTze f = 125 n ZaliT, moZraobis sawinaaRmdego Tavisufalia da umniSvnelo masa aqvs. risi mimarTulebiT moqmedebs. gamoTvale yuTis tolia Tokis daWimuloba? aCqareba, Tu a) misi masaa 310 kg da b) misi wonaa 310 n. sur. 5.63. amocana 67. 68. 15000 kg masis vertmfreni 4500 kg satvirTo manqanas 1,4 m/wm2 aCqarebiT swevs. gamoTvale a) haeridan vertmfrenze moqmedi, zeviT mimarTuli tolqmedi Zala, b) vertmfrenisa sur. 5.61. amocana 63. da satvirTo manqanis damakavSirebeli bagiris 64. birTvi masSi moxvedrili moxetiale daWimuloba. neitronis gasaCereblad masze Zlieri Sina 69. 52 kg masis cirkis msaxiobi Tokze unda gani ZaliT moqmedebs. es Zala, romelic birTvebs erTmaneTTan àwebebs~, birTvis gareT daaxloebiT nulis tolia. davuSvaT, 1,4×107 m/wm siCqariT moZravi neitroni gavides, romelic gawydeba, Tu daWimuloba 425 n-s gadaaWarbebs. a) ra moxdeba, Tu msaxiobi Toks uZravad Camoekideba? b) aCqarebis ra sididiT aicilebs msaxiobi Tokis gawyvetas? 164 Zala da moZraoba Tavi 5 70. 5.64 suraTze naCvenebia m = 4 kg moZraobs. risi tolia a) Tokis daWimuloba da masis 75. 5.66 suraTze m2=1 kg masis yuTze  12 n sididis F Za yuTi, b) m2 masa? ro melzec xuTi Zala moqmedebs. Zalebis sidideebia: F1=11 n, la moqmedebs. Za F2 = 17 n, F2 = 3 n, F4 = 14 n da F5 = 5 n, xolo θ4 = 30°. mo la sur. 5.64. amocana 70. Zebne yuTis aCqarebis a) erTeulovan-veqto ruli Canaweri, b) sidide da g) kuTxe x RerZis dadebiTi mimarTulebisadmi. daxrili sibrtyis sur. 5.66. amocana 75. gaswvrivaa mimarTuli. yuTi TokiT iatakze dadebul m1=3kg masis meore yuTTanaa dakav Sirebuli. iataki, sibrtye da bloki xaxunisgan Tavisufalia, xolo blokisa da Tokis masa 71. 3 kg masis y RerZis dadebiTi mimarTulebiT 3 m/wm2 aCqarebiT moZrav sxeulze ori Zala moqmedebs. Tu erTi Zala x RerZis dadebiTi mimarTulebiT moqmedebs da aqvs 8 n sidide, maSin risi tolia meore Zalis sidide? 72. θ=37° kuTxiT umniSvneloa. risi tolia Tokis daWimuloba? 76. 100 kg masis yuTi satvirTo liftis iatakze Zevs. t=0 momentSi lifti Senobis pirveli sarTulidan moZraobas iwyebs da 8 wamis Semdeg bolo sarTulze adis. 5.67 suraTze naCvenebia kosmosuri liftis siCqaris droze damokidebulebis erT-erT grafiki. risi tolia a) liftze moqmedi Zalis Tanamgzavrs, kalistos uaxlovdeba. Tu misi Fl sidide, b) liftis iatakidan yuTze moqmedi Zrava gamoimuSavebs zeviT mimarTul 3260 n Zalis mimarTuleba (zeviT Tu qveviT), rodesac Zalas, maSin xomaldi Tanabari siCqariT eSveba. t = 1,8 wm? risi tolia g) Fl-is sidide da d) mimarTuleba, rodesac t = 4,4 wm? risi tolia e) Fl-is sidide da v) mimarTuleba, rodesac t = 6,8 wm? warmoidgine xomaldi, sadesanto romelic iupiteris Tu motori gamoimuSavebs mxolod 2200 n Zalas, maSin xomaldi qveda mimarTulebiT 0,39 m/wm2-iT Cqardeba. a) risi tolia xomaldis wona kalistos zedapirTan axlos? b) risi tolia xomaldis masa? g) risi tolia Tavisufali vardnis aCqareba kalistos zedapirTan axlos? 73. m1 masis sxeuls garkveuli Zala 12 m/wm aCqarebas 2 aniWebs, xolo m2 sur. 5.67. amocana 76. masis sxeuls – 3,3 77. 5.68 suraTze daxril zedapirze mdebare m/wm2 aCqarebas. ra aCqarebas mianiWebs Zala a) m 2 –m 1 da b) m 2 +m 1 masis sur. 5.65. amocana 74. Tunuqis WurWeli (m1=1kg) Tunuqis meore WurWelTanaa (m2=2kg) sxeulebs? bloki mibmuli. xaxunisgan 74. 5.65 suraTze naCvenebia m1=3 kg masis konteineri, romelic m2 masis yuTTan Tavisufalia xaxunisgan Tavisufal blokze Semoxveuli aqvs. zeviT mimar TokiTaa dakavSirebuli. Tokisa da blokis Tuli F = 6 n Zala masebi moqmedebs umniSvneloa. rodesac konteineri uZravi mdgomareobidan gamodis, is xaxunisgan Tavisufal zedapirze 1 m/wm 2 aCqarebiT umniSvnelo WurWelze, melsac da masa meore ro qveviT sur. 5.68. amocana 77. 165 mimarTuli 5,5 m/wm2 aCqareba aqvs. risi tolia a) motociklistze moqmedi Zalebis tolqmedi a) Tokis daWimuloba da b) β kuTxe? da b) motociklidan motociklistze moqmedi 78. sxeuli liftis Werze damagrebul zam Zala? barian kabinis 84. rodesac 17 kn wonis avtomobili 3,66 m/wm2- uZraobis dros saswori 65 n-s aCvenebs. risi iT Cqardeba, risi tolia masze moqmedi Zale tolia sasworis Cveneba, Tu kabina zeviT bis tolqmedi? sasworzea Camokidebuli. 7,6 m/wm mudmivi siCqariT? b) 7,6 m/wm siCqaris 2,4 m/wm2 SenelebiT? moZraobs a) 5.17 85. 85 kg masis adamiani paraSutiT xteba da 2,5 m/wm2 qveviT mimarTuli aCqarebiT vardeba. paraSutis masaa 5 kg. a) risi tolia haeridan suraTze me-2 Toksa da Wers Soris kuTxe me- gaxsnil paraSutze moqmedi zeviT mimarTuli 2 Tokis sigrZis cvlilebiT SevcvaloT da Zala? b) risi tolia adamianidan paraSutze vuwodoT mas θ2 (1-el Toksa da Wers Soris moqmedi qveviT mimarTuli Zala? 79. 5.6 amocanis amoxsnis nimuSis kuTxea 28°). a) amocanis am nimuSSi mocemuli monacemebis safuZvelze aage pirveli Tokis T1 θ2-ze damokidebulebis grafiki, rodesac 0°<θ2<90°. daaxloebiT risi tolia T1-is a) maqsimaluri da b) minimaluri daWimulobis sidideebi? aris SesaZlebeli g) Tu ara es maqsimaluri fizikurad sidide da d) minimaluri sidide? 86. 1400 kg masis reaqtiuli Zrava samgzavro TviTmfrinavis WanWikiTaa davuSvaT, korpusze mimagrebuli TiToeul mxolod (rogorc WanWikze sami wesi). datvirTis erTi mesamedi modis. a) gamoTvale TiToeul WanWikze moqmedi Zala, rodesac Tvimfrinavi gafrenisTvis emzadeba. b) frenis dros TviTmfrinavze moqmedebs Zala, romelic mas 80. gamoTvale 75 kg masis kosmosur xomald zeviT mimarTul 2,6 m/wm2 aCqarebas aniWebs. `reinjeris~ wona a) dedamiwaze, b) marsze, gamoTvale sadac g=3,8m/wm , da g) Ria kosmosSi, sadac moqmedi Zala. 2 g=0. d) risi tolia `reinjeris~ masa TiToeul mdebareobaSi? TiToeul WanWikze amjerad 87. 12 kg masis javSnosani (cxoveli) sanadirod tbis gayinul, xaxunisgan Tavisufal 81. wertili im adgilze, sadac g = 9,8 m/wm2, 22 n-s iwonis. risi tolia wertilis a) wona da b) masa iq, sadac g = 4,9 m/wm2? risi tolia RerZis dadebiTi mimarTulebiT. javSnosanis misi g) wona da d) masa, Tu wertili imyofeba sawyisi mdebareoba aTvlis saTaved miiCnie. kosmosSi, sadac g = 0? cxoveli yinulze srials iwyebs, roca masze 82. 1,2×106 kg masis kosmosuri xomaldi varskvlavebis sistemis mimarT uZrav mdgomareobaSia. a) ra aCqareba dasWirdeba xomaldis 0,1c siCqariT (c sinaTlis siCqarea zedapirze Serbis. javSnosanis sawyisi siCqare, romlis sididea 5 m/wm, mimarTulia x y RerZis dadebiTi mimarTulebis 17 n Zalis qari moqmedebs. erTeulovan-veqtorul CanawerSi, risi tolia javSnosanis a) siCqare da b) mdebareoba 3 wm srialis Semdeg? amoZravebas 88. 50 kg masis mgzavri liftiT moZraobs, 3 dReSi? romelic pirveli sarTulidan moZraobas b) risi tolia es aCqareba g-ebSi? g) ra Zalaa iwyebs t=0 momentSi da bolo sarTulze 10 wm- saWiro am aCqarebisTvis? d) Tu 0,1c siCqaris is Semdeg adis. 5.69 suraTze naCvenebia kabinis da 3×108 m/wm-isa) tolia varskvlavebis miRwevisas sistemis Zravebi mimarT gamorTes (siCqare Tanabaria), ra dro dasWirdeba xomalds 5 sinaTlis Tvis gasavlelad (manZili, romelsac sinaTle 5 TveSi gaivlis)? 83. motocikli da 65 kg masis motociklisti 3 m/wm2 aCqarebiT horizontisadmi 10° kuTxiT daxril zedapirze moZraoben. risi tolia sur. 5.69. amocana 88. 166 Tavi 5 Zala da moZraoba aCqarebis droze damokidebulebis grafiki. aCqarebis dadebiTi sidideebi asaxavs mis zeda mimarTulebas. rogoria liftis iatakidan mgzavrze moqmedi maqsimaluri Zalis a) sidide da b) mimarTuleba (zeviT Tu qveviT)? rogoria iatakidan mgzavrze moqmedi minimaluri Zalis g) sidide da d) mimarTuleba? rogoria mgzavridan iatakze moqmedi maqsimaluri Zalis e) sidide da v) mimarTuleba?  89. F Zalis moqmedebiT cdilob xaxunisgan Tavisufal iatakze macivris gadaadgilebas.  F an horizontaluria (pirveli SemTxveva), an θ kuTxiT daxrili (meore SemTxveva). a) risi  95. Tu 1 kg masis sxeuli mxolod F1 =(3 n) î +  +(4 n) ˆj da F2 =(–2 n) î +(–6 n) ˆj Zalebis zemo qmedebiT Cqardeba, maSin risi tolia tolqmedis a) erTeulovan-veqtoruli Cana weri, b) sidide da g) x RerZis dadebiTi mimar  Tulebisadmi kuTxe? risi tolia a -s d) sidide da e) kuTxe? 96. kosmosuri xomaldi mTvaridan, sadac g=1,6 m/wm2, vertikalurad frindeba. Tu afrenisas xomaldis aCqarebaa 1 m/wm2, maSin ra sididis ZaliT moqmedebs xomaldi pilotze, romelic dedamiwaze 735 n-s iwonis? ganmavlobaSi ubiZgeb? b) risi tolia es 97. laboratoriul eqsperimentSi Tavda pirvelad uZravi eleqtroni (masa=9,11×10–31 kg) 1,5 sm-is manZilze Tanabar aCqarebas ganicdis da distanciis dasasruls 6×106 m/wm fardoba, Tu macivars garkveul d manZilze siCqares aRwevs. risi tolia a) eleqtronis ubiZgeb? amaCqarebeli Zalis sidide da b) eleqtronis tolia meore SemTxvevis Sesabamisi siCqaris fardoba pirveli SemTxvevis Sesabamis siCqaresTan, Tu macivars garkveuli t drois 90. 1 kg masis sxeuli 4 m/wm2 aCqarebiT x RerZis dadebiTi mimarTulebisadmi 160° kuTxiT moZraobs. sxeulze ori Zala moqmedebs.  erTia F1 =(2,5 n) î +(4,6 n) ˆj . risi tolia meore wona? 98. gamoTvale 1,3×104 kg masis raketis sawyisi, zeviT mimarTuli aCqareba, Tu Zravis mier warmoqmnili sawyisi zeviT mimarTuli Zalis Zalis a) erTeulovan-veqtoruli Canaweri, (biZgis) sididea 2,6×105 n. ar ugulebelyo b) sidide da g) kuTxe? raketaze moqmedi mizidulobis Zala. 91. raketisa da misi sasargeblo tvirTis saerTo masaa 5×104 kg. ra sididisaa Zravis mier 99. 5.70a suraTze naCvenebia Werze Camo kidebuli konstruqcia. is liTonis ori warmoqmnili Zala (dartyma), rodesac raketa nawilisgan (m1=3,5 kg da m2=4,5 kg) Sedgeba, a) afrenis adgilis Tavze haerSi `hkidia~ da b) romlebic zeviT Cqardeba 20 m/wm2-iT? TokiTaa gadabmuli. risi tolia a) qveda Tokis 92. 80 kg masis kaci 0,5 m simaRlis fanjridan ezoSi xteba. gadmoxtomisas is muxlebs ar xris da gaCerebisTvis 2 sm sWirdeba. a) risi tolia misi saSualo aCqareba ezos miwis pirvelad Sexebidan gaCerebamde? b) ra saSualo Sema Cerebeli ZaliT moqmedebs Sida ezo? erTmaneTze umniSvnelo daWimuloba da b) zeda Tokis daWimuloba? 5.70b suraTze naCvenebia liTonis sami nawilisgan Semdgari konstruqcia. ori maTganis masaa m3=4,8 kg da m5=5,5 kg. zeda Tokis daWimulobaa 199 n. risi tolia g) qveda Tokis daWimuloba da d) Sua Tokis daWimuloba? 93. 2 kn wonis motocikli 0-dan 88,5 km/sTmde 6 wamSi Cqardeba. risi tolia a) Tanabari aCqareba da b) aCqarebis gamomwvevis Zalebis tolqmedi? 94. davuSvaT, 5.14 suraTze sxeulebis masebia 2 kg da 4 kg. a) ra masa unda hqondes Camokidebul sxeuls, raTa aCqareba maqsimaluri iyos? rogori iqneba am SemTxvevaSi b) aCqarebis sidide da g) Tokis daWimuloba? masis (a) (b) sur. 5.70. amocana 99. 167 100. 29 kg masis pingvini aRmosavleTisken 6 m/wm siCqariT moZraobs da didi farTobis yinulze Serbis. yinulze curvis dros qari pingvinze ucvleli sididis da mimarTulebis ZaliT moqmedebs. 5.71 suraTze naCvenebia yinulze curvis dros pingvinis mdebareobebi 1-wamian SualedebSi. x RerZis dadebiT mimarTulebad arCeulia aRmosavleTi. pin gvini yinuls pirvelad t = 0 momentSi exeba. rogoria qaris pingvinze moqmedi Zalis a) sidide da b) mimarTuleba? 101. 0,2 kg masis Saiba gayinuli tbis zedapirze 2 m/wm siCqariT aRmosavleTisken micuravs. risi toli unda iyos 0,5 wamis ganmavlobaSi Saibaze moqmedi saSualo Zalis sidide da mimarTuleba, raTa Saibis siCqare gaxdes a) 5 m/wm dasavleTisken da b) 5 m/wm samxre Tisken? sur. 5.71. amocana 100. 6 Zala da moZraoba bolidis usafrTxo moxveva rbolis dros xaxunzea damokidebuli. Zalian swrafad moZraobis SemTxvevaSi, xaxuni ver asrulebs Tavis funqcias da bolidi trasaze curdeba. adre mosaxvevSi Sesvlisas mrbolels Zalian unda Seenelebina bolidi, dResdReobiT ki ukeTesi konstruqciis Sedegad haeri bolids zemodan awveba da didi siGqaris SemTxvevaSic mocurebis albaToba mcirea. qvemoT mimarTul biZgs uaryofiTi weva ewodeba. uaryofiT wevas bolidis zemodan da qvemodan gamavali haeris nakadi iwvevs. Tanamedrove rbolebSi uaryofiTi weva sakmaod Zlieria. SesaZlebelia Tu ara amoyiravebuli bolidis Werze gatareba? pasuxs am TavSi SeityobT 6.1. ra aris fizika? mocemul TavSi yuradRebas sami saxeobis Zalas mivapyrobT. esenia xaxunis, wevis da centriskenuli Zalebi. indianapolis-500-sTvis sarbolo manqanis momzadebisas inJinerma samive saxeobis Zala unda gaiTvaliswinos. saburavebze moqmedi xaxunis Zala gadamwyvet zegavlenas axdens manqanis aCqarebaze, rodesac piloti pit-stopidan gamodis an mosaxvevSi Sedis. haeridan manqanaze moqmedi wevis Zala minumamde unda Semcirdes, winaaRmdeg SemTxvevaSi manqana bevr sawvavs moixmars da pit-stopSi Sesvla maleve mouwevs (pit-stopSi 14 wamiT gaCerebac ki pilots rbolis wagebad SeiZleba daujdes). centriskenul Zalas didi mniSvneloba mosaxvevebSi aqvs (arasakmarisi centriskenuli Zalis SemTxvevaSi manqana kedels Seejaxeba). Cvens ganxilvas xaxunis ZalebiT viwyebT. 6.2. xaxuni xaxuni Cveni yoveldRiuri cxovrebis ganuyofeli nawilia. maTTvis winaaRmdegobis gawevis gareSe yvela moZravi sagani gaCerdeboda. avtomobilSi moxmarebuli sawvavis 20% Zravasa da transmisiaSi arsebuli xaxunis winaaRmdeg gamoiyeneba. meores mxriv, xaxunis ararsebobis SemTxvevaSi avtomobili saerTod ver imoZravebda da, ufro metic, fexiT an velosipediT gadaadgilebac ki SeuZlebeli iqneboda. ver daviWerdiT fanqars da, Sesabamisad, verc davwerdiT. WanWikebi da lursmnebi gamousadegarebi gaxdeboda, moqsovili tansacmeli nawilnawil daiSleboda da kvanZebic ki TavisiT gaixsneboda. amjerad Cven SevexebiT mSral, myar zedapirebs Soris moqmed xaxunis Zalebs, rodesac zedapirebi 168 6.2. xaxuni 169 erTmaneTisadmi uZravni arian an mcire siCqariT moZraoben. CavataroT sami martivi eqsperimenti: 1. wigni grZel, horizontalur zedapirze gavacuroT. rogorc mosalodnelia, wigni Seneldeba da Semdeg gaCerdeba. es niSnavs, rom wigns mxebi zedapiris paraleluri da sakuTari (a) siCqaris sawinaaRmdego mimarTulebis mqone aCqareba aqvs. niutonis meore kanonidan gamomdinare, wignze mxebi zedapiris paraleluri da misi siCqaris sawinaaRmdego mimarTulebis Zala moqmedebs. es aris xaxunis Zala. moZraoba ar gvaqvs (b) 2. wigns horizontalurad ubiZgeT, raTa man mxebi zedapiris gaswvriv mudmivi siCqariT imoZraos. SeiZleba Tu ara Tqveni Zala wignze moqmedi erTaderTi horizontaluri Zala iyos? ara, vinaidan aseT SemTxvevaSi wigni aCqardeba. niutonis meore kanonidan gamomdinare, wignze meore Zala moqmedebs, romelic Tqveni Zalis sawinaaRmdego mimarTulebis da (g) igive sididisaa. maSasadame, Zalebi erTmaneTs awonasworebs. wignze moqmedi meore Zala xaxunisaa da is mxebi zedapiris paraleluria. 3. mZime yuTs horizontalurad ubiZgeT. yuTi ar moZraobs. niutonis meore kanonidan gamomdinare, yuTze meore Zala (d) moqmedebs da Tqvens Zalas ewinaaRmdegeba. ufro metic, es Zala Tqvens sawinaaRmdegod aris mimarTuli da igive aCqareba sididisaa, radgan Zalebi erTmaneTs awonasworebs. yuTze moqmedi meore Zala kvlav xaxunis Zalaa. Semdeg scadeT da yuTs ufro Zlierad ubiZgeT. yuTi kvlav ar moZraobs. (e) cxadia, xaxunis Zalis sidide Seicvala da Zalebi kvlav erTmaneTs awonasworebs. amjerad mTeli Tqveni Zala CarTeT mudmivi siCqare maqsimaluri sidide aqvs da Tu am sidides gadaaWarbebT, yuTi amoZravdeba. (v) fs-is maqsimaluri mniSvneloba xaxunis Zalis sidide saqmeSi. yuTi moZraobas iwyebs. maSasadame, xaxunis Zalas fk miaxloebiT mudmivia moZraobis dasawyisi (z) 6.1 suraTze naCvenebia zemoT aRwerilis msgavsi mdgomareoba. 6.1a suraTze bloki magidaze Zevs da  Fg mizidulobis Zalas  FN drekadobis Zala awonasworebs. 6.1b suraTze gamosaxul  blokze F Zala moqmedebs da mas marcxnisken ubiZgebs.  sapasuxod f s xaxunis Zala marjvniskenaa mimarTuli da Zalas  awonasworebs. f s Zalas statikuri xaxunis Zala ewodeba. bloki ar moZraobs. sur. 6.1. a) uZrav blokze moqmedi 6.1g da 6.1d suraTebze naCvenebia, Tu ra xdeba, rodesac Tqven Zalebi. b)-d)  blokze modebul gare Zalas f s statikuri xaxunis Zala awonasworebs. -is zrdasTan erTad fs Zalac izrdeba, vidre fs garkveul maqsimalur mniSvnelobas SeiZens. e) amis Semdeg bloki is mimarTulebiT Cqardeba. v) Tu bloki mudmivi siCqariT moZraobs, maSin F Zala kvlav im maqsimalur mniSvnelobamde Semcirdeba, rome lic blokze iyo modebuli amoZra vebamde. z) a)-v) movlenebis eqsperi mentuli Sedegebi. mier modebuli Zala izrdeba. Tavdapirvelad  f s statikuri xaxunis Zalis sidide izrdeba da bloki uZravia. rodesac Tqveni Zala garkveul sidides miaRwevs, bloki marcxnisken aCqarebas iwyebs (6.1e). am SemTxvevaSi xaxunis Zala moZraobas ewinaaRmdegeba da mas  f k kinetikuri xaxunis Zala ewodeba. Cveulebriv, moZraobis dros moqmedi kinetikuri xaxunis Zalis sidide moZraobis ararsebobis dros moqmedi statikuri xaxunis Zalis sidideze metia. aqedan gamomdinare, Tu gsurT, rom bloki zedapirze mudmivi siCqariT gadaadgildes, maSin 170 Tavi 6 Zala da moZraoba moZraobis dawyebisas modebuli Zalis sidide unda SeamciroT. 6.1z suraTze naCvenebia eqsperimentis Sedegebi, rodesac moZraobis dawyebamde blokze modebuli Zala nel-nela gaizarda. yuradReba miaqcieT, rom Zalis Semcireba blokis (a) mudmivi siCqariT moZraobisTvis aris saWiro. xaxunis Zala sxeulebis zedapirebis atomebs Soris moqmedi Zalebis veqtoruli jamia. Tu gaprialebuli da kargad gawmendili zedapirebis mqone liTonebs vakuumSi erTmaneTTan (b) mivaaxlovebT, isini erTmaneTze ar gasrialdeba, vinaidan erT zedapirze ganlagebuli mravali atomi meore zedapirze sur. 6.2. srialis xaxunis me ganlagebul mraval atomTan urTierTqmedebs, zedapirebi qanizmi. a) zeda zedapiri qveda zedapirze marjvnisken misria lebs. b) naCvenebia civi SeduRebis ori adgili. SenaduRi nakerebis garRvevisa da moZraobis gagrZe lebisTvis Zalaa saWiro. wamierad civad SeduRdebian da liTonis erTi naWeri warmoiqmneba. specialurad gaprialebuli dagraduirebuli Suasadebebis erTmaneTTan miaxloebisas (haerSi), atomebs Soris kontaqti naklebia, magram Suasadebebi erTmaneTs myarad ekroba da maTi gancalkeveba mxolod specialuri xelsawyoTi SeiZleba. Cveulebriv pirobebSi atomTa Soris mWidro kontaqti ar xdeba, vinaidan Zalian gaprialebuli zedapiris mqone liTonic ki atomur skalaze ar aris idealurad swori. ufro metic, yoveldRiuri sagnebis zedapirebze Jangis da sxva gamWuWyianeblebis fenebia dadebuli da civi SeduRebis albaTobac mcirea. ori Cveulebrivi zedapiris erTmaneTTan miaxloebisas, mxolod SemaRlebuli wertilebi exeba erTmaneTs. Sexebis realuri, mikroskopuli farTobi xilul, makroskopul farTobze daaxloebiT 104-jer mcirea. miuxedavad amisa, Sexebis bevri wertili mainc ganicdis civ SeduRebas. SeduRebis Sedegad statikuri xaxuni Cndeba da modebuli Zala zedapirebis erTmaneTze gasrialebas cdilobs. rodesac modebuli Zala sakmarisad didia, SenaduRi nakerebi irRveva damoZraobasTan erTad SemTxveviTi Sexebis adgilebi Cndeba (sur. 6.2). moZraobis sawinaaRmdego fk kinetikuri xaxunis Zala SemTxveviTi Sexebis mraval adgilze moqmedi Zalebis veqtoruli jamia. zedapirebs Soris ufro myari kontaqtis dros, civi SeduRebis wertilebis raodenoba metia da zedapirebis erTmaneTisadmi gacureba met Zalas moiTxovs. Sesabamisad,  f s statikuri xaxunis Zalis maqsimaluri mniSvneloba ufro metia. zedapirebis srialisas wamieri civad SeduRebis wertilebis raodenoba metia da, maSasadame,  f k kinetikuri xaxunis Zalis sididec izrdeba. xSirad erTi zedapiris meoreze srials, biZgebi axlavs Tan, vinaidan zedapirebi erTmaneTs ekroba. msgavsi moZraobisas wripinis an RrWialis xma ismis. magaliTad, saburavebis xma mSral trasaze damuxruWebisas, frCxilebis moZraoba carciT gasvril dafaze an daJanguli karebis gaReba. ar aris aucilebeli xma arasasiamovno iyos, magaliTad violinos xemis simze gasmac gamodgeba. 6.3. xaxunis Tvisebebi eqsperimentis Tanaxmad, rodesac  F Zala mSrali da gaupoxavi sxeulis igive mdgomareobaSi myof zedapirze gasrialebas cdilobs, Sedegad miRebul xaxunis Zalas sami Tviseba aqvs: pirveli Tviseba. Tu sxeuli ar moZraobs, maSin   f s statikuri xaxunis Zala da F Zalis zedapirisadmi paraleluri komponenti erTmaneTs awonasworebs. Zalebi sididiT tolia, xolo mimarTulebiT urTierTsawinaaRmdegoa. 6.3. xaxunis Tvisebebi meore Tviseba. 171  f s Zalis fs,max maqsimaluri mniSvnelobaa: fs,max=μsFN (6.1) sadac μs aris statikuri xaxunis koeficienti, xolo FN - zedapiridan sxeulze moqmedi drekadobis Zala. Tu  F Zalis zedapirisadmi paraleluri komponenti fs,max mniSvnelobaze metia, maSin sxeuli zedapiris gaswvriv srials iwyebs. mesame Tviseba. Tu sxeuli zedapiris gaswvriv srials iwyebs, xaxunis Zala fk mniSvnelobamde swrafad mcirdeba: f k =μ k F N (6.2)  sadac μk aris kinetikuri xaxunis koeficienti. amis Semdeg, 6.2 formuliT mocemuli f k kinetikuri xaxunis Zala moZraobas ewinaaRmdegeba. xaxunis Zalis meore da mesame TvisebebSi moyvanili FN drekadobis Zalis sidide gviCvenebs, Tu ramdenad Zlierad awveba sxeuli zedapirs. niutonis mesame kanonidan gamomdinare, rac ufro metia dawola, miT ufro metia FN. pirveli da meore Tvisebebi erTaderT  F Zalas axasiaTebs, magram isini sxeulze modebuli ramdenime Zalis tolqmedsac miesadageba. 6.1 da 6.2 formulebis veqtoruli Cawera SeuZlebelia, radgan srials ewinaaRmdegeba, xolo   f s da f k Zalebi yovelTvis zedapiris paraleluria da  FN drekadobis Zala zedapirisadmi perpendikularulia. μs da μk koeficientebs erTeuli ar gaaCnia da isini eqsperimentis saSualebiT ganisazRvreba. maTi mniSvneloba sxeulis da zedapiris maxasiaTeblebzea damokidebuli da amitom xSirad moixseniebian windebul `Soris~-Tan erTad. magaliTad: `kvercxsa da tefloniT dafarul patara qvabs Soris μs-is mniSvnelobaa 0,04, xolo kldeebze sacoc fexsacmelsa da kldes Soris - 1,2~. vuSvebT, rom μk-is mniSvneloba sxeulis srialis siCqareze damokidebuli ar aris. sakontrolo wertili 1. bloki zedapirze Zevs. a) ra xaxunis Zala moqmedebs blokze zedapiridan? b) Tu blokze 5n horizontaluri Zala moqmedebs, magram bloki uZravia, ra xaxunis Zala moqmedebs blokze? g) Tu blokze moqmedi statikuri xaxunis Zalis fs,max maqsimaluri mniSvnelobaa 10n, imoZravebs Tu ara bloki masze 8n horizontaluri Zalis modebisas? d) gaeciT igive SekiTxvas pasuxi, Tu blokze modebuli Zalaa 12n. e) ra xaxunis Zala moqmedebs g) SekiTxvis SemTxvevaSi? 6.1 amocanis amoxsnis nimuSi avariuli damuxruWebis dros borblebi ar brunavs da manqana gzis gaswvriv srialdeba. saburavis nakuwebi da gzis safaris patara gamdnari nawilebi kvals toveben da es (a) cxadyofs, rom srialis dros civi SeduReba moxda. 1960 wels, inglisis M1 trasaze, iaguarma 290 m sigrZis kvali datova da rekordi daamyara (6.3a suraTi). davuSvaT μk = 0,6 da damuxruWebis dros manqanis aCqareba mudmivi iyo. ra siCqariT moZraobda manqana borblebis gaCerebis momentSi? amoxsna: vinaidan aCqarebas mudmivad (b) miviCnevT, 2.1 cxrilSi mocemuli formulebis gamoyenebiT ν0 sawyisi siCqare SegviZlia davadginoT. modiT 2.16 formulis gamoyeneba sur. 6.3. a) manqana marjvnisken misrialebs da 290mis gavlis Semdeg Cerdeba. b) Tavisufali sxeulis grafiki manqanisTvis 172 Zala da moZraoba Tavi 6 vcadoT da davuSvaT, rom manqana x RerZis dadebiTi mimarTulebiT moZraobda: ν 2 =ν 0 2 +2a(x-x 0 ) 6.2 formulidan gamomdinare, xaxunis Zalis sididea fk=μkFN , sadac FN aris gzidan manqanaze (6.3) Cven ukve viciT, rom x-x0 gadaadgileba 290mia, saboloo ν siCqared 0-s miviCnevT da ν0-is gamoTvla gvsurs. manqanis a aCqareba CvenTvis Zala manqanaze moqmedi ucnobia. tolia. aqedan gamomdinare, FN=mg. a-s gamoTvlisTvis gaviTvaliswinoT, rom Tu manqanaze haeris moqmedebas ugulvebelvyofT, maSin a aCqarebas mxolod gzidan manqanaze moqmedi moqmedi drekadobis Zala. vinaidan manqana vertikalurad ar Cqardeba, 6.3b suraTidan da niutonis meore kanonidan gamomdinare  FN  Fg mizidulobis Zalis amjerad 6.4 formula a-sTvis amovxsnaT da fk-s nacvlad fk=μkFN=μkmg CavsvaT: f  f k kinetikuri xaxunis Zala iwvevs, a = – mk = – μkmg m = – μkg, romelic manqanis moZraobis sawinaaRmdegod minus niSani asaxavs, rom aCqareba x RerZis aris mimarTuli (6.3b suraTi). Zalas da uaryofiTi aCqarebas Soris kavSiri niutonis meore mimarTulebis sawinaaRmdegoa. amis Semdeg, kanonis (ft,x=max) saSualebiT Caiwereba: –f k =ma xaxunis Zalis mimarTulebas da ν0-sTvis amovxsnaT: ν0 = = 58 m/w = 210 km/sT gamosaxuli qali (pasuxi) 6.2 amocanis amoxsnis nimuSi suraTze siCqaris formuliT SevcvaloT, gavixsenoT, rom ν=0 (6.4) asaxavs. 6.4a anu 6.3 formulaSi Sesuli a zemoT miRebuli sadac m aris manqanis masa. minus niSani kinetikuri mimarTulebisaa m = 75 kg masis datvirTul marxils horizontalur zedapirze mudmivi siCqariT miacurebs. marxilis kavebsa da Tovls Soris μk kinetikuri xaxunis koeficientia 0,1, xolo φ kuTxea 42°. a) ra  T Zala moqmedebs Tokidan marxilze? φ amoxsna: 1. vinaidan marxili mudmivi siCqariT moZraobs, misi aCqareba nulis tolia. (a)  2. aCqarebas Tovlidan marxilze moqmedi f k kinetikuri xaxunis Zala uSlis xels. 3. marxilis aCqarebisa da masze moqmedi Zalebis dakavSireba  kanoniT SeiZleba ( Ft niutonis  = ma ). φ meore marxili da tvirTi 6.4b suraTze naCvenebia marxilze moqmedi  Fg mizidulobis Zala da  Tovlis zedapiridan moqmedi FN drekadobis Zalebi, maT Soris (b) Zala. am ZalebisTvis niutonis meore kanoni (imis gaTvaliswinebiT, rom  a =0) gvaZlevs: (6.5)  6.5 formulas pirdapir T -sTvis ver amovxsniT, vinaidan sxva veqtorebi ar viciT. maSasadame, es formula 6.4b suraTze gamosaxuli x da sur. 6.4. a) qali datvirTul marxils Tovlian zedapirze mudmivi siCqariT miacurebs; b) datvirTuli marxilisTvis Tavisufali sxeulis grafiki. 173 6.3. xaxunis Tvisebebi y RerZebis gaswvrivi komponentisTvis unda amoxsna: 6.2 formulidan gamomdinare fk sidide FNdrekadobis Zalis sididezea damokidebuli. Caiweros: Tx + 0 + 0 – fk = 0 maSasadame, SekiTxvaze pasuxis gacemisTvis FN-sa da T-s Soris kavSiri unda davadginoT. anu T cos φ – μ k F N = 0 (6.6) am formulaSi 6.2 formulis gamoyenebiT fk-s nacvlad μkFN CavsviT. y RerZisTvis gveqneba: Ty +F N –F g +0=0 es kavSiri 6.7 formuliT gamoisaxeba. 6.7 formula Semdegi saxiT gadavweroT: F N = mg – Tsin φ rom T-s yuradReba (6.8) miaqcieT, gazrdis SemTxvevaSi FN mcirdeba (Tokis moqaCvisas anu T sin φ + F N – mg = 0, (6.7) sadac Fg-is nacvlad mg CavsviT. 6.6 da 6.7 formulebSi ori ucnobia - T da FN . Tovlidan marxilze moqmedi maRla mimarTuli Zala naklebia). vinaidan fk=μkFN , amitom fk mcirdeba. T-s dasadgenad jer 6.6 formula FN-sTvis sakontrolo unda amovxsnaT da Semdeg miRebuli Sedegi 6.7 gamosaxul yuTze 10 n sididis formulaSi CavsvaT: moqmedebs, magram μk mg T= cos φ+μksin φ (0,1)(75kg)(9,8m/w2) = cos42° +(0,1)(sin42°) = 91 n Semdeg  F2 wertili yuTi vertikaluri 2. suraTze  F1 Zala uZravia. Zala amis zrdas iwyebs. vidre yuTi gasrialdeba, a) yuTze moqmedi xaxunis Zala, b) zedapiridan (pasuxi) b) Tu qali Toks ufro Zlierad eqaCeba da T Zala 91 n-ze metia, fk xaxunis Zala metia, naklebia Tu igive? yuTze moqmedi drekadobis Zala da g) yuTze moqmedi statikuri xaxunis Zalis fs,max maqsimaluri mniSvneloba izrdeba, mcirdeba Tu igive rCeba? amocanis amoxsnis nimuSi 6.5a suraTze naCvenebia θ kuTxiT daxrili 6.3 moneta wigni da masze uZravad mdebare m masis moneta. θ kuTxis 13°-mde gazrdisas mone ta Camosrialebis zRvarzea anu am kuTxis umniSvnelod gazrdac ki monetis (a) Camosrialebas gamoiwvevs. risi tolia wignsa da monetas Soris μs statikuri xaxunis koeficienti? amoxsna: wigns rom xaxunisgan Tavisufali moneta zedapiri hqondes, moneta nebismieri daxris SemTxvevaSi Camosrialdeboda, vinaidan masze mizidulobis Zala moqmedebs. aqedan gamomdinare,  fs monetaze xaxunis Zala moqmedebs da mis moZraobas xels uSlis. garda amisa, vinaidan moneta Camosrialebis zRvarzea, xaxunis Zalis sidide maqsimaluria da mimarTulia gaswvriv. 6.1 maRla, wigniszedapiris formulidan gamomdinare fs,max=μsFN , sadac FN aris wignidan monetaze moqmedi drekadobis Zala. gvaZlevs: fs=fs,max= μsFN , zemoTqmuli (b) sur. 6.5. a) wignze mdebare moneta Camosrialebis zRvarzea. b) Tavisufali sxeulis grafiki mone  tisTvis. naCvenebia monetaze moqmedi sami Zala. Fg mizidulobis Zala daSlilia x da y RerZebis gaswvriv komponentebad da maTi mimarTulebebi amocanis amoxsnas amartivebs. Fgsin θ komponenti monetas Fgcos θ komponenti Camosrialebisken ubiZgebs. monetas wignze aCerebs. 174 Zala da moZraoba Tavi 6 komponentebisTvis CavweroT. Fg-s nacvlad mg saidanac viRebT: f μs= s FN am formulis gamoTvlisTvis (6.9) fs FN da CavsvaT da x RerZisTvis miviRebT: fs + 0 – mg sin θ = 0 anu mniSvnelobebi gvesaWiroeba. gavixsenoT, rom moneta Camosrialebis zRvarzea, uZravia da y RerZisTvis gveqneba: 0 + FN – mg cos θ = 0 misi  a aCqareba nulis tolia. aCqareba da monetaze moqmedi Zalebi niutonis meore kanonis saSualebiT SegviZlia erTmaneTTan davakavSiroT. 6.5b suraTze gamosaxulia Tavisufali sxeulis grafiki monetisTvis  da Cans, rom monetaze f s xaxunis Zala,   FN drekadobis Zala da mg sididis Fg mizidulobis Zala moqmedebs. niutonis meore kanonidan gamomdinare gvaqvs: fs da FN    f s + FN + Fg = 0 (6.10) sidideebis gamoTvlisTvis, 6.10 formula 6.5b suraTze gamosaxuli koor dinatTa sistemis x da y RerZebis gaswvrivi fs = mg sin θ (6.11) anu FN = mg cos θ (6.12) 6.11 da 6.12 formulebi 6.9 formulaSi CavsvaT da miviRebT: μs= mg sin θ , mg cos θ (6.13) saidanac gamomdinareobs, rom: μs = tg 13° = 0,23 (pasuxi) μs-is misaRebad θ kuTxis gazomva saWiro ar aris. Tqven SegiZliaT 6.5a suraTze naCvenebi ori sigrZe gazomoT da Semdeg 6.13 formulaSi tg θ-s nacvlad h/d CasvaT. 6.4. wevis Zala da saboloo siCqare yvelaferi, rasac dineba SeuZlia, siTxea, iqneba es airi Tu Txevadi Soris nivTiereba. fardobiTi rodesac siCqare siTxesa arsebobs da (sxeuli sxeuls moZraobs siTxeSi an siTxe sxeulSi gadis), sxeulze fardobiTi moZraobis sawinaaRmdego da sxeulisadmi siTxis moZraobis mimarTulebis Cven  D wevis Zala moqmedebs. ganvixilavT mxolod iseT SemTxvevebs, rodesac siTxe haeria, sxeuli blagvia (beisbolis burTiviT) da ara pirwamaxuli (xelSubiviT), xolo fardobiTi moZraoba imdenad swrafia, rom sxeulis ukan haeri turbulenturi xdeba. aseT dros erTmaneTs  D wevis Zala da ν fardobiTi siCqare eqsperimentulad gansazRvruli C wevis sur. 6.6. moTxilamure ikuzeba, kvercxis formas iRebs da misi ganivi kveTis faqtiuri farTobi mcirdeba, rac masze moqmedi haeris wevis Zalis Semcirebas iwvevs. koeficientis saSualebiT ukavSirdebian: D= 1 CρAν 2 , 2 (6.14) sadac ρ aris haeris simkvrive (masa moculobis erTeulze), xolo A - sxeulis ganivi kveTis faqtiuri farTobi ( siCqaris perpendikularuli ganivi kveTis farTobi). C wevis koeficientis mniSvneloba 0,4-dan 1-mde meryeobs da mocemuli sxeulisTvis mudmivi ar aris. ν siCqaris mniSvnelovnad Secvlis dros C-s mniSvnelobac SeiZleba Seicvalos, Tumca Cven amgvar garTulebebs mxedvelobaSi ar viRebT. moTxilamureebma kargad ician, rom wevis Zala A-sa da ν2-zea damokidebuli. maRali siCqaris ganviTarebisTvis moTxilamure `kvercxis formas~ iRebs da A-s Semcirebis Sedegad D-sac amcirebs. 6.4. wevis Zala da saboloo siCqare 175  D wevis Zala rodesac blagvi sxeuli haerSi vardeba, maRlaa mimarTuli. misi sidide sxeulis siCqaris zrdasTan erTad TandaTanobiT izrdeba. maRla mimarTuli qvemoT mimarTul  D Zala  Fg mizidulobis Zalas ewinaaRmdegoba. am Zalebis da sxeulis aCqarebis erTmaneTTan dakavSireba kvlav niutonis meore kanoniT SeiZleba, romelic y vertikaluri RerZisTvis Semdegi saxiT Caiwereba: (a) (b) (g) sur. 6.7. haerSi vardnil sxeulze moqmedi Zalebi a) vardnis dawyebisas da b) odnav mogvianebiT, wevis Zalis gaCenisas. g) wevis Zala izrdeba da sxeulze moqmed mizidulobis Zalas awonasworebs. amis Semdeg sxeuli mudmivi siCqariT ecema. D – Fg = ma , (6.15) sadac m aris sxeulis masa. 6.7 suraTze Cans, rom xangrZlivi vardnis SemTxvevaSi D gautoldeba Fg-s. 6.15 formulidan gamomdinare es niSnavs, rom a = 0 da sxeulis siCqare aRar izrdeba. amis Semdeg sxeulis siCqare mudmivia da mas νt saboloo siCqare aqvs. νt - s gamoTvlisTvis 6.15 formulaSi a = 0 CavsvaT da 6.14 formulidan D aviRoT: 1 CρAν 2 – t 2 Fg = 0 es formula gvaZlevs: νt (6.16) 6.1 cxrilSi mocemulia νt -s mniSvnelobebi zogierTi sagnisTvis. cxrili 6.1 cxrili 6.1. saboloo siCqareebi haerSi saboloo siCqare (m/wm) manZilis 95% (m)a paraSutisti paraSutis gaxsnamde 145 60 2500 430 beisbolis burTi 42 210 CogburTis burTi 31 115 kalaTburTis burTi 20 47 magidis CogburTis burTi 9 10 wvimis wveTi (1,5 mm radiusis) 7 5 6 3 sagani tyvia paraSutisti gaxsnili paraSutiT a es aris manZili, romelic sxeulma unda gaiaros saboloo siCqaris 95%-is ganviTarebisTvis. 6.14 formulaze dafuZnebuli gamoTvlebis mixedviT, katam daaxloebiT eqvsi sarTuli unda ifrinos, raTa saboloo siCqare ganaviTaros. manamde Fg>D da qvemoT mimarTuli Zalis moqmedebis Sedegad kata Cqardeba. meore paragrafidan gavixsenoT, rom Tqveni sxeuli aCqarebas grZnobs da ara siCqares. vinaidan katac igives ganicdis, mas eSinia, fexebs da Tavs Sekecavs, zurgs amozniqavs, A-s amcirebs, xolo νt-s zrdis, rac dazianebis safrTxes ufro realurs xdis. miuxedavad amisa, xangrZlivi vardnis SemTxvevaSi kata νt siCqares ganaviTarebs, aCqareba qreba da moSvebuli cxoveli saxsrebSi iSleba (am dros kata ciyvs emsgavseba). am dros A farTobi da 6.14 formulidan gamomdinare D wevac izrdeba. vinaidan D>Fg, kata Senelebas iwyebs da axal, ufro mcire νt siCqares iZens. saboloo siCqaris Semcireba miwaze daSvebisas dazianebis safrTxes amcirebs. miwaze daSvebamde kata kvlav kecavs fexebs, raTa dacemisTvis moemzados. 176 Zala da moZraoba Tavi 6 paraSutistebi xSirad Zalian didi simaRlidan xtebian. 1987 wels, paraSutistma gregori robinsonma SeniSna, rom ufro qvemoT mfrinavi debi viliamsi mesame paraSutists Seejaxa, goneba dakarga da paraSuts ver xsnida. 4km simaRlidan momavalma robertsonma Tavi qvemoT mimarTa, raTa A Semcirebuliyo, xolo qvemoT mimarTuli siCqare maqsimaluri gamxdariyo. SeiZina ra 320km/sT saboloo siCqare, man viliamss xeli staca da gaSlilfrTebiani arwivis forma miiRo, raTa D gazrdiliyo da viliamsis xelSi daWera SesaZlebeli yofiliyo. dacemamde 10 wamiT adre robertsonma viliamsis paraSuti gaxsna, Semdeg mas xeli gauSva da sur. 6.8. paraSutisti gaSlilfrTebiani arwivis formas iRebs, raTa haeris weva maqsimaluri gaxdes dacemamde 10 wamiT adre sakuTari paraSutis gaxsnac moaswro. ugonod myofma viliamsma sakmaod mZime dazianebebi miiRo, magram gadarCa. amocanis amoxsnis nimuSi simaRlidan Camovardnili kata 97 km/sT saboloo siCqares iZens, Semdeg saxsrebSi iSleba da A ormagdeba. risi toli iqneba axali 6.4 saboloo siCqarea νt0 da νtn, xolo ganivi kveTis sawyisi da axali farTobia A0 da An, maSin 6.16 formula gvaZlevs: saboloo siCqare? amoxsna: 6.16 formulis Tanaxmad, katis saboloo siCqare cxovelis ganivi kveTis A farTobzea damokidebuli. Tu sawyisi da axali amocanis amoxsnis nimuSi es niSnavs, rom ν tn≈0,7ν to anu axali saboloo siCqare daaxloebiT 68 km/sT-ia. 6.5 R = 1,5 mm radiusis wvimis wveTi dedamiwidan h = 1200 m simaRleze myofi Rrublidan ecema. wveTisTvis wevis koeficienti C aris 0,6. 1) Fg=mg, sadac m aris wveTis masa; 2) wvimis dauSviT, rom wvimis wveTi sferos formisaa. ρw=m/V. maSasadame:L wylis ρw simkvrivea 1000 kg/m3, xolo haeris ρa simkvrivea 1,2 kg/m3. wveTis moculobaa V=3 πR3 da 3) wveTSi wylis 4 simkvrive masaa moculobis erTeulze anu Fg = Vρwg = 3 πR3ρwg 4 zemoT miRebuli formula, A-s gamosaxuleba a) risi tolia wveTis saboloo siCqare? da monacemebi 6.16 formulaSi CavsvaT. ρa haeris amoxsna: wvimis wveTi saboloo νt siCqares simkvrive da ρw wylis simkvrive erTmaneTSi ar maSin SeiZens, rodesac mizidulobis Zala da avurioT da miviRebT: haeris wevis Zala erTmaneTs gaawonasworebs da aCqareba nulis toli gaxdeba. amis Semdeg, niutonis meore kanonis da wevis Zalis formulis daxmarebiT νt-s gamoTvla SesaZlebeli gaxdeba, magram 6.16 formula yvelafers amartivebs. 6.16 formulis gamoyenebisTvis wvimis wveTis ganivi kveTis faqtiuri A farTobi da Fg mizidulobis Zala gvWirdeba. vinaidan wveTi sferos formisaa, wris A farTobia (πR2), romlis radiusi sferos radiusis tolia. Fg-s gamoTvlisTvis gamoviyenoT sami faqti: = = 7,4 m/w ≈ 27 km/sT yuradReba Rrublis miaqcieT, simaRle ar (pasuxi) rom gamoTvlebSi dagvWirvebia. 6.1 cxrilidan gamomdinare, wvimis wveTi saboloo siCqares ramdenime metris gavlis Semdeg iZens. 6.5. Tanabari wriuli moZraoba 177 b) ra siCqare eqneba dacemamde wvimis wveTs wevis Zalis ararsebobis SemTxvevaSi? amoxsna: Tu vardnisas wevis Zala wveTis siCqares ar amcirebs, is g Tavisufali vardnis aCqarebiT imoZravebs da 2.1 cxrilSi moyvanili formulebi gamosadegi gaxdeba. vinaidan viciT, rom aCqareba g-s tolia, sawyisi siCqare nulia da x–x0 gadaadgileba –h-ia, ν-s gamoTvlisTvis 2.16 formula gamoviyenoT: =153 m/w ≈ 550 km/sT sakontrolo wertili (pasuxi) 3. dedamiwis zedapirTan axlos wvimis didi wveTebis siCqare mcire wveTebis siCqareze metia, naklebia Tu igivea? dauSviT, rom wvimis yvela wveTi sferos formisaa da maTi wevis koeficientebic Tanabaria. 6.5. Tanabari wriuli moZraoba 4.7 paragrafidan gaixseneT, rom rodesac sxeuli wreze (an wriul rkalze moZraobs) ν mudmivi siCqariT moZraobs, amgvar moZraobas Tanabarwriuli moZraoba ewodeba. am dros sxeuls centriskenuli (wris centrisken mimarTuli) aCqareba aqvs, romlis sididea: a= ν2 R (centriskenuli aCqareba), (6.17) sadac R aris wris radiusi. modiT Tanabari wriuli moZraobis ori magaliTi ganvixiloT: 1. moZraoba manqaniT. Tqven manqanis ukana savarZlis centrSi zixarT da manqana swor gzaze mudmivi siCqariT moZraobs. Semdeg mZRoli marcxniv uxvevs da wriul rkals Semowers. am dros Tqven savarZlis marjvena mxareze gacurdebiT da mTeli mosaxvevis ganmavlobaSi fanjaras aekrobiT. ra aris amis mizezi? rodesac manqana wriul rkals Semowers, is Tanabarwriulad moZraobs. es niSnavs, rom manqanas wris centrisken mimarTuli aCqareba aqvs. niutonis meore kanonidan gamomdinare, aCqarebas Zala iwvevs. garda amisa, es Zala mimarTulia wris centrisken da maSasadame centriskenulia. moyvanil magaliTSi centriskenuli Zala gzidan saburavebze moqmedi xaxunis Zalaa. Tu manqanasTan erTad Tanabar wriul moZraobas ganicdiT, maSin centriskenuli Zala Tqvenzec moqmedebs. savarZlidan Tqvenze moqmedi xaxunis Zala ar aris sakmarisad didi, raTa manqanasTan erTad wriulad imoZraoT. aqedan gamomdinare, savarZeli Tqvens qvemoT curdeba da manqanis marjvena fanjaras ejaxebiT. Semdeg fanjridan miRebuli biZgi saWiro centriskenul Zalas qmnis da Tqven manqanis Tanabar wriul moZraobas uerTdebiT. 2. dedamiwis garSemo moZraoba. amjerad warmoidgineT, rom kosmosur xomald atlantisis mgzavri xarT da dedamiwis garSemo moZraobisas kabinaSi tivtivebT. ra aris amis mizezi? kosmosuri xomaldi da Tqven Tanabar wriul moZraobas asrulebT da wris centrisken mi marTuli aCqarebebi gaqvT. niutonis meore kanonidan gamomdinare, am aCqarebebs centris kenuli Zalebi iwveven. amjerad centriskenuli Zala dedamiwidan moqmedi mizidulobaa, romelic dedamiwis centrisken aris mimarTuli. orive zemoT moyvanil SemTxvevaSi, Tqven Tanabrwriulad moZraobT da Tqvenze centris kenuli Zala moqmedebs, Tumca SegrZnebebi erTmaneTisgan gansxvavebulia. manqanaSi jdomis dros fanjaras ekrobiT da Zalas grZnobT, xolo kosmosuri xomaldSi Zalis SegrZneba ar arsebobs. ratom xdeba ase? amgvari gansxvaveba ori centriskenuli Zalis bunebiTaa gamowveuli. manqanaSi centriskenuli Zala Tqveni sxeulis im nawils waubiZgebs, romelic fanjaras ekroba, xolo kosmosur xomaldSi 178 Zala da moZraoba Tavi 6 centriskenuli Zala dedamiwis mizidulobaa da is Tqveni sxeulis yvela atomze moqmedebs. swored amitom sxeulis arc erTi nawili masze moqmed Zalas ar SeigrZnobs (amgvar mdgomareobas ùwonoba~ ewodeba, Tumca dasaxeleba ar niSnavs, rom dedamiwa Tqvenze ar moqmedebs). centriskenuli Zalis kidev erTi magaliTi 6.9 suraTzea naCvenebi. hokeis Saiba ν mudmivi siCqariT wriulad moZraobs da masze mibmuli Toki wris centrzea damagrebuli. centriskenuli Zala radialurad centriskenaa mimarTuli da Tokidan Saibaze moqmedebs. am Zalis gareSe Saiba wrfivad gasrialdeboda da wreze ar imoZravebda. yuradReba miaqcieT, rom centriskenuli Zala ar aris Zalis axali saxeoba. saxelwodeba ubralod Zalis mimarTulebas asaxavs. centriskenuli Zala SeiZleba iyos xaxunis Zala, mizidulobis Zala, manqanis fanjridan an sur. 6.9. horizontalur, xaxunisgan Tavisufal zedapirze R radiusis wriuli traeqtoriiT da ν mudmivi siCqariT moZravi hokeis Saibis zed  xedi. Saibaze Tokidan T centri skenuli Zala moqmedebs, romelic mimarTulia Saibaze gamavali r ra dialuri RerZis gaswvriv. Tokidan moqmedi Zala da saerTod nebismieri saxis Zala. yvela SemTxvevaSi: centriskenuli Zala sxeulis veqtoruli siCqaris da ara skalaruli siCqaris mimarTulebas cvlis da ase aCqarebs sxeuls. niutonis meore kanonidan da 6.17 formulidan (a = ν 2 /R) gamomdinare, centriskenuli Zalis F sididea: F=m ν2 R (6.18) (centriskenuli Zalis sidide) vinaidan ν siCqarea mudmivia, Zalis da aCqarebis sidideebic mudmivebia. centriskenuli aCqarebis da Zalis mimarTuleba mudmivi ar aris. is sinusoidalurad icvleba da yovelTvis wris centrisken aris mimarTuli. swored amitom, Zalis da aCqarebis veqtorebs xSirad gamosaxaven r radialuri RerZis gaswvriv, romelic sxeulTan erTad moZraobs da 6.9 suraTis msgavsad wris centridan sxeulisken aris gadaWimuli. RerZi mimarTulia radiusis gaswvriv centridan wrewirisaken, xolo aCqarebis da Zalis veqtori radialurad SigniT aris mimarTuli. amocanis amoxsnis nimuSi 6.6 igori saerTaSoriso kosmosuri sadguris = kosmonavtia. sadguri h = 520 km simaRleze ν = 7,6 km/wm mudmivi siCqariT = 8,38 m/w2 ≈ 8,4 m/w2 dedamiwis garSemo wriul orbitaze moZraobs. igoris m masaa 79 kg. (7,6 × 103 m/w)2 6,37 × 106 m + 0,52 × 106 m (pasuxi) miRebuli pasuxi igoris yofnis simaRleze Tavisufali vardnis aCqarebaa. Tu igori a) risi tolia igoris aCqareba? amoxsna: igori Tanabarwriulad moZraobs da, maSasadame, 6.17 formuliT (a = ν2/R) mocemuli centriskenuli aCqareba aqvs. misi moZraobis radiusia RE+h, sadac RE aris dedamiwis radiusi (6,37 × 106 m). aqedan gamomdinare: a= ν2 ν2 = R RE + h am simaRleze asvlis ar imoZravebs, is Semdeg dedamiwis orbitaze centrisken vardnas daiwyebs da misi aCqarebis sawyisi mniSvneloba zemoT miRebuli pasuxis toli iqneba. gansxvaveba imaSia, rom igori drois nebismier momentSi traeqtoriis mxebis mimarTulebiTac moZraobs, vardnasTan erTad gverdiTac gadaadgildeba da dedamiwis garSemo mrud traeqtorias Semowers. 179 6.5. Tanabari wriuli moZraoba b) ra ZaliT moqmedebs dedamiwa igorze? amoxsna: Tanabarwriuli moZraobis gamo igorze centriskenuli Zala moqmedebs. es Zala dedamiwidan moqmedi romelic brunvis  Fg mizidulobaa, centrisken (dedamiwis centrisken) aris mimarTuli. niutonis meore kanonidan gamomdinare, am Zalis sididea: amocanis amoxsnis nimuSi Fg = ma = (79 kg)(8,38 m/wm2) = 662 n ≈ 660 n (pasuxi) Tu igori h = 520 km simaRlis koSkze dadgeba, sasworis Cveneba 660 n iqneba, magram orbitaze moZraobisas saswori da igori Tavisufal vardnas erTdroulad ganicdian, kosmonavtis fexi saswors ar awveba da Sesabamisad Cvenebac nulia. 6.7 1901 wels Semdgari sacirko warmodgenisas, alo diavolom rgolis formis trasaze velosipediT gaiara (sur. 6.10a). davuSvaT rgoli R=2,7m radiusis wre iyo. ra umciresi ν siCqare unda hqonoda diavolos rgolis zeda nawilSi, raTa qvemoT ar Camovardniliyo? amoxsna: diavolos triuki warmovidginoT, rogorc nivTieri wertilis Tanabari wriuli moZraoba rgolSi. aqedan gamomdinare, rgolis zeda nawilSi nivTier wertils 6.17 formuliT mocemuli  a aCqareba aqvs, romelic rgolis centrisken aris mimarTuli. rgolis zeda nawilze yofnisas nivTier wertilze moqmedi Zalebi 6.10b suraTze gamosaxul Tavisufali sxeulis grafikzea  Fg mizidulobis Zala  da rgolidan nivTier wertilze moqmedi FN drekadobis naCvenebi. Zala y RerZis gaswvriv, qvemoTaa mimarTuli. niutonis meore kanoni y komponentebisTvis gvaZlevs: –FN – Fg = m(–a) da rodesac –FN – mg = m(–ν2/R) wertils rgolTan (6.19) sur. 6.10. a) diavolos reklama da b) Tavisufali sxeulis grafiki cirkis msaxiobisTvis. kontaqtis SenarCunebisTvis saWiro ν umciresi siCqare aqvs, is rgolidan Camovardnis zRvarzea anu velosipedis masas mniSvneloba ar aqvs, FN=0. 6.19 formulaSi FN=0 CavsvaT, ν-sTvis Camovardnis Tavidan acilebisTvis mxolod amovxsnaT da Semdeg cnobili monacemebi zemoT SevitanoT: saWiro. = 5,1 m/w miRebuli siCqaris gadaWarbebaa sakontrolo wertili 4. èSmakis bor (pasuxi) balze” mudmivi siCqariT moZraobisas ra rgolis zeda nawilSi 5,1m/wm-ze meti iqneboda  a aCqarebas da Tqvenze  moqmed FN drekadobis Zalas, rodesac da Camovardnis safrTxe ar daemuqreboda. a) umaRles wertils da b) udables wertils yuradReba miaqcieT, rom diavolos da misi CauvliT? diavolo winaswar darwmunda, rom misi siCqare mimarTuleba eqneba 180 Tavi 6 Zala da moZraoba amocanis amoxsnis nimuSi 6.8 atraqcionebis gamobrZmedili moyvarulebic ki fiTrdebian rotoris gagonebaze, romelic sakuTari centraluri RerZis garSemo mrbunavi, didi, carieli cilindria (sur. 6.11). atraqcionis dawyebamde adamiani cilindris gverdiTi karidan Sedis da tiloTi dafarul kedelTan dgeba. karebis CaketvisTanave cilindri brunvas iwyebs, xolo adamiani, kedeli da iataki masTan erTad moZraobs. garkveuli siCqaris ganviTarebis Semdeg iataki uecrad vardeba, xolo cilindri brunvas ganagrZobs. gasaocaria, magram adamiani dacemis nacvlad kedels ekroba, TiTqos raRac uxilavi Zala akeTebs amas. mogvianebiT iataki Tavis adgilze brundeba da cilindris brunvac neldeba. davuSvaT adamianis tansacmelsa da tilos sur. 6.11. rotori gasarTob parkSi. naCvenebia  adamianze moqmedi Zalebi. centriskenuli Zala FN drekadobis Zalaa, romliTac kedeli adamianze moqmedebs. Cven gvWirdeba ν siCqare, rodesac adamiani Soris μs statikuri xaxunis koeficientia 0,4, dabla Cavardnis zRvarzea. xolo cilindris R radiusia 2,1 m. Tavdapirvelad adamianze gamavali y verti a) ra minimaluri ν siCqare unda hqondeT kaluri RerZi gavavloT da Semdeg niutonis cilindrsa da adamians, raTa iatakis gamoclis Semdeg adamiani dabla ar Cavardes? amoxsna: Tavdapirvelad davsvaT SekiTxva: `ra Zala icavs adamians Cavardnisgan da rogor ukavSirdeba es Zala siCqares, rodesac adamiani cilindris centraluri RerZis garSemo wriulad moZraobs?”. am SekiTxvaze meore kanoni y komponentebisTvis Semdegi saxiT CavweroT: fs – mg = m(0), sadac m aris adamianis masa, xolo mg _ Zalis sidide. am formulaSi fs-is nacvlad μsFN maqsimaluri mniSvneloba CavsvaT da miviRebT: μsFN – mg = 0 pasuxis gasacemad sami faqti moviSvelioT:  1. adamianze moqmedi Fg mizidulobis Zala mis qvemoT Cagdebas cdilobs, magram am Zalas kedlidan moqmedi da maRla mimarTuli xaxunis Zala ewinaaRmdegeba. 2. Tu adamiani Cavardnis zRvarzea, maRla mimarTuli Zala μsFN maqsimaluri mniS  vnelobis mqone f s statikuri xaxunis Zalaa, sadac FN aris cilindridan adamianze  Fg anu mg FN = μ s (6.20) Semdeg adamianze gamavali r radialuri RerZi gavavloT da niutonis meore kanoni am RerZis gaswvrivi komponentebisTvis CavweroT: ( νR ) – FN = m – 2 (6.21) FN-is nacvlad 6.20 formulis Casma da ν-sTvis amoxsna mogvcems: moqmedi drekadobis Zala. 3. drekadobis Zala cilindris centraluri RerZisken horizontaluradaa mimarTuli da, maSasadame, centriskenuli Zalaa, rac wriuli traeqtoriiT moZraobs iwvevs. centriskenuli aCqarebaa a=ν2/R da Zala wris centriskenaa mimarTuli. = 7,17 m/w ≈ 7,2 m/w (pasuxi) yuradReba miaqcieT, rom pasuxi adamianis masaze damokidebuli ar aris da sumoistidan dawyebuli patara bavSviT damTavrebuli yvelas SeuZlia rotoriT garToba. 181 6.5. Tanabari wriuli moZraoba b) Tu adamianis masaa 49kg, ra centriskenuli nebis mizezi is aris, rom adamiani aTvlis Zala moqmedebs masze? arainerciuli amoxsna: 6.21 formulis Tanaxmad: (7,17 m/w)2 ν2 FN = m = (49 kg) 2,1 m R ≈ 1200 n sistemaa da am sistemidan gazomili Zala realuri ar aris. sakontrolo wertili 5. Tu rotori (pasuxi) es Zala mimarTulia centraluri RerZisken, magram adamians iseTi SegrZneba aqvs, TiTqos masze moqmedi Zala radialurad centridan moSorebiT aris mimarTuli. amgvari SegrZ amocanis amoxsnis nimuSi Tavdapirvelad arCavardnisTvis saWiro minimaluri siCqariT moZraobs da Semdeg Cqardeba, a)   f s -is, b) FN -is da g) fs,max-is mniSvnelobebi izrdeba, mcirdeba Tu ar icvleba? 6.9 6.12a suraTze gamosaxuli, m=600kg masis bolidi R=100m radiusis wriul, brtyel trasaze moZraobs. bolidis formisa da masze damagrebuli frTebis gamo haeri qvemoT mimarTuli  FL uaryofiTi weviT moqmedebs. saburavebsa da trasas Soris statikuri xaxunis 0,75 (dauSviT, rom oTxive koeficientia (a) saburavze moqmedi Zalebi identurebia). a) Tu bolidi mosaxvevSi mocurebis zRvarzea, rodesac misi siCqare 28,6m/wm-s aRwevs, ra  sididisaa FL ? amoxsna: Cven  FL vertikaluri Zala bolidis (b) sur. 6.12. a) bolidi brtyel, uswormasworo trasaze  ν mudmivi siCqariT gadaadgildeba. f s xaxunis Zala r radialuri RerZis gaswvriv mimarTul centriskenul Zalas uzrunvelyofs. b) Tavisufali sxeulis grafiki bolidisTvis. wriul horizontalur moZraobas da im faqts unda davukavSiroT, rom bolidi mocurebis es Zala simrudis centridan manqanis gavliT zRvarzea. gadaWimuli r radialuri RerZis uaryofiTi Tavdapirvelad moZraobasTan horizontalur dakavSirebuli oTxi faqti mimarTulebisaa da ν2/R sididis centriskenul CamovweroT: aCqarebas iwvevs. Zalasa da aCqarebas Soris 1. bolidze centriskenuli Zala moqmedebs, kavSiri r RerZis gaswvrivi komponentebisTvis vinaidan is wriul rkalze moZraobs. es niutonis meore kanonis Caweris saSualebiT Zala rkalis simrudis centrisken aris davadginoT: mimarTuli. 2. bolidze moqmedi erTaderTi horizon taluri Zala trasidan saburavebze moqmedi xaxunis Zalaa. maSasadame, centriskenuli Zala xaxunis Zalaa. 3. vinaidan bolidi ar curdeba, xaxunis Zala f s statikuri xaxunis Zalaa (sur. 6.12a). 4. bolidi mocurebis zRvarzea, amitom fs Zalis mniSvnelobaa fs,max=μsFN, sadac FN aris trasidan bolidze moqmedi drekadobis Zala. 6.12b ( νR ) –fs = m – 2 (6.22) fs-is nacvlad fs,max=μsFN CavsvaT da miviRebT: μsFN = m ( νR ) 2 (6.23) amis Semdeg bolidze moqmedi vertikaluri Zalebi ganvixiloT. 6.12b suraTze,  FN y   RerZis dadebiTi mimarTulebisken. Fg = mg  mizidulobis Zala da FL uaryofiTi weva drekadobis Zala mimarTulia maRla, mimarTula qvemoT. y RerZis gaswvriv bolidis aCqareba nulia, amitom y RerZis gaswvrivi suraTze  gamosaxul Tavisufali sxeulis grafikze f s xaxunis Zalaa naCvenebi. komponentebisTvis niutonis meore kanoni ase SeiZleba Caiweros: 182 Zala da moZraoba Tavi 6 FN – mg – FL = 0 ZAalis Sefardeba ukve miRebul FL wevis Zalaze FN = mg + FL anu (6.24) (ν=28,6m/wm siCqareze) Semdegia: (90 m/w)2 FL,90 = (28,6 m/w)2 FL amjerad SegviZlia ori RerZisTvis miRebuli Sedegi gavaerTianoT. amisTvis 6.23 formulaSi FN-is nacvlad 6.24 formula CavsvaT da FLsTvis amovxsnaT: FL = m ( ν2 –g μsR = (600 kg) ( ) CavsvaT FL=663,7n da FL,90-sTvis amovxsnaT: FL,90 = 6572 n ≈ 6600 n (pasuxi) aris Tu ara amgvari uaryofiTi weva sakmarisi (28,6 m/w)2 – 9,8 m/w2 (0,75)(100 m) = 663,7 n ≈ 660 n ) (pasuxi) bolidis Werze gatarebisTvis? Tavdapirvelad gaviTvaliswinoT, bolidze moqmedi mizidulobis Zalaa Fg = mg = (600 kg)(9,8 m/wm2) b) uaryofiTi wevis FL sidide da wevis Zala bolidis siCqaris kvadratzea damokidebuli rom =5880 n. bolidis amotrialebis SemTxvevaSi uaryofiTi (6.14 formula). maSasadame, uaryofiTi weva weva 6600n sididis maRla mimarTuli Zalaa. metia, rodesac manqana swrafad moZraobs. es Zala 5880n sididis qvemoT mimarTul ra sididisaa uaryofiTi weva 90m/wm siCqaris SemTxvevaSi? lids Teoriulad SeuZlia amotrialebulad amoxsna: FL proporciulia ν -is. maSasadame, 2 ν=90m/wm siCqareze FL,90 uaryofiTi wevis amocanis amoxsnis nimuSi trasis mizidulobis Zalas aWarbebs. maSasadame, bo mosaxveviani nawili gadaadgileba, Tu misi siCqare daaxloebiT 90m/wm-ia (324km/sT). 6.10 yovelTvis daxrilia, raTa manqana ar mocurdes. mSral amindSi saburavebsa da gzis safars Soris moqmedi xaxunis Zala mocurebas xels uSlis, magram wvimian amindSi xaxunis Zala SesaZloa umniSvnelo gaxdes da am SemTxvevaSi trasis daxra aucilebelia. 6.13a suraTze (a) gamosaxulia R=190m radiusis wriul, daxril trasaze ν=20m/wm mudmivi siCqariT moZravi m masis manqana (es manqana sportuli ar aris da amitom haeridan moqmedi vertikaluri Zala umniSvneloa). Tu trasidan moqmedi xaxunis Zala umniSvneloa, ra θ kuTxiT daxraa saWiro mocurebis Tavidan asacileblad? amoxsna: manqana wriuli traeqtoriiT (b) moZraobs da amitom masze centriskenuli Zala moqmedebs. am SemTxvevaSi trasa daxrilia da manqanaze moqmedi  FN drekadobis Zala wris centrisken aris gadaxrili (sur. 6.13b). aqedan gamomdinare,  FN Zalas FNr sididis centriskenuli komponenti aqvs, romelic r radialuri RerZis gaswvriv, SigniTken aris mimarTuli. Cven unda davadginoT daxris θ kuTxis sidide, rodesac centriskenuli sur. 6.13. a) manqana wriul, daxril trasaze ν mudmivi siCqariT moZraobs. daxris kuTxe sicxadis mizniT gadidebulia. b) Tavisufali sxeulis grafiki manqanisTvis. daSvebulia, rom saburavebsa da trasas Soris moqmedi xaxuni nulia da manqanaze uaryofiTi weva ar moqmedebs. drekadobis Zalis radialurad SigniTken mimarTuli (r radialuri RerZis gaswvriv) FNr komponenti saWiro centriskenul Zalas da radialur aCqarebas uzrunvelyofs. 183 6.5. Tanabari wriuli moZraoba komponenti, xaxunis aucileblobis gareSe, xolo y RerZis gaswvriv manqanis aCqareba nulis manqanas wris formis trasaze Seakavebs. tolia. yovelive zemoTqmulidan gamomdinare, 6.13b suraTidan Cans, rom vertikalsa da y RerZis gaswvrivi komponentebisTvis niuto  FN Zalas Soris kuTxe trasis daxris θ nis meore kanoni aseT saxes iRebs: kuTxis tolia. maSasasadame, FNr radialuri komponenti FNsinθ-is tolia. amis Semdeg r FN cos θ – mg = m(0) RerZis gaswvrivi komponentebisTvis niutonis saidanac gamomdinareobs: meore kanoni Semdegi saxiT Caiwereba: ( νR ) 2 –FN sin θ = m – (6.25) am formulas θ-sTvis ver amovxsniT, vinaidan FN da m ucnobia. modiT y RerZis gaswvriv arsebuli Zalebi da aCqareba ganvixiloT. drekadobis Zalis FN cos θ = mg (6.26) miRebuli formula kvlav Seicavs FN da m ucnobebs, magram 6.25 formulis 6.26 formulaze gayofa orive ucnobs abaTilebs. amis Semdeg (sin θ)/(cos θ) SevcvaloT tan θ-iT da θ-sTvis amovxsnaT: θ = tg–1 vertikaluri komponentia FN,y = FN cosθ, man qanaze moqmedi mizidulobis Zalis sididea mg, = tg–1 ν2 gR (20 m/w)2 = 12° (pasuxi) (9,8 m/w2)(190 m) mimoxilva da Sejameba  xaxuni. rodesac F Zala sxeulis zedapirze gasrialebas cdilobs, zedapiridan sxeulze swrafad mcirdeba: fk = μkFN (6.2) xaxunis Zala moqmedebs. xaxunis Zala zedapiris sadac μk aris kinetikuri xaxunis koefi paraleluria da srials ewinaaRmdegeba. am cienti. Zalas sxeulisa da zedapiris Sewebeba iwvevs. Tu sxeuli ar moZraobs, xaxunis Zala  fs statikuri xaxunis Zalaa. moZraobis Sem  TxvevaSi xaxunis Zala f k kinetikuri xaxunis Zalaa. arsebobs, sxeulze moZraobis sawinaaRmdego da siTxis dinebis mimarTulebis xaxunis Zala  F -is zedapirisadmi da paraleluri komponenti tolia, xolo  f s Zalis mimarTuleba F-is komponentis sapirispiroa. komponentis zrdisas fs-ic izrdeba.  f s Zalis fs,max mniSvnelobaa: fs,max = μsFN , sadac μs aris statikuri (6.1) xaxunis  F -is zedapirisadmi paraleluri fs,max-s  D Zala da ν fardobiTi siCqare aWarbebs, wevis koeficientis saSualebiT ukavSirdeba: D = 1 CρAν2, (6.14) 2 sadac ρ aris siTxis simkvrive (masa moculobis erTeulze), xolo A – sxeulis ganivi kveTis faqtiuri farTobi ( siCqaris perpen dikularuli ganivi kveTis farTobi). saboloo siCqare. rodesac blagvi sagani koeficienti, xolo FN - drekadobis Zala. komponenti  D wevis Zala erTmaneTs eqpserimentulad gansazRvruli C 1. Tu sxeuli ar moZraobs, f s statikuri  Tu raime sxva siTxes) Soris fardobiTi moZraoba moqmedebs. xaxunis sami Tviseba: 2. wevis Zala. rodesac sxeulsa da haers (an sxeuli zedapirze moZraobs. 3. Tu sxeuli zedapirze moZraobs, xaxunis Zalis sidide fk mudmiv mniSvnelobamde haerSi didi xnis ganmavlobaSi vardeba,   D wevis Zala da sxeulze moqmedi Fg mizidulobis Zala Tanabrdeba. amis Semdeg sxeuli νt saboloo siCqariT vardeba: νt Tanabarwriuli (6.16) moZraoba. Tu wertili wreze an R radiusis wriul rkalze ν mudmivi siCqariT moZraobs, wertili Tanabrwriulad 184 Zala da moZraoba Tavi 6 moZraobs. am SemTxvevaSi wertils  a cen a= F=m triskenuli aCqareba aqvs: centriskenuli Zala iwvevs: ν R 2 (6.17) centriskenul aCqarebas wertilze moqmedi ν2 , R (6.18)   sadac m aris wertilis masa. a da F veqtoruli sidideebi wertilis traeqtoriis simrudis centrisken aris mimarTuli. SekiTxvebi 1. bloki zedapirze Zevs da masze sam eqspe 5. meoTxe SekiTxvas Tavidan gaeciT pasuxi, riments atareben, romelTa drosac blokze roca wina variantisgan gansxvavebiT. sami sxvadasxva horizontaluri Zala mo maRlaa mimarTuli. qmedebs. ZalTa sidideebia F1 = 12 n, F2 = 8 n 6. da F3 = 4 n. arcerT eqsperimentSi bloki moZraobas ar iwyebs. daalageT Zalebi a) zeda piridan blokze moqmedi fs statikuri xaxunis Zalis mixedviT da b) am Zalis fs,max maqsimaluri mniSvnelobis mixedviT dididan mcireze. dan moqmedi xaxunis Zala m masis bloks uZravad sur. 6.16. SekiTxva 6.  F Zala iwyebs mo qmedebas. am Zalis zrdis dros rogor icvleba blokze moqmedi xaxunis Zalis sidide da F1 mimarTuleba? Zala yuTzea mo sur. 6.14. SekiTxva 2. debuli, romelic  ar moZraobs. amis Semdeg vertikaluri F2  Zala iwyebs moqmedebas. a) yuTze moqmedi f s xaxunis Zala, b) zedapiridan yuTze moqmedi  FN suraTze daxrili sibrtyi aCerebs. amis Semdeg blokze 2. 6.14 suraTze 10n sididis hori  zontaluri 6.16 F Zala drekadobis Zala, g) yuTze moqmedi statikuri xaxunis Zalis fs,max maqsimaluri 7. meeqvse SekiTxvas pasuxi Tavidan gaeciT, F Zala sapirispirod  mimarTeT. F -is zrdasTan erTad rogor oRond amjerad icvleba blokze moqmedi xaxunis Zalis sidide da mimarTuleba? 8. 1987 wels, orma paraSutistma helouinis* mniSvneloba izrdeba, mcirdeba Tu ucvleli aRsaniSnavad gogrebi daiWira da ise gadmoxta. rCeba? d) iwyebs Tu ara yuTi moZraobas? mogvianebiT, ukan momavalma paraSutistma 3. Tu yuTs kedelTan miitanT da xels ise ra Tze miaWerT, rom mimarTuleba moqmed  fs yuTi eqneba ar a) Camovardes, kedlidan statikuri xaxunis yu Za  las da b) kedlidan yuTze moqmed FN drekadobis Zalas? ufro Zlierad miwo lis SemTxvevaSi rogor Seicvleba g) fs , d) FN da e) fs,max? 4. 6.15 Soris saxlis saxuravs daeca, samzareulos fanjaraSi Sevarda da yvelaferi mosvara. ratom gaeSva xelidan gogra paraSutists gogris da paraSutistis TvalTaxedvidan? 9. 6.17 suraTze 10kg masis firfita xaxunisgan Tavisufal zedapirze uZravad devs da masze moTavsebuli. bloksa da firfitas Soris horizontalsa da Zalas gausxlta. gogram daaxloebiT 0,5 km ifrina, 100n sididis horizontaluri Zala iwyebs moqmedebs. firfitaze 10kg masis blokia suraTze yuTi uZravia da  F sakuTari paraSuti gaxsna da gogra xelidan μ xaxunis koeficienti ucnobia da bloki sur. 6.15. SekiTxva 4. θ kuTxe izrdeba. a) Fx , b) fs , g) FN , d) fs,max izrdeba, mcirdeba Tu ucvleli rCeba? e) Tu yuTi moZraobs da θ sur. 6.17. SekiTxva 9. izrdeba, yuTze moqmedi xaxunis Zala izrdeba, mcirdeba Tu ucvleli rCeba? * helouin (Halloween) _ yvela wmindanis dRis wina dRe. 185 SeiZleba gasrialdes. a) mniSvnelobaTa ra arealSi SeiZleba moxvdes firfitis af eqtoria R0, 2R0 da 3R0 radiusebis xuTi wriuli rkalisgan Sedgeba. daalageT rkalebi aCqarebis sidide? (miniSneba: mravalricxovani adamianze gamoTvlebi mixedviT dididan mcireze. saWiro ar aris. ubralod μ-s ukiduresi mniSvnelobebi ganixileT). b) mniSvnelobaTa ra arealSi SeiZleba moxvdes blokis ab aCqarebis sidide? moqmedi centriskenuli Zalis 11. eSmakis borbalze mjdomi adamiani zeda, qveda da Sua mdebareobebs gadis. borbali Tanabari tempiT moZraobs. daalageT 10. 6.18 suraTze es sami mdebareoba a) adamianze moqmedi naCvenebia parkSi centriskenuli aCqarebis, b) adamianze moqmedi moseirne adamianis jamuri centriskenuli Zalis sididis da g) traeqtoria. miani ada adamianze moqmedi drekadobis Zalis sididis sur. 6.18. SekiTxva 10. mudmivi mixedviT dididan mcireze. siCqariT moZraobs da misi moZraobis tra amocanebi SSM amoxsnebi mocemulia Student Solutions Manual-Si. www amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday ILW Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley. com/college/halliday. • – ••• wertilebis raodenoba gviCvenebs amocanis sirTules paragrafi 6.3. xaxunis Tvisebebi sur. 6.19. amocana 4. ram gadaadgila qva? •1. WurWelsa da masSi Calagebul kvercxebs Soris statikuri xaxunis koeficientia 0,04. kaliforniaSi, `sikvdilis velSi~, qvebi udabnos qviSaze kvals avleben da STabeWdileba horizontisadmi ra umciresi kuTxe gamoiwvevs rCeba, rom isini gadaadgildebian (sur. 6.19). WurWlis fskerze kvercxebis gacurebas? wlebis ganmavlobaSi es movlena saidumloebiT •2. m = 79 kg masis beisbolis moTamaSe meore bazaSi misrialebs da 470 n sididis xaxunis iyo moculi. erT-erTi ganmartebis Tanaxmad Zalis winaaRmdegobas ganicdis. risi tolia gadaadgilebdnen qvebs da udabnos gaSrobis μk Semdeg kvali rCeboda. gazomvis Sedegad moTamaSesa da miwas Soris arsebuli Zlieri qarebi wvimisgan darbilebul miwaze dadginda, rom qvebsa da svel miwas Soris kinetikuri xaxunis koeficienti? •3. sawol oTaxSi 45 kg masis komodi uZravad Zevs. komodi gamosawevi ujriTa da tansacmlis CamosakidiT aris aRWurvili. a) Tu komodsa da iataks Soris statikuri xaxunis koeficientia 0,45, ra minimaluri horizontaluri Zalaa kinetikuri xaxunis koeficienti daaxloebiT 0,8-ia. ra horizontalurma Zalam unda imoqmedos 20 kg masis qvaze, raTa qarisgan amoZravebulma qvam moZraoba ganagrZos? (istoria 35-e amocanaSi grZeldeba). saWiro komodis gadaadgilebisTvis? b) komods •5. adamiani 55 kg masis yuTs 220 n ZaliT 17 kg masis gamosawevi ujra da tansacmlis horizontalurad ubiZgebs. kinetikuri Camosakidi moxsnes. amjerad ra minimaluri xaxunis koeficientia 0,35. a) ra sididisaa Zalaa saWiro komodis gadaadgilebisTvis? xaxunis Zala? b) risi tolia yuTis aCqareba? SSM WWW SSM ILW •4. idumalebiT moculi moZravi qvebi. •6. vagoni-platforma yuTebiT aris 186 Zala da moZraoba Tavi 6 datvirTuli, romelTa statikuri xaxunis koeficienti zedapirTan 48 km/sT matarebeli aris siCqariT 0,25. Tu moZraobs, ra manZilze SeiZleba matarebeli mudmivi aCqarebiT ise gaCerdes, rom yuTebi gasrialdnen? ar da zedapirs Soris xaxunis koeficientebia μs = 0,4 da μk = 0,25.gamoTvaleT blokze moqmedi xaxunis Zala, Tu P Zalis sididea a) 8n, b) 10 n da g) 12 n. •11. muSam 35 kg masis yuTs 110 n ZaliT horizontalurad ubiZga. yuTsa da zedapirs  •7. 3,5 kg masis bloks 15n sididis F Zala horizontisadmi θ = 40° kuTxiT ubiZgebs Soris statikuri xaxunis koeficientia 0,37. (sur. 6.20). bloksa da zedapirs Soris kine g) risi tolia zedapiridan yuTze moqmedi tikuri xaxunis xaxunis Zala? d) warmoidgineT, rom meore a) risi tolia fs,max? b) amoZravdeba Tu ara yuTi? 0,25. muSa yuTs maRla awveba da pirvels exmareba. gamoTvaleT a) zeda ra umciresi vertikaluri biZgia saWiro, piridan raTa pirveli muSisgan modebulma 110 n Zalam koeficientia blokze yuTi gadaaadgilos? e) Tu meore muSa yuTs moqmedi xaxunis Za sur. 6.20. amocana 7. la da b) blokis aC pirvels, ra umciresi biZgi aamoZravebs yuTs? qareba. •8. 6.21 lis •12. 1915 wels, fi suraTze gamosaxulia sria ladelfieli moyvaruli Rori, 35° horizontalurad ubiZgebs da ise exmareba sincoski romelic henri koWebze orive xeliT Camo daxris mqone ekida. aRsaniSnavi is sasrialoze or aris, rom sincoskis met droSi cera TiTebi koWebis srialdeba, vidre erT mxareze hqonda jer mas moqceuli, dasWirdeboda xolo Tavisu danarCeni TiTebi - fali zedapiris mqo meore mxareze (sur. xaxunisgan 6.23). sincoskis masa ne igive sasrialoze. sur. 6.21. amocana 8. risi tolia Rorsa 79 kg iyo. Tu xelsa da sasrialos Soris da arsebuli kinetikuri xaxunis koeficienti? statikuri •9. Saflbordis TamaSis dros, studentebma koeficientia saerTo sacxovreblis gadaadgilebisTvis koridorSi cocxis wignis gamoyeneba gadawyvites. maT 3,5 kg masis wignze 25 n horizontaluri ZaliT imoqmedes, wignma koWas Soris xaxunis sur. 6.23. amocana 12. 0,7, ra umciresi drekadobis Zala moqmedebda TiToeuli cera TiTidan koWaze? (Camokidebis Semdeg sincoski xelebze aiwia da koWebze gadaadgileba daiwyo. Tu ggoniaT, rom es 0,9 m gaiara da Semdeg 1,6 m/wm siCqare SeiZina. Zalian advilia, scadeT da igive gaimeoreT). risi tolia wignsa da iataks Soris arsebuli •13. 68 kg masis yuTze horizontisadmi 15°-iT kinetikuri xaxunis koeficienti? daxrili Tokia mobmuli da yuTs zedapirze •10. miaxoxeben. 2,5kg masis bloki horizontalur zedapirze uZravad Zevs. Semdeg blokze 6n sididis  F ho Tu statikuri xaxunis imoqmedos Tokidan, raTa yuTi amoZravdes? b) Tu μk = 0,35, ra sawyis aCqarebas SeiZens rizontaluri Zala  da P vertikaluri yuTi? SSM ZalebiT imoqmedes (sur. 6.22). bloksa a) koeficientia 0,5, ra umciresma Zalam unda sur. 6.22. amocana 10. •14. 6.24 suraTze naCvenebia mTis gverdze gamavali gzis ganivi kveTi. AA´ wrfe sibrtyes 187 asaxavs, ro yuTsa da zedapirs Soris statikuri xaxunis melzec sriali koeficientia 0,35. a) ra kuTxe unda iyos SesaZlebelia. gzis horizontalsa da Toks Soris, raTa qviSis Tavze ganlagebuli udidesi SesaZlo raodenobis gadatana gaxdes B bloki zeda kldi SesaZlebeli? b) risi tolia yuTisa da qviSis sgan didi napraliT aris sur. 6.24. amocana 14. gamoyofili da amitom mxolod bloksa da sibrtyes Soris arsebuli xaxuni uSlis xels blokis Camovardnas. blokis masaa 1,8 × 107kg, sibrtyis θ daxris kuTxea 24°, xolo bloksa da sibrtyes Soris statikuri xaxunis koeficientia 0,63. a) daamtkiceT, rom bloki ar Camovardeba. b) napralSi wyali Cavida da gaiyina. gayinvis Sedegad gafarToebuli  wyali blokze AA´-s paraleluri F ZaliT moqmedebs. ra minimaluri F Zala gamoiwvevs blokis wona am mdgomareobaSi? ••18. avariis dros A manqana wiTel Suqze gaCerebul B manqanas uknidan Seejaxa. gza gorakze Cadis da misi daxrilobaa θ = 12°. manqanebs Soris d = 24 m manZili iyo, rodesac A manqanis mZRolma daiwyo damuxruWeba. A manqanis siCqare damuxruWebis dawyebisas ν = 18 m/wm iyo. ra siCqariT Seejaxa A manqana B-s, Tu kinetikuri xaxunis koeficienti iyo a) 0,6 (mSral gzaze) da b) 0,1 (sveli foTlebiT dafarul gzaze)? Camovardnas? ••15. 5n wonis bloks horizontaluri Zala 12n F sur. 6.27. amocana 18. vertikaluri kedlisken awveba (sur. 6.25). kedelsa da sur. 6.25. amocana 15. bloks Soris statikuri xaxunis koeficientia ••19.  6.28 suraTze gamosaxul 45n wonis blokze P Zala moqmedebs. bloki horizontalisadmi θ = 15°-iT daxril 0,6, xolo kinetikuri xaxunis koeficientia 0,4. davuSvaT bloki Tavdapirvelad uZravia. sibrtyeze uZravad a) amoZravdeba Tu ara bloki? b) risi tolia sibrtyes Soris xa- kedlidan blokze moqmedi Zala erTeulovan- xunis koeficiente- veqtorul CanawerSi? SSM da sur. 6.28. amocana 19. bia μs = 0,5 da μk = 0,34. ra xaxunis Zala mo sibrtyidan veqtorul CanawerSi, a) (–5 n) , b) (–8n) izontaluri Zala blokze erTeulovan- rodesac  P aris da g) (–15n) ? gada ••20. marxili da masze damjdari pingvini 80 n-s iwonis. marxili horizontalisadmi θ = 20°-iT 6.26 daxril sibrtyeze uZravad Zevs (sur. 6.29). moqmedebs da ze adgilebs. bloksa qmedebs ••16. 4,1kg masis blokze 40 n hor dapirze Zevs. suraTze gamo saxulia blo sur. 6.26. amocana 16. marxilsa da sibrtyes Soris statikuri xaxunis koeficientia 0,25, xolo kinetikuri xaxunis kis ν siCqaris t droze damokidebulebis koeficientia 0,15. a) sibrtyis paraleluri grafiki, ra umciresi rodesac gavlebuli risi tolia bloki zedapirze x RerZis gaswvriv moZraobs. bloksa da zedapirs Soris arsebuli kinetikuri xaxunis koeficienti? ••17. Tavdapirvelad uZravi qviSis yuTi Tokis meSveobiT unda gadaaadgilon da Tokis daWimulobam 1100n-s ar unda gadaaWarbos.  F Zalaa saWiro, raTa marxili ar Camosrialdes? b) ra umciresi sididis F Zala aitans marxils maRla? g) ra F Zalaa saWiro, raTa marxilma mudmivi siCqariT imoZraos maRla? sur. 6.29. amocana 20. 188 Zala da moZraoba Tavi 6 ••21. 6.30 suraTze gamosaxuli B blo kis wonaa 711n. ••25. 3,6 n da 7,2 n wonis ori bloki umniSvnelo masis TokiTaa gadabmuli da 30°-iT daxril bloksa magi Soris kinetikuri xaxunis koeficientia 0,1, da sibrtyeze srialebs. msubuq bloksa da sibrtyes stati xolo mZime bloksa da sibrtyes Soris igive kuri xaxunis koe koeficienti 0,2-ia. davuSvaT msubuqi bloki 0,25, θ mZimeze win aris. gamoTvaleT a) blokebis das Soris ficientia kuTxea 30°, xolo B-sa da kvanZs Soris aCqareba da b) Tokis daWimuloba. SSM sur. 6.30. amocana 21. gabmuli Toki horizontaluria. gamoTvaleT A blokis maqsimaluri wona, romlisTvisac sistema uZraobas SeinarCunebs. SSM WWW ••26. 6.34 suraTze gamosaxuli ori yuTi erTmaneTTan blokze gadakidebuli TokiT aris dakavSirebuli. A yuTis masaa 10kg, xolo A-sa da daxril ••22. 6.31 suraTze gamosaxulia `Ciriouzebis~ yuTi (mc=1kg) da ùitiebis~ yuTi (mw=3kg).  sibrtyes Soris Zala moqmedebs da daxris `Ciriouzebis~ yuTze horizontaluri F kinetikuri xaxunis 0,2. koeficientia θ kuTxea yuTebi aCqarebulad 30°. A yuTi mudmivi moZraobs. `Ciriuo siCqariT zebis~ yuTze moqme sur. 6.31. amocana 22. di xaxunis Zalaa 2n, xolo ùitiebis~ yuTze 4n sididis xaxunis  F Zalis sididea 12n, qvemoT srialdeba. ra masa aqvs B yuTs? sur. 6.34. amocana 26 da 27. sami ••27. 6.34 suraTze gamosaxuli A yuTi 102n-s iwonis, xolo B yuTis wona 32 n-ia. A-sa da daxril sibrtyes Soris xaxunis koeficientebia μs = 0,56 da μk = 0,25. θ kuTxea 40°. risi tolia A yuTis aCqareba erTeulovanveqtorul CanawerSi, Tu Tavdapirvelad a) A bloki 0,5m/wm2 aCqa uZravia, b) daxril sibrtyeze maRla adis da rebas iZens. pirveli b) qvemoT srialdeba? SSM Zala moqmedebs. Tu ra Zala moqmedebs `Ciriouzebis~ yuTidan ùitiebis~ yuTze? ••23. 6.32 suraTze gamosaxuli masaa M, meore da blokis xolo sur. 6.32. amocana 23 da 79. mesame blokebis masebia 2M. risi tolia meore bloksa da magidas Soris arsebuli kinetikuri xaxunis koeficienti? ••28. 6.35 suraTze gamosaxuli A blo kis wonaa 44 n, xolo B bloki 22 n-s iwonis. a) ra ••24. 6.33 suraTze minimaluri gamosaxul sam yuTs unda 440 n sididis hori bloks, raTa A ar zontaluri F Zala sur. 6.33. amocana 24. betonis zedapirze hqondes C sur. 6.35. amocana 28. gasrialdes? magidasa da A-s Soris statikuri xaxunis koeficientia 0,2. b) davuSvaT C bloki yuTs Soris kinetikuri xaxunis koeficientia A-dan aiRes. ra aCqarebas SeiZens A bloki, Tu A-sa da magidas Soris kinetikuri xaxunis koeficientia 0,15? 0,7. a) ra F32 Zala moqmedebs mesame yuTidan ••29. 6.36 suraTze gadaadgilebs. yuTebis masebia m1=30kg, wona m2=10kg da m3=20kg. zedapirsa da TiToeul meoreze? zedapirze b) Tu yuTebi moZraoben, sadac gaprialebul kinetikuri xaxunis koeficienti 0,7-ze naklebia, F32 Zalis sidide metia, naklebia Tu ar icvleba? gamosaxuli bloki ori (m1 = 16 kg da M = 88 kg) erT maneTze damagre sur. 6.36. amocana 29. 189 buli ar aris. blokebs Soris statikuri xaxunis 0,38, koeficientia xolo ••34. davuSvaT 6.14 formula pilotsa da mo katapultirebad savarZelze moqmed wevis zrdili bloki xaxunisgan Tavisufal ze  dapirzea ganlagebuli. ra minimaluri F Zalas gvaZlevs, rodesac piloti 1300 km/ horizontaluri Zalaa saWiro, raTa patara gadmoxteba. savarZlisa da pilotis masa da bloki dididan ar Camosrialdes? ILW wevis •••30. 6.37 suraTze gamosaxuli m1 = 40 kg masis firfita xaxunisgan zedapirze uZravad Zevs, sur. 6.37. amocana 30. xo lo firfitaze m2 = 10 kg masis blokia gan lagebuli. bloksa siCqariT moZravi koeficientebi TviTmfrinavidan Tanabaria. pilotis masa Tqven SearCieT, xolo Sesabamisi νt 6.1 cxrilidan aiReT. gamoTvaleT a) pilotze da savarZelze moqmedi wevis Zala da b) maTi Tavi sufal sT da firfitas Soris statikuri xaxunis koeficientia 0,6, xolo kinetikuri xaxunis koeficientia 0,4. blok horizontaluri Seneleba TviTmfrinavidan gadmoxtomis dros. ••35. meoTxe amocanis gagrZeleba. amjerad 6.14 formula 20 kg masis qvaze moqmed haeris wevis Zalas gvaZlevs. qvas 0,4 m2 farTobis vertikaluri ganivi kveTi aqvs da misi C wevis  ze 100 n sididis F horizontaluri Zala koeficientia 0,8. haeris simkvrivea 1,21 kg/ moqmedebs. risi tolia a) blokis da b) m3, xolo kinetikuri xaxunis koeficientia firfitis aCqareba erTeulovan-veqtorul 0,8. a) dedamiwis gaswvriv qaris ra V siCqarea CanawerSi? saWiro, raTa qva moZraobis dawyebis Semdeg ar •••31. 1000 kg masis navi 90 km/sT siCqariT moZraobs, rodesac misi Zrava gamorTulia.  wyalsa da navs Soris arsebuli f k xaxunis Zala navis ν siCqaris proporciulia da fk = 70ν, sadac ν m/wm-ebSia, xolo fk – niutonebSi. ra droSi Semcirdeba navis siCqare 45 km/sT-mde? SSM gaCerdes? dedamiwis gaswvriv mqrolav qars dedamiwa anelebs, amitom qaris siCqare xSirad 10 m simaRleze izomeba. davuSvaT qaris siCqare dedamiwis gaswvriv siCqaresTan SedarebiT 2-jer metia. b) a) kiTxvaze pasuxisTvis qaris ra siCqare gaizomeba Stormis dros? g) aris Tu ara es siCqare realuri? (istoria 53-e amocanaSi grZeldeba). paragrafi 6.4. wevis Zala da saboloo siCqare paragrafi 6.5. Tanabari wriuli moZraoba •36. saburavebsa da gzas Soris statikuri •32. gaSlilfrTebiani arwivis mdgomareobaSi paraSutistis saboloo siCqarea 160 km/sT, xolo pikirebis dros igive siCqare 310 km/ sT-s aRwevs. davuSvaT paraSutistis C wevis xaxunis koeficientia 0,6 da manqanaze uar koeficienti sxvadasxva mdgomareobaSi ar •37. ra umciresi radiusis mqone mosaxvevSi SeZlebs Sesvlas velosipedisti, Tu is 29 km/ icvleba. gamoTvaleT orive mdgomareobis ganivi kveTis faqtiuri A farTobebis Sefar yofiTi weva ar moqmedebs. ra siCqaris dros iqneba manqana mosrialebis zRvarze, Tu is 30,5 m radiusis rkals Semowers? sT siCqariT moZraobs, saburavebsa da trasas deba. Soris statikuri xaxunis koeficientia 0,32 ••33. gamoTvaleT 1000 km/sT siCqariT da 10 km da trasa daxrili ar aris? ILW simaRleze mfrinav reaqtiul TviTmfrinavze moqmedi wevis Zalis Sefardeba amis naxevar •38. bobsleis Sejibris dros iamaikelebis gundi 7,6 m radiusis mosaxvevSi 96,6 km/sT siCqareze da simaRleze mfrinav propelerian siCqariT Sedis. gamoTvaleT am gundis aCqareba TviTmfrinavze moqmed wevis ZalasTan. 10 km g-s erTeulebSi. simaRleze haeris simkvrivea 0,38 kg/m , xolo 5 km simaRleze – 0,67 kg/m3. davuSvaT, rom Tvlems orive TviTmfrinavis ganivi kveTis faqtiuri manZiliT aris daSorebuli. Semdeg atraqcioni 3 farTobi da, agreTve, C wevis koeficienti erTi da igivea. •39. kata uZravad gaCerebul atraqcionze da atraqcionis centridan 5,4 m CarTes da is yovel 6 wamSi erT srul bruns 190 Tavi 6 akeTebs. ra Zala da moZraoba umciresi koeficienti unda statikuri arsebobdes xaxunis katasa da atraqcions Soris, raTa kata uZravad darCes? Zelakidan kabinaze, Tu kabinis siCqarea ν = 5m/wm? ra g) sididis da d) mimarTulebisaa igive Zala, Tu ν = 12 m/wm? •40. atraqcion `rusuli mTebis~ manqana 1200 kg-s iwonis, rodesac mgzavrebiT savsea. 18 m radiusis wriuli borcvis mwvervalze zis. gavlisas manqanis siCqare ar icvleba. ra 556 n-ia. a) am dros studenti `simsubuqes~  a) sidide da b) mimarTuleba aqvs trasidan ••45. 660n wonis studenti èSmakis borbalze~ umaRles wertilSi savarZlidan  studentze moqmedi FN drekadobis Zala grZnobs Tu `simZimes”? b) ra sididis FN Zala manqanaze moqmed drekadobis Zalas borcvis moqmedebs atraqcionis udables wertilSi? mwvervalze, Tu manqana ν = 11 m/wm siCqariT Tu borblis siCqares gavaormagebT, ra iqneba moZraobs? ra g) sidide da d) mimarTuleba aqvs FN sidide g) umaRles da d) udables wertilSi? drekadobis Zalas, Tu ν = 14 m/wm? SSM ILW •41. 6.38 suraTze gamosaxuli manqana wriuli formis goraksa da velze mudmivi siCqariT ••46. manqana 9,1 m radiusis brtyel mosaxvevSi 16 km/sT siCqariT Sedis. vertikalisadmi ra moZraobs. kuTxes adgens Tavisuflad Camokidebuli gorakis mwvervalze, manqanis savarZlidan mZRolze moqmedi drekadobis Zala nulis tolia. mZRolis masaa 70 kg. ra drekadobis ZaliT imoqmedebs manqanis savarZeli mZRolze, rodesac manqana velis udables wertils gaivlis? usafrTxoebis Rvedi? ••47. TviTmfrinavi horizontalur wre ze 480km/sT siCqariT mifrinavs (sur. 6.39). misi frTebi hori zontisadmi 40° θ = kuTxiT aris daxrili. ri sur. 6.39. amocana 47. si tolia   wris   ra sur. 6.38. amocana 41. diusi? dauSviT, rom saWiro Zalas frTis ••42. damnaSavisTvis dadevnebuli policiis zedapirisadmi perpendikularuli àerodina oficeri 300 m radiusis mosaxvevSi 80 km/sT mikuri weva~ uzrunvelyofs. SSM WWW mudmivi siCqariT Sedis. misi masa 55 kg-ia. ••48. WanWiki viw- risi tolia oficridan manqanis savarZelze ro, WanWiki horizonta moqmedi jamuri Zalis a) sidide da b) kuTxe luri Zelakis erT (vertikalis mimarT)? (miniSneba: ganixileT bolozea damagre horizontaluri da vertikaluri Zalebi). buli ••43. 80 kg masis adamiani èSmakis borbalze~ 10 m radiusis vertikalur wreze 6,1 m/wm meore bolos gar mudmivi siCqariT moZraobs. a) risi tolia lurad moZraobis periodi? ra drekadobis Zala inJineri stroboskopul sanaTs Zelaksa da moqmedebs savarZlidan adamianze, rodesac WanWiks savarZeli ganaTebis tempi imgvaria, rom Zelaki da WanWiki wriuli traeqtoriis umaRles wertils da g) udables wertils gaivlis? ••44. atraqcionis kabina vertikalur wreze moZraobs. kabina xist Zelakzea damagrebuli. manqanis da mgzavrebis saerTo wonaa 5 kn, xolo wris radiusia 10 m. wris umaRles wertilSi ra a) sididis da b) mimarTulebis Zala moqmedebs Semo da Zelaki Zelaki ganaTebis poziciebi horizonta sur. 6.40. amocana 48. brunavs. anaTebs da moZraobas akvirdeba. erTi sruli Semobrunebis ganmavlobaSi 8-jer naTdeba (sur. 6.40). ganaTebis tempi wamSi 2000 ganaTebaa. WanWikis masaa 30 g, xolo radiusia 3,5 sm. ra Zala moqmedebs Zelakidan WanWikze? ••49. m = 1,5kg masis Saiba xaxunisgan Tavisufal magidaze r = 20 sm radiusis wreze srialebs 191 da M = 2,5kg masis balansiris Camokidebul cilin masa umniSvneloa da misi sigrZea L=0,9 m. drzea damagrebuli. balansiri 0,94 m sigrZis wriul traeqtorias Toki magidis Sua Semowers. risi tolia a) Tokis daWimuloba da gulSi b) moZraobis periodi? gakeTebul xvrelSi gadis xolo Tokis amoucnobi movlenis meore axsna imaSi mdgo siCqaris SemTxvevaSi cilindri 0,04 kg, 53. me-4 da 35-e amocanebis gagrZeleba. (sur. 6.41). Saibis ra iqneba masaa sur. 6.41. amocana 49. uZravi? SSM ••50. trasis mosaxveviani nawili daxrilia da 60km/sT siCqariT moZraobisTvis aris gankuTvnili. mosaxvevis radiusia 200m. manqana wvimian dRes 40km/sT siCqariT moZraobs. ra minimaluri xaxunis koeficienti unda iyos saburavebsa da trasas Soris, raTa manqana mosaxvevSi ar mosrialdes? (davuSvaT, rom mareobs, rom qvebi mxolod maSin moZraobs, rodesac Stormis Sedegad darCenili wya li yinulis did, Txel firfitad iqceva. qvebi yinulSi rCeba. Semdeg, rodesac qari wamouberavs, yinulze da qvebze moqmedi haeris wevis Zala orives gadaadgilebs da qvebic nakvalevs tovebs. haeris wevis Zala Dice = 4Cice ρAice ν2, sadac Cice aris wevis koeficienti (2×10–3), ρ - haeris simkvrive (1,21 kg/m3), Aice – yinulis horizontaluri far manqanaze uaryofiTi weva ar moqmedebs). Tobi da ν – qaris siCqare yinulis gaswvriv. •••51. 6.42 suraTze gamosaxuli 1,34  kg davuSvaT yinulis firfitis zomebia ZelakTan masis ar 400 m × 500 m × 4 mm, miwasTan misi kinetikuri xaxunis koeficientia 0,1 da simkvrivea 917 kg/m3. aseve davuSvaT, rom meoTxe amocanaSi mocemuli qvebis identuri 100 qva mqone, L = 1,7 m si yinulSia darCenili. firfitis moZraobis grZis ori TokiTaa SesanarCuneblad ra siCqaris qaria saWiro masis burTi tikalur, ver mbrunav dakavSirebuli. a) firfitasTan axlos da b) 10 m simaRleze? Zelakze g) aris Tu ara realuri miRebuli mniSvnelo Tokebis damagrebebs ris So sur. 6.42. amocana 51. manZilia d = 1,7 m. zeda Tokis daWimulobaa 35 n. bebi Stormis SemTxvevaSi? 54. trasis manqana mosaxvevis mosaxvevSi daproeqteba. metismetad Tu swrafad risi tolia a) qveda Tokis daWimuloba, Sedis, mocurebis albaToba didia. daxrili b) Tokidan burTze moqmedi mosaxvevis  F jamuri Zala da  g) burTis siCqare? d) ra mimarTuleba aqvs F Zalas? SSM ILW 6.43 statikuri xaxunis koeficientia μs . manqana konu- mosaxvevSi moZraobs brun vasTan erTad Toki Semowers). Sedis. νmax-is θ kuTxeze damokidebulebis grafiki μs = 0,6 (mSrali trasa) da μs = 0,5 (sveli an yinuliani trasa) koeficientebisTvis. νmax siCqare θ = 10° daxrisTvis da g) μs = 0,6 da d) μs = 0,5 koeficientebisTvis km/sT-ebSi wreze mudmivi si konuss msgavsad zRvarzea. b) aageT 0°-dan 50°-mde arealSi horizontalur (balansiris suraTis gamosaxuleba, rodesac manqana mocurebis Camokidebuli CqariT 6.13 a) daadgineT manqanis maqsimaluri νmax siCqaris mlis balansiri Tok da manqanaze xaxunis Zala moqmedebs da mocurebas xe, rodesac saburavebsa da trasas Soris suri qanqara, ro zea moZrav ewinaaRmdegeba. ganvixiloT R = 200 m radiu suraTze gamosaxulia swrafad sis wriuli mosaxvevi da daxrilobis θ kuT damatebiTi amocanebi. 52. SemTxvevaSi sur. 6.43. amocana 52. 192 Zala da moZraoba Tavi 6 gamoTvaleT (axla xom xvdebiT ratom xdeba avariebi moyinul gzaze, rodesac mZRolebi dapiris gapoxva θ1-s gazrdis, Seamcirebs Tu ar Secvlis? d) risi tolia θ1, rodesac μs = 0,6? safrTxes ar iTvaliswineben?) 59. 55. 556 n wonis karada iatakze uZravad Zevs. mosamsaxure daudevari massa da iataks Soris statikuri xaxunis  Sva bris tars F Za koeficientia 0,68, xolo kinetikuri xaxunis liT koeficientia 0,56. karadas a) 222 n, b) 334 n, vertikalisadmi   θ g) 445 n da d) 556 n horizontaluri ZalebiT kuTxes adgens, xo ubiZgeben. gamoTvaleT iatakidan karadaze lo μs da μk moqmedi xaxunis Zala TiToeul SemTxvevaSi. brasa da e) romeli mcdelobis Sedegad daiZvreba Soris karada? da 56. manqana SuadRisas quCazea gaCerebuli, romelic horizontisadmi 35°-iT daxril Svabras gorakze Cadis. am dros quCasa da saburavebs Soris statikuri xaxunis koeficientia 0,725. SEebindebisas wvima wamovida, temperatura daeca da xaxunis koeficientic Sesabamisad Semcirda. ra procentiT unda Semcirdes awveba. tari Sva iataks sur. 6.46. amocana 59. statikuri kinetikuri taris xaxunis masa koeficientebia. ugulebelyaviT da dauSviT, rom mTeli m masa Svabras bolozea koncentrirebuli. a) Tu Svabra iatakze mudmivi siCqariT moZraobs, risi tolia F? b) daamtkiceT, rom Tu θ kuTxe garkveul θ0 mniSvnelobaze naklebia,  F Zala Svabras gadaadgilebas ver SeZlebs. gamoTvaleT θ0. koeficienti, raTa manqanas Camosrialebis 60. Surdulis msroleli 0,01kg masis Sur safrTxe gauCndes? dulSi 0,25kg masis qvas debs da qvis sasrolad 57. 6.44 0,65 m radiusis vertikalur wres akeTebs. suraTze gamosaxul Surdulis gaWimuli Tokis masa umniSvneloa bloksa da gawydeba, Tu misi daWimuloba 33 n-s da 60°-iT daxril sibrtyes gadaaWarbebs. davuSvaT Surdulis msroleli Soris kinetikuri xaxunis koeficientia sur. 6.44. amocana 57. 0,2. ra a) sidide da b) mimarTuleba aqvs blokis aCqarebas, Tu bloki qvemoT curdeba? ra g) sidide da d) mimarTuleba aqvs blokis aCqarebas, Tu bloki sibrtyeze maRla adis? 58. 6.45 qmedebs statikuri xaxunis gawydeba Toki? 61. mt = 4 kg masis bloki mb = 5 kg masis sasrialeblad (qve das daZvris gareSe) yuTsa da zedapirs Soris wertilSi? b) qvis srolis ra siCqareze blokis qvedaze ga  F Zala mo da umaRles wertilSi gawydeba Toki Tu udables blokze Zevs. zeda suraTze gamosaxul m masis yuTze qvis srolis siCqares nel-nela zrdis. a) wris sur. 6.47. amocana 61. zeda blokze sul mcire 12 n horizontalurma sur. 6.45. amocana 58. koeficientia μs. θ kuTxe Tavdapirvelad nu lis tolia, xolo Semdeg ise izrdeba, rom Zalis veqtori saaTis isris moZraobis mimar TulebiT brunavs. brunvis dros Zalis F sidide ise icvleba, rom yuTi yovelTvis gasrialebis zRvarzea. Tu μs = 0,7, a) aageT F/mg-s θ-ze damokidebulebis grafiki da Zalam unda imoqmedos. blokebi ganla gebulebia horizontalur, xaxunisgan Tavi sufal magidaze (sur. 6.47). gamoTvaleT: a) maqsimaluri  F horizontaluri Zala, romliTac qveda blokze unda vimoqmedoT, raTa orive bloki erTdroulad amoZravdes da b) Sedegad miRebuli blokebis aCqareba. b) gansazRvreT θ1 kuTxe, rodesac es fardoba 62. 6.48 suraTze m1 = 2 kg masis pirveli yuTi da m2 = 3kg masis meore yuTi umniSvnelo masis usasrulo mniSvnelobas uaxlovdeba. g) ze TokiT aris gadabmuli da ar moZraobs. meore 193 yuTi θ = 30° kuTxiT horizontalur zedapirs Soris kinetikuri daxril, xaxunisgan xaxunis koeficientia 0,2. risi tolia Tokis T a v i s u f a l daWimuloba? zedapirzea 67. lagebuli. gan pirvel sur. 6.48. amocana 62. yuTsa da horizon talur zedapirs Soris kinetikuri xaxunis koeficientia 0,25. blokis masa da xaxuni 6.50 suraTze yuTi daxril RarSi misrialebs. yuTsa da Rars Soris kinetikuri xaxunis koeficientia μk. risi tolia yuTis aCqareba μk-s, θ-s da g-s erTeulebSi? umniSvneloa. yuTebis amoZravebis Semdeg risi tolia Tokis daWimuloba? 63. foladis 8kg masis bloki horizontalur magidaze uZravad devs. bloksa da magidas Soris 0,45. statikuri xaxunis koeficientia risi tolia blokze moqmedi Zala, Tu a) horizontalurad mimarTuli, b) ho rizontalidan maRla 60°-iT gadaxrili da g) horizontals qvemoT 60°-iT gadaxrili Zala bloks gacurebis zRvarze aCerebs. 64. konservebiT datvirTuli sardafSi Cavarda. 68. 6.51 daRmarTi horizontalisadmi 40° kuTxes adgens. risi suraTze gamosaxuli m1 = 1,65 kg masis pirveli yuTi da m2 = 3,3 kg masis yuTi daRmarTze Casrialda da 0,75 m/wm2 aCqarebiT sabaylos sur. 6.50. amocana 67. meore daxril yuTi sibrtyeze sur. 6.51. amocana 68. misrialebs. yuTebi masis armqone da sibrtyis paraleluri ZelakiT aris gadabmuli. daxris kuTxea 30°. tolia daRmarTsa da yuTs Soris arsebuli pirvel yuTsa da zedapirs Soris kinetikuri kinetikuri xaxunis koeficienti? xaxunis koeficientia μ1 = 0,226, xolo meore 65. aeroportSi tvirTis gadatana eskala yuTsa da sibrtyes Soris kinetikuri xaxunis μ2 = 0,113. toriT xdeba. erT adgilze tvirTi daRmarTze koeficientia Cadis, romelic horizontisadmi 2,5° kuTxes Zelakis daWimuloba da b) yuTebis aCqareba. adgens. g) rogor Seicvleba wina kiTxvebze pasuxi, Tu dros davuSvaT, tvirTi ar amgvari mcire srialebs. daxris gansazRvreT 69. xaxunis Zala, rodesac yuTi daxril monakveTs gamosaxuli b) 0,65 m/wm da mudmivi, g) 0,65 m/wm da 0,2 m/wm2 tempiT zrdadi, d) 0,65 m/wm da 0,2 m/wm2 tempiT klebadi da e) 0,65 m/wm da 0,57 m/wm tempiT 2 6.52 mB = 2 kg. B yuTsa da horizontalur iqneba kinetikuri daRmarTis gaswvriv 6.49 sur. 6.52. amocana 69. Soris xaxunis koeficientia μk = 0,5. sibrtyis daxrilobaa 30° da is xaxunisgan mimarTuli? 66. A yu xolo B yuTis masaa sibrtyes Zala suraTze Tis masaa mA = 4 kg, zrdadi. v) zemoT CamoTvlil romel variantSi xaxunis a) yuTebi adgilebs gacvlian? eskalatoridan 69 n wonis yuTze moqmedi gadis da eskalatoris siCqarea a) 0 da mudmivi, gamoTvaleT Tavisufalia. suraTze gamosaxuli m1 = 2 kg masis pirveli blo ki da m2 = 1 kg ma bloki mxolod yuTebis damakavSirebuli Tokis mimarTulebas cvlis. gamoTvaleT sur. 6.49. amocana 66. a) Tokis daWimuloba da b) yuTebis aCqareba. sis meore bloki erTmaneTze umniSvnelo 70. studentma ficarsa da yuTs Soris masis TokiT aris gadabmuli. meore blokze arsebuli kinetikuri da statikuri xaxunis 20  n sididis da θ = 35° kuTxiT daxrili koeficientebi unda gansazRvros. man yuTi F Zala moqmedebs. TiToeul bloksa da ficarze dado da ficris erTi bolo aswia. 194 Tavi 6 Zala da moZraoba rodesac daxris kuTxem 30°-s miaRwia, yuTi la kedelze da T) qvemoT kedelze? i) ro 4 wamis ganmavlobaSi 2,5 m manZilze mudmivi mel aCqarebiT Camosrialda. risi tolia a) sta mimarTuli? tikuri xaxunis koeficienti da b) kinetikuri 75. xaxunis koeficienti yuTsa da ficars Soris? eqsperimentSia gzis xaxunis mSeneblobis Zala adgilze qvemoT TamaSis dros bavSvi horizontisadmi 35°-iT daxril 71. 5 kg masis qviT ormoSi Cavarda. Casrialebis ganmavlobaSi xexaven Weris hori bavSvis aCqarebaa 0,5m/wm2 da misi mimarTuleba zontalur zedapirs srialis sawinaaRmdegoa. ra kinetikuri xaxu (sur. 6.53). Tu kine sur. 6.53. amocana 71. tikuri xaxunis koe ficientia 0,65 da qvaze moqmedi Zala θ = 70° kuTxiTaa daxrili, ra Zalam unda imoqmedos qvaze, raTa man mudmivi siCqariT imoZraos? nis koeficienti arsebobs bavSvsa da daxril zedapirs Soris? 76. orTqlmavali 25 vagonian matarebels swor liandagze aCqarebs. TiToeuli vagonis masaa 5×104kg da yvela maTganze f=250ν xa 72. trasis mosaxveviani monakveTi 60 km/sT si xunis Zala moqmedebs, sadac ν siCqare m/wm- Cqarezea gaTvlili. davuSvaT, mosaxvevSi Se ebSia, xolo f Zala niutonebSia. rodesac sul manqanebze uaryofiTi weva ar moqmedebs. matareblis siCqarea 30km/sT, misi aCqareba a) Tu mosaxvevis radiusia 150 m, ra kuTxiT unda 0,2m/wm2-s aRwevs. a) risi tolia orTqlmavalsa iyos daxrili trasa? b) trasis dauxrelobis da pirvel vagons Soris jaWvis daWimuloba? SemTxvevaSi b) ra minimaluri xaxunis koe Tu es daWimuloba moqmedi orTqlmavalidan ficienti unda arsebobdes saburavebsa da matarebelze trasas Soris, raTa 60 km/sT siCqariT moZravi tolia, ra maqsimaluri siswrafiT SeiZleba maqsimaluri Zalis manqanebi mosaxvevSi ar mocurdnen? orTqlmavalma ubiZgos 30km/sT-iT moZrav 73. θ kuTxiT daxril sibrtyeze mudmivi siCqariT mosriale bloks ν0 sawyisi siCqariT matarebels? 77. 2 g masis moneta horizontaluri mbrunavi ukan ubiZgeben. a) ra manZils gaivlis sibrtyeze magidis centridan 5 sm manZilze moaTavses, bloki gaCerebamde? b) gaCerebis Semdeg bloki romelic 3,14 wamSi sam bruns asrulebs. aseT kvlav pirobebSi moneta ar curavs. risi tolia Camosrialdeba Tu ara? SekiTxvas argumentirebuli pasuxi gaeciT. monetis a) siCqare, aCqarebis b) sidide da 74. 6.54 suraTze 22 n wonis bloki  60 n sididis F ho g) mimarTuleba da monetaze moqmedi xaxunis rizontaluri liT centridan uZravad gaCerebuli. 10 sm manZilze ganlagebis dros moneta mocurebis zRvarzea. v) risi Za tolia monetasa da magidas Soris arsebuli vertikalur kedelze aris Zalis d) sidide da e) mimarTuleba? magidis statikuri xaxunis koeficienti? sur. 6.54. amocana 74. 78. 6.1 cxrilis monacemebze dayrdnobiT 16 kedelsa da bloks Soris statikuri xaxunis funtiani tyviis diametri daadgineT, Tu koeficientia 0,55, C = 0,49 da haeris simkvrivea 1,2 kg/m3. xolo kinetikuri xa xunis koeficientia 0,38. blokze eqvsi eqs perimenti tardeba da masze kedlis pa raleluri da sxvadasxva sididis meore  P Zala moqmedebs. am Zalis sidideebi da mimarTulebebia: a) 34 n, maRla, b) 12 n, maRla, g) 48 n, maRla, d) 62 n, maRla, e) 10 n, qvemoT, v) 18 n, qvemoT. TiToeul eqsperimentSi ra 79. 6.32 suraTze pirveli, meore da mesame blokebi umniSvnelo masis TokebiTaa gada bmuli. blokebis amoZravebis Semdeg maTi aCqa rebebia 1,5 m/wm2. blokebis masebia m1 = 2 kg, m2 = 4 kg da m3 = 4 kg. ra xaxunis Zala moqmedebs meore blokze? romel 80. oTxkaciani bobslei (630 kg mTliani masis) eqsperimentSi gadaadgileba bloki z) maR moZraobas swor trasaze iwyebs. trasis am xaxunis Zala moqmedebs blokze? 195 monakveTis sigrZe 80 m-ia da horizontisadmi mudmivi aCqarebiT misrialebs. yuTis mdeba 0,5 wm-is 10,2° kuTxiT aris daxrili. davuSvaT xaxuni reoba da haeris weva bobsleize 62 n mudmivi ZaliT a) ra aCqareba aqvs yuTs? b) Tu yuTi 240 n-s moqmedebs, romelic daxris paralelurad iwonis, ra sididis kinetikuri xaxunis Zala da bobsleis moZraobis sawinaaRmdegod aris moqmedebs yuTze? mimarTuli. 83. Tovlze dadebuli Txilamuri droTa upasuxeT Semdeg SekiTxvebs: a) Tu moZraobis dawyebisas bobsleis siCqarea 6,2 m/wm, ra droSi gaivlis is trasis swor monakveTs? b) davuSvaT sportsmenebma xaxunis da haeris wevis moqmedeba 42 n-mde Seamcires. igive sawyisi siCqaris pirobebSi ra droSi gaivlis bobslei trasis swor monakveTs? ganmavlobaSi SualediT Tovls fiqsirdeba. ewebeba, magram mo Zraobis dros Txilamuri Tovls aTbobs da nawilobriv adnobs, rac kinetikuri xaxunis koeficients srials amcirebs aadvilebs. da, Sesabamisad, Txilamuris cviliT dafarva mas wyalgaumtars xdis da wylis 81. saxli gorakis wverzea aSenebuli, xolo axlomdebare daRmarTi θ = 45° kuTxes adgens fenasaTan (sur. 6.55). Seswavlis Sedegad dadginda, rom Txilamuri gansakuTrebulad wyalgaumtaria es daRmarTi unda Semcires, vinaidan masze da alpebis 200 m-ian saTxilamuro trasas ganlagebuli miwis zeda fenebi qveda fenebze axali curdeba. Tu or amgvar fenas Soris statikuri 42 wamSi dafaravT. gamoTvaleT saSualo xaxunic mcirdeba. reklamaSi amtkiceben, rom axali saxeobis plastmasis TxilamurebiT 61 wamis nacvlad xaxunis koeficientia 0,5, ra umciresi φ daxra aCqareba a) standartuli Txilamurebis da unda hqondes daRmarTs, raTa miwis Cacureba b) axali Txilamurebis pirobebSi. davuSvaT Tavidan aviciloT? saTxilamuro trasis daxrilobaa 3°. gamo TvaleT g) kinetikuri standartuli xaxunis koeficienti TxilamurebisTvis da d) axali TxilamurebisTvis. 84. risi tolia 6kg masis, 3sm radiusisa da 1,6 wevis koeficientis mqone sferos formis burTis saboloo siCqare? burTi haerSi vardeba da haeris simkvriva 1,2 kg/m . 3 sur. 6.55. amocana 81. 82. yuTebi sardafSi tvirTisgan daxrili dacales sibrtyis da saSualebiT Caasriales. 6.56 suraTze gamosaxulia 2,5m simaRlis daxril sibrtyeze Casrialebuli yuTis gadaadgilebis grafiki. davuSvaT yuTi moZraobas iwyebs, rodesac t = 0 da 85. 1,5kg masis yuTi horizontalurzedapirze F =(1,8 t) n uZravad Zevs. rodesac t = 0, masze horizontaluri Zala iwyebs moqmedebs. yuTis  a = 0, rodesac 0 ≤ t ≤ 2,8 wm da  a = (1,2t – 2,4) m/wm2, rodesac t > 2,8 wm. a) risi aCqarebaa tolia yuTsa da zedapirs Soris arsebuli statikuri xaxunis koeficienti? b) risi tolia yuTsa da zedapirs Soris arsebuli kinetikuri xaxunis koeficienti? 86. 26kg masis bavSvi èSmakis borbalis~ 12 m radiusis vertikalur, wriul traeqtoriaze 5,5 m/wm mudmivi siCqariT moZraobs. ra Zala moqmedebs bavSvze savarZlidan a) borblis umaRles wertilSi da b) borblis udables wertilSi? 87. 10,7 kn wonis da 13,4 m/wm siCqariT moZravi manqana 61 m radiusis swor (daxris gareSe) sur. 6.56. amocana 82. 196 Tavi 6 Zala da moZraoba mosaxvevSi Sedis. a) ra xaxunis Zalam unda koeficientebi SeiZleba arsebobdes amocanis imoqmedos saburavebze, raTa manqanam wriuli pirobidan gamomdinare? traeqtoria SeinarCunos? b) Tu saburavebsa da 93. 25kg masis skamze horizontisadmi θ kuTxiT daxrili 100 n Zala moqmedebs. Tu θ = 0°, risi tolia a) modebuli Zalis Fh horizontaluri gzas Soris statikuri xaxunis koeficientia 0,35, SeZlebs Tu ara manqana mosaxvevSi Sesvlas? 88. komponenti da b) iatakidan skamze moqmedi bavSvma kalaTa atraqcionis gare fersoze dado. atraqcionis radiusia 4,6 m da yovel 30 wamSi erT srul bruns akeTebs. a) ra siCqariT moZraobs ferso? b) ra umciresi statikuri xaxunis koeficienti unda arsebobdes kalaTasa da atraqcions Soris, FN drekadobis Zala? Tu θ = 30°, risi tolia g) Fh da d) FN? Tu θ = 60°, risi tolia e) Fh da v) FN ? amjerad davuSvaT, rom skamsa da iataks Soris statikuri xaxunis koeficientia 0,42. rodesac θ aris z) 0°, T) 30° da i) 60°, skami iatakze gasrialdeba Tu adgilze darCeba? raTa kalaTa ar Camovardes? 94. 220 n wonis Cemodani uZravad Zevs. 89. Cemodansa 6.57 suraTze manqana horizontalurma gviZlia miviCnioT. ra udidesi siCqariT Sei moZraobdes manqana, raTa gorakis wverze gzas ar acdes? 90. studenti 5 kg masis daxrili kuTxiT  P ZaliT unda imoqmedos daZvris Semdeg ra Zalam unda imoqmedos, raTa man mudmivi siCqare SeinarCunos? g) Tu Cemodnis daZvrisas modebuli Zala ar 95. 40 n wonis bloki horizontisadmi 25°iT daxril sibrtyeze 0,8 m/wm2 aCqarebiT ganerviulebuli da θ = 70° Zalam Seicvleba, ra aCqareba eqneba Cemodans? gamocdebisgan bloks 80 n sididis statikuri Cemodanze, raTa is daiZras? b) Cemodnis sur. 6.57. amocana 89. radiusis wred Se Zleba Soris xaxunis koeficientia 0,32. a) ra minimalurma R = 250 m kveTi iataks xaxunis koeficientia 0,41, xolo kinetikuri gorakze adis, romlis gani vi da moZraobs. aCqareba mimarTulia aRmarTisken. ra kinetikuri xaxunis koeficienti arsebobs sur. 6.58. amocana 90. bloksa da sibrtyes Soris? Werze awveba (sur. 6.58). Tu bloksa da Wers Soris kinetikuri xaxunis koeficientia 0,4, ra iqneba blokis aCqareba? 96. foladis 11kg masis bloki horizontalur magidaze uZravad Zevs. bloksa da magidas Soris statikuri xaxunis koeficientia 0,52. a) ra horizontaluri Zala miiyvans bloks 91. maRaziis gamyidveli 40 kg masis yuTze 85 n horizontaluri ZaliT moqmedebs. Tav amoZravebis zRvarze? b) ra 60°-iT maRla dapirvelad yuTi horizontalur zedapirze zRvarze? g) Tu Zala 60°-iT qvemoT aris uZravad Zevs. 1,4 m-iT gadaadgilebis Semdeg gadaxrili, ra maqsimaluri sidide SeiZleba yuTis siCqarea 1 m/wm. risi tolia yuTsa da hqondes mas ise, rom blokma ar imoZraos? zedapirs Soris moqmedi kinetikuri xaxunis koeficienti? 97. 6.59 suraTze gamosaxul 5kg masis blokze 50 n sididis horizontaluri F Zala 92. 140 n wonis bavSvi horizontalisadmi 25°- moqmedebs da bloki iT daxril sasrialoze zis da wonasworobas xelebiT inarCunebs. Camosrialebis Semdeg bavSvis mudmivi aCqarebaa 0,86m/wm2. a) risi tolia sasrialosa da bavSvs Soris arsebuli kinetikuri xaxunis koeficienti? b) ra maqsimaluri da minimaluri statikuri xaxunis gadaxrili Zala miiyvans bloks amoZravebis θ = 37° kuTxiT da xril sibrtyeze maRla adis. bloksa da sibrtyes Soris kinetikuri xaxunis koeficientia 0,3. sur. 6.59. amocana 97. 197 ra a) sididis da b) mimarTulebisaa blokis statikuri xaxunis koeficientia 0,51, xolo aCqareba? blokis sawyisi siCqarea 4 m/wm. kinetikuri xaxunis koeficientia 0,32. Tqven g) ra manZils gaivlis bloki sibrtyeze? d) am gan yuTze modebuli Zala horizontaluria. manZilis gavlis Semdeg bloki gaCerdeba Tu a) ra Zala miiyvans yuTs gaxoxebis zRvarze? ukan Camocurdeba? b) Semdeg ra ZaliT unda waubiZgoT yuTs, raTa 98. vagoni mudmivi siCqariT 470 m radiusis man mudmivi siCqariT imoZraos? g) Tu yuTze brtyel, horizontalur wreze moZraobs. Tavdapirvelad modebuli Zala ucvlelad vagoni 51kg masis mgzavrze ZaliT moqmedebs, moqmedebs, ra iqneba misi aCqareba? romlis horizontaluri da vertikaluri 104. 110 g masis Saiba yinulze srialebs da 15 m- komponentebi Sesabamisad aris 210n da 500n. is gavlis Semdeg xaxunis Zalis moqmedebiT a) ra jamuri Zala moqmedebs mgzavrze? b) ra Cerdeba. a) Tu Saibis sawyisi siCqarea 6 m/wm, ra siCqariT moZraobs vagoni? xaxunis Zala moqmedebs masze? b) risi tolia 99. velosipedisti 9m/wm mudmivi siCqariTY 25m radiusis wreze moZraobs da velosipedTan erTad misi masaa 85kg. gamoTvaleT Saibasa da yinuls Soris arsebuli xaxunis a) gzidan velosipedze moqmedi xaxunis Zala da iataks Soris statikuri xaxunis koeficientia b) gzidan velosipedze moqmedi jamuri Zala. 0,42. 6.61 suraTze gamosaxuli bavSvi Tokis 100. 53sm diametris raketa dabal simaRleze, sadac haeris simkvrivea 1,2kg/m3, 250m/wm siCqariT gadaadgildeba. dauSviT, rom C=0,75 saSualebiT yuTis gadaadgilebas cdilobs. da gamoTvaleT raketaze moqmedi wevis Zala. 101. manqana ν = 94 km/sT maqsimaluri SesaZlo siCqariT R = 220 m radiusis brtyel mosaxvevSi Sedis. ra jamuri Zala moqmedebs m = 85 kg koeficienti? 105. saTamaSoebis yuTis wonaa 180 n. yuTsa da a) Tu θ = 42°, ra  F ZaliT unda imoqmedos bavSma Tokze, raTa yuTi moZraobis zRvramde mivides? b) dawereT am F Zalis θ kuTxeze damokidebulebis funqciis gamosaxuleba. gamoTvaleT g) θ-is mniSvneloba, rodesac F minimaluria da d) F-is minimaluri mniSvneloba. masis mgzavridan manqanis savarZelze? 102. 49 kg masis mTamsvleli napralSi adis (sur. 6.60). fexsacmelebsa da kldes Soris statikuri xaxunis koeficientia 1,2, xolo zurgsa da kldes Soris igive koeficienti 0,8-ia. Semdeg sur. 6.61. amocana 105. mTamsvlelma sax srebi da 106. warmoidgineT, rom standartuli kilo moaduna gramis masis sagani dedamiwis ekvatorze Camovardnis moaTavses, sadac is dedamiwis brunvis Sedegad zRvarze mivida. a) 465 m/wm mudmivi siCqariT 6,4×106 m radiusis aageT Tavisufali sxeulis sur. 6.60. amocana 102. grafiki mTamsvlelisTvis. b) ra ZaliT awveba mTamsvleli kldes? g) misi wonis ra nawils exmareba fexsacmelebze moqmedi xaxunis Zala? 103. Tqven 165 n wonis yuTi navsayudelisken unda gaaxoxialoT. yuTsa da zedapirs Soris wreze moZraobs. a) ra sididis centriskenuli Zala imoqmedebs brunvis dros saganze? davuSvaT, igive sagani zambarian sasworzea Camokidebuli da dedamiwis brunvis gareSe misi wona zustad 9,8 n-ia. b) ras aCvenebs saswori anu ra sididis Zala moqmedebs sagnidan sasworze? 198 Tavi 6 Zala da moZraoba 107. Toki 40 n maqsimalur daWimulobas uZlebs. bavSvma am Toks 0,37kg masis qva moaba da Tokis meore boloTi qvas 0,91m radiusis mudmivi vertikalur asrulebs wreze atrialebs. bavSvi siCqariT mifrinavs. Tokis horizontalur meore bolo wreze miwazea damagrebuli. TviTmfrinavi wuTSi 4,4 bruns da frTebis Sedegad horizontaluri qvis siCqares Tokis gawyvetamde zrdis. a) ganlagebis traeqtoriis ra adgilze iqneba qva Tokis vertikalurad maRla awveba. a) risi tolia gawyvetisas? b) ra iqneba qvis siCqare Tokis TviTmfrinavis aCqareba? b) risi tolia Tokis gawyvetisas? daWimuloba? g) ra jamuri maRla mimarTuli 108. 0,75kg masis saTamaSo TviTmfrinavi 30m sigrZis Tokzea gamobmuli da 18m simaRleze (wevis) Zala frTebze? haeri moqmedebs TviTmfrinavs TviTmfrinavis kinetikuri energia da muSaoba 7 mSobiarobis dros qirurgs an anesTeziologs gautkivarebis wamali pacientis sxeulSi nemsiT Seyavs. nemsis wveri zurgis tvinis garSemo ganlagebul arxSi arsebul epiduralur sivrceSi unda moxvdes. es procedura sakmaod xangrZliv praqtikas moiTxovs, raTa eqimi mixvdes Tu rodis Sevida nemsi epiduralur sivrceSi. winaaRmdeg SemTxvevaSi, sakmaod seriozul garTulebebs SeiZleba hqondes adgili. dResdReobiT, eqimebi gamocdilebas praqtikaSi iZenen, magram momavalSi albaT SesaZlebeli gaxdeba virtualur modelze varjiSi da mxolod amis Semdeg darTaven eqims nebas cocxal pacients Seuyvanos ineqcia. rogor unda aigos virtualuri modeli, raTa eqimma sworad Caataros epiduraluri procedura? pasuxs am TavSi SeityobT 7.1. ra aris fizika? fizikis erT-erTi fundamentaluri amocana energiis Seswavlaa. marTlac, Cveni civilizacia energiis miRebasa da efeqturad gamoyenebaze Zalian aris damokidebuli. magaliTad, yvelam icis, rom nebismieri saxis moZraoba energias saWiroebs. wynar okeaneze gadafrena, Senobis zeda sarTulze tvirTis atana, kosmosuri xomaldis moZraoba dedamiwis orbitaze, burTis srola da a.S. energias moiTxovs. energiis miRebasa da gamoyenebaze uzarmazari resursebi ixarjeba. Tanamedrove omebi swored energoresursebis gamo iwyeba da erT-erTi mxaris mier resursis Warbad gamoyenebis Sedegad sruldeba kidec. nebismierma adamianma icis energiis da misi gamoyenebis magaliTebi, magram ras niSnavs termini energia realurad? 7.2. ra aris energia? energiis cneba Zalze farToa da misi zustad gansazRvra sakmaod rTulia. formalurad energia aris erTi an meti raodenobis sagnis mdgomareobasTan asocirebuli skalaruli sidide, Tumca es gansazRvreba metismetad bundovania da Znelad Tu dagvexmareba energiis aRqmaSi. modiT zemoT moyvanili gansazRvreba odnav ganvavcroT da energiis Seswavlac ase daviwyoT. energia erTi an meti raodenobis sagnis sistemasTan dakavSirebuli ricxvia. Tu Zala erT-erT sagans cvlis (magaliTad, amoZravebs), maSin energiis ricxvi icvleba. uricxvi eqsperimentebis Sedegad mecnierebi mixvdnen, rom Tu energiis ricxvebis miniWebis sqema kargad aris dagegmili, ricxvebi eqsperimentebis Sedegebis winaswarmetyvelebisTvis SegviZlia gamoviyenoT da, rac 199 200 Tavi 7 kinetikuri energia da muSaoba ufro mniSvnelovania, SesaZlebeli xdeba meqanizmebis, magaliTad, safreni mowyobilobebis ageba. mecnierebis warmateba Cveni samyaros erT SesaniSnav Tvisebas efuZneba: energia erTi saxeobidan meoreSi da erTi sagnidan meoreze gadadis, magram misi mTliani raodenoba ucvlelia (energia inaxeba). energiis Senaxvis kanonis gamonaklisi aRmoCenili ar aris. energiis sxvadasxva saxeobebi sxvadasxva saxis sabanko angariSebze Senaxuli fulis amsaxvel ricxvebad SegviZlia warmovidginoT. arsebobs wesebi, Tu ras niSnavs fulis amsaxveli ricxvi da rogor icvleba is. fulis raodenoba erTi angariSidan meoreze an erTi sistemidan meoreSi SegviZlia gadavitanoT, magram mTliani raodenoba yovelTvis inaxeba. mocemul TavSi aqcents mxolod erTi saxeobis energiaze (kinetikur energiaze) da energiis gadacemis erT saSualebaze (muSaobaze) gavakeTebT. momdevno TavSi ramdenime saxis energias ganvixilavT da vnaxavT, Tu rogor SeiZleba energiis Senaxvis kanonis formulebis saxiT Cawera. 7.3. kinetikuri energia K kinetikuri energia moZrav mdgomareobaSi myof sagnebs axasiaTebs. rac ufro swrafad moZraobs sagani, miT ufro metia misi kinetikuri energia. rodesac sagani uZravia, misi kinetikuri energia nulis tolia. m masis da sinaTlis siCqareze naklebi ν siCqariT moZravi sagnisTvis: K = 1 mν2 2 (kinetikuri energia) (7.1) magaliTad, rodesac 3kg masis ixvi 2m/wm siCqariT mifrinavs, misi kinetikuri energiaa 6kg · m2/wm2 anu ixvis moZraoba am ricxviT xasiaTdeba. SI sistemaSi kinetikuri energiis (da yvela sxva saxeobis energiis) erTeuli joulia (j). es erTeuli me-19 saukuneSi moRvawe ingliseli mecnieris - jeims preskot joulis sapativcemulod aris SemoRebuli. 7.1 formulidan gamomdinare jouli masis da siCqaris erTeulebiT ganisazRvreba: 1 jouli = 1kg · m2/wm2 maSasadame, mfrinavi ixvis kinetikuri energiaa 6 j. amocanis amoxsnis nimuSi (7.2) 7.1 1896 wels, texasis Statis qalaq vakoSi, uiliam kraSma ori orTqlmavali 6,4km sigrZis liandagis mopirdapire boloebze gaaCera, daqoqa da erTmaneTisken mTeli siCqariT daZra (sur. 7.1). am yvelafers daaxloebiT 30 000 TviTmxilveli Seeswro. aTasobiT ada miani orTqlmavlebis namsxvrevebma daaziana, ramdenime daiRupa kidec. davuSvaT TiToeuli orTqlmavalis wona 1,2×106n iyo da isini 0,26 m/wm2 mudmivi aCqarebiT moZraobdnen. ra jamuri kinetikuri energia hqonda orive orTqlmavals Sejaxebamde? amoxsna: 7.1 formulis saSualebiT TiTo euli orTqlmavalis SegviZlia kinetikuri gamovTvaloT, magram energia amisTvis sur. 7.1. orTqlmavalebis Sejaxebis Sedegi 7.4. muSaoba 201 orTqlmavlebis siCqare Sejaxebamde da maTi TiToeuli orTqlmavalis masis dasadgenad masa gvWirdeba. vinaidan davuSviT, rom TiTo maTi wonebi g-ze gavyoT: 1,2 × 106 n euli orTqlmavalis aCqareba mudmivia, 2.1 m= cxrilis saSualebiT ν siCqareebis gamoTvla = 1,22 × 105 kg SesaZlebelia. Cven 2.16 formulas virCevT, Semdeg radgan ν-s garda yvela mniSvneloba viciT: orTqlmavalis jamur kinetikur energias ν2 = ν02 + 2a(x – x0), ν = 40,8 m/wm (daaxloebiT 150 km/sT) saSualebiT orive K = 2(1 mν2) = (1,22 × 105 kg)(40,8 m/wm)2 2 = 2 × 108 j mTliani manZilis naxevari): anu formulis gamovTvliT: sadac ν0 = 0 da x – x0 = 3,2 × 103 m (liandagis ν2 = 0 + 2(0,26 m/wm2)(3,2 × 103m) 7.1 9,8 m/w2 (pasuxi) miRebuli Sedegi cxadyofs, rom orTql mavalebis Sejaxeba bombis afeTqebis magvari iqneboda. 7.4. muSaoba Tu saganze ZaliT imoqmedebT da mis siCqares gazrdiT, sagnis kinetikuri energiac gaizrdeba. aseve, Tu saganze ZaliT imoqmedebT da mis siCqares SeamcirebT, kinetikuri energiac Sesabamisad Semcirdeba. kinetikuri energiis cvlilebis SefasebisTvis vambobT, rom Zalam energia Tqvenidan saganze an piriqiT gadaitana. Zalis saSualebiT energiis gadacemis dros Zala saganze W muSaobas asrulebs. muSaobis gansazRvreba Semdegia: W muSaoba aris saganze moqmedi Zalis saSualebiT sagnidan an sagnisTvis gadacemuli energia. sagnisTvis gadacemuli energia dadebiTi muSaobaa, xolo sagnidan gamosuli energia - uaryofiTi muSaoba. maSasadame, `muSaoba~ gadacemuli energiaa, xolo `muSaobis Seasruleba~ - energiis gadacemis procesi. muSaobis erTeuli energiis erTeulia da muSaoba skalaruli sididea. termini „gadacema“ sakmaod damabneveli SeiZleba gaxdes. is ar niSnavs, rom raime nivTiereba saganze gadadis an sagnidan gamodis. energiis gadacema, magaliTad, wylis dinebas, sulac ar gavs. es ufro bankis erTi angariSidan meoreze fulis eleqtronuli gadatanis msgavsia. erTi angariSis ricxvi mcirdeba, xolo meorisa izrdeba, sinamdvileSi ki or angariSs Soris araviTari nivTiereba ar moZraobs. yuradReba miaqcieT, rom sityva `muSaoba~ ar niSnavs fizikurad an gonebrivad Sesrulebul samuSaos. magaliTad, rodesac kedels Zlierad awvebiT, kunTebis permanentuli daZabvis Sedegad daiRlebiT anu muSaobT, magram es ar niSnavs, rom Tqvensa da kedels Soris energiis gadacema xdeba anu kedelze muSaoba sruldeba. gaugebrobis Tavidan asacileblad W simbolos mxolod muSaobisTvis gamoviyenebT, xolo wonas misi eqvivalenturi mg simboloTi warmovadgenT. 7.5. muSaoba da kinetikuri energia muSaobis gamosaxuleba modiT, ganvixiloT gaxvretili burTula, romelic horizontaluri x RerZis gaswvriv gadaWimul, xaxunisgan Tavisufal mavTulze misrialebs (sur. 7.2). burTulaze mavTulisadmi  φ kuTxiT daxrili, mudmivi F Zala moqmedebs da mas mavTulis gaswvriv aCqarebs. Zalasa da aCqarebas Soris kavSiris dasadgenad x RerZis gaswvrivi komponentebisTvis niutonis meore 202 Tavi 7 kinetikuri energia da muSaoba kanoni CavweroT: Fx = max , (7.3)  sadac m aris burTulis masa. Zala d gadaadgilebis gavlis Semdeg burTulis siCqaris 0 sawyis mniSvnelobas raime sxva mniSvnelobiT cvlis. vinaidan Zala mudmivia, aCqarebac mudmivia. aqedan gamomdinare, x RerZis gaswvrivi sur. 7.2.  mavTulze mosriale burTu laze d gadaadgilebisadmi φ kuTxis  mdgeneli F mudmivi Zala moqmedebs da mis siCqares -dan -ze cvlis. 0 `kinetikuri energiis sawyao~ burTulis kinetikuri energiis cvlilebas asaxavs. Ki sawyisi kinetikuri energiaa, xolo Kf – saboloo kinetikuri energia. komponentebisTvis 2.16 formula SeiZleba Caiweros: ν 2 = ν02 + 2axd. (7.4) am formulis ax-sTvis amoxsna, Semdeg 7.3 formulaSi Casma da gardaqmnebi mogvcems: 1 mν2 2 – 1 mν02 = Fxd 2 (7.5) formulis marcxena mxaris pirveli wevri d gadaadgilebis bolos burTulis Kf kinetikuri energiaa, xolo meore wevri - Ki kinetikuri energia gadaadgilebis dasawyisSi. 7.5 formulis marcxena mxare gvauwyebs, rom kinetikuri energia Zalis moqmedebiT Seicvala, xolo formulis marjvena mxaris Tanaxmad es cvlileba Fxd-s tolia. maSasadame, Zalis mier burTulaze Sesrulebuli W muSaobaa: W = Fxd (7.6) Tu Fx da d sidideebi viciT, maSin am formulis gamoyenebiT Zalis mier burTulaze Sesrulebul W muSaobas gamovTvliT. moZraobisas Zalis mier saganze Sesrulebuli muSaobis gamoTvlisTvis mxolod gadaadgilebis gaswvriv Zalis komponents viyenebT. gadaadgilebis perpendikularuli Zalis komponenti muSaobas ar asrulebs. 7.2 suraTze Cans, rom Fx = F cos φ, sadac φ aris kuTxe   d gadaadgilebis mimarTulebasa da F Zalas Soris. 7.6 formula ufro zogadi saxiTac SeiZleba Caiweros: W = Fd cos φ (mudmivi Zalis mier Sesrulebuli muSaoba) (7.7)   es formula gamosadegia, Tu F, d da φ sidideebi viciT. vinaidan formulis marjvena mxare F ⋅ d skalaruli namravlis tolia, amitom:   W = F ⋅d (mudmivi Zalis mier Sesrulebuli muSaoba), (7.8)  sadac F aris F Zalis mniSvneloba (sidide). 7.8 formula gansakuTrebulad maSin xdeba   mosaxerxebeli, rodesac F da d erTeulovan-veqtorul CanawerSi aris mocemuli. gafrTxileba: 7.6-7.8 formulebis gamoyenebaze ori SezRudva arsebobs: 1) Zala unda iyos mudmivi anu misi sidide sagnis moZraobis dros ar unda icvlebodes da 2) sagani nivTieri wertilis magvari unda iyos anu is unda iyos myari da misi yvela nawili erTdroulad moZraobdes. mocemul TavSi mxolod nivTieri wertilis magvar sagnebs da mudmivi Zalis SemTxvevebs ganvixilavT. muSaobis niSani. Zalis mier saganze Sesrulebuli muSaoba SeiZleba iyos dadebiTi an uaryofiTi. magaliTad Tu 7.7 formulaSi φ kuTxe 90°-ze naklebia, maSin cos φ da Sesabamisad muSaobac dadebiTia. rodesac φ kuTxe 90°-ze sur. 7.3. sawolebiT rbolis monawile. sawoli da misi mgzavri nivTier wertilad SegviZlia miviCnioT, raTa monawilis Zalis mier Sesrulebuli muSaoba gamovTvaloT. metia (180°-mde), cosφ da Sesabamisad muSaobac uaryofiTia. zemoT miRebul Sedegebs martiv wesamde davyavarT. muSaobis niSnis gasagebad Zalis veqtoris gadadaadgilebis paraleluri komponenti unda ganvixiloT. 203 7.5. muSaoba da kinetikuri energia Zala dadebiT muSaobas asrulebs, rodesac misi veqtoruli komponentis mimarTuleba gadaadgilebis mimarTulebas emTxveva da uaryofiT muSaobas asrulebs, rodesac veqtoruli komponentis mimarTuleba gadaadgilebis mimarTulebis sapirispiroa. veqtoruli komponentis ararsebobis SemTxvevaSi Zala muSaobas ar asrulebs. muSaobis erTeulebi. SI sistemaSi muSaobis da kinetikuri energiis erTeuli joulia. 7.6 da 7.7 formulebidan Cans, rom joulis eqvivalenturi erTeulia n · m, xolo britanul sistemaSi futi · funti. gavSaloT 7.2 formula da miviRebT: 1j = 1kg · m2/wm2 = 1n · m = 0,738 futi · funti (7.9) ramdenime Zalis mier Sesrulebuli jamuri muSaoba. rodesac saganze ori an meti Zala moqmedebs, jamuri muSaoba calkeuli Zalis mier Sesrulebuli muSaobaTa jamis tolia. jamuri muSaoba ori meTodiT SegviZlia gamovTvaloT: 1) gamovTvaloT TiToeuli Zalis mier  Sesrulebuli Ft gamovTvaloT,   Semdeg ki 7.7 formulis gamoyenebiT F-is nacvlad Ft CavsvaT, xolo φ-s nacvlad - Ft -sa da d -s muSaoba da Semdeg SevkriboT, 2) Tavdapirvelad am Zalebis tolqmedis sidide Soris kuTxe. muSaoba-kinetikuri energiis Teorema 7.5 formula burTulis kinetikuri energiis cvlilebas burTulaze Sesrulebul W (=Fxd) muSaobasTan akavSirebs. nivTieri wertilis magvari sagnebisTvis am formulis ganzogadeba SesaZlebelia. davuSvaT, ΔK aris sagnis kinetikuri energiis cvlileba, xolo W – masze Sesrulebuli jamuri muSaoba. amis Semdeg SegviZlia davweroT: ΔK = Kf – Ki = W 7.10 formulis Tanaxmad: wertilis kinetikuri ( energiis cvlileba ) ( = wertilze Sesrulebuli jamuri muSaoba (7.10) ) . 7.10 formula asec SeiZleba daiweros: es formula gvauwyebs: kinetikuri energia jamuri ( muSaobis Sesrulebis Semdeg Kf = Ki + W ) ( = kinetikuri energia jamuri muSaobis Sesrulebamde ) ( = (7.11) Sesrulebuli jamuri muSaoba ) . zemoT moyvanili mtkicebebi muSaoba-kinetikuri energiis Teoremis saxeliT aris cnobili. is gamosadegia, rogorc dadebiTi, aseve uaryofiTi muSaobisTvis. Tu wertilze Sesrulebuli jamuri muSaoba dadebiTia, maSin wertilis kinetikuri energia muSaobis odenobiT izrdeba. Tu jamuri muSaoba uaryofiTia, maSin wertilis kinetikuri energia muSaobis odenobiT mcirdeba. magaliTad, Tu wertilis kinetikuri energiaa 5j da mas 2j energia gadaecema, saboloo kinetikuri energia 7j iqneba. 2 j energiis gamosvlis SemTxvevaSi wertilis saboloo energia 3 j gaxdeba. sakontrolo wertili 1. wertili x RerZis gaswvriv moZraobs. rodesac wertilis siCqare icvleba a) –3m/wm-dan –2m/wm-mde da b) –2m/wm-dan 2m/wm-mde, misi kinetikuri energia izrdeba, mcirdeba Tu ar icvleba? g) TiToeul SemTxvevaSi wertilze Sesrulebuli muSaoba dadebiTia, uaryofiTia Tu nulis tolia? 204 kinetikuri energia da muSaoba Tavi 7 7.2 amocanis amoxsnis nimuSi ori Spioni 225kg masis seifs d=8,5m manZilze 001 12n sididis da  horizontalidan qvemoT 30°-iT daxrili F1 ZaliT moqmedebs, xolo 002 Spioni 10n sididis  da horizontalidan zemoT 40°-iT daxrili F2 miacurebs. Spioni ZaliT cdilobs seifis mitanas daniSnulebis adgilamde. moZraobis dros am Zalebis sidideebi da mimarTulebebi ar icvleba, xolo seifsa da zedapirs Soris xaxuni ar arsebobs.  F1 da F2 Zalebi  seifze d gadaadgilebis ganmavlobaSi? a) ra jamur muSaobas asruleben amoxsna: ori Zalis mier seifze Sesrulebuli W jamuri muSaoba calkeuli muSaobebis jamis tolia. garda amisa, seifi nivTieri wertilia da Zalebis sidideebi da mimarTulebebi ar icvleba. maSasadame, muSaobis SegviZlia b) Tavisufali sxeulis grafiki seifisTvis mimarTuleba ar icvleba, maT mier Sesru lebuli muSaoba 7.7 formuliT SegviZlia gamovTvaloT. mizidulobis Zalis sididea mg da gveqneba: gamoTvli sTvis 7.7 (W = Fd cos φ) an 7.8 formula (W=   F ⋅d )  sur. 7.4. a) ori Spioni seifs d manZilze gadaadgilebs. (pasuxi) WN = FN d cos 90° = FN d(0) = 0 (pasuxi) vinaidan amgvari Sedegis winaswar gamocnoba namdvilad Zalebis sidideebi da mimarTulebebi viciT, SegveZlo. vinaidan es Zalebi seifis gadaad 7.7 am gilebis perpendikularulad aris mimarTuli, formulidan da 7.4b suraTze gamosaxuli seifze muSaobas ar asruleben da energiis Tavisufali sxeulis grafikidan gamomdinare  gadacema ar xdeba. formula gamoviyenoT. da Wgmgd cos 90° = mgd(0) = 0 ufro mosaxerxebelia. F1 Zalis mier Sesrulebuli muSaobaa: W1 = F1d cos φ1 = (12 n)(8,5 m)(cos 30°) = 88,33 j  F2 Zalis mier Sesrulebuli muSaobaa: W2 = F2d cos φ2 = (10n)(8,5 m)(cos 40°) = 65,11j maSasadame, jamuri W muSaobaa: νf siCqare 8,5 m-is gavlis Semdeg? amoxsna: seifis siCqare icvleba, vinaidan  da  F1 F2 Zalis mier energiis gadacemis dros misi kinetikuri energia icvleba. siCqaresa da W = W1 + W2 = 88,33j + 65,11j = 153,4j ≈ 153j g) seifi Tavdapirvelad uZravia. ra iqneba misi muSaobas Soris kavSiris dasadgenad 7.10 da 7.1 (pasuxi) 8,5m manZilze gadaadgilebis ganmavlobaSi Spionebi seifs 153j energias gadascemen. b) gadaadgilebis ganmavlobaSi ra Wg muSaobas  asrulebs Fg mizidulobis Zala da ra WN  muSaobas asrulebs FN drekadobis Zala formulebi gavaerTianoT: W = Kf – Ki = 1 mνf2 – 1 mνi2 2 2 sawyisi νi siCqare nulis tolia, xolo mTliani Sesrulebuli muSaobaa 153,4 j. zemoT miRe buli formulis νf-sTvis amoxsna da monacemebis Casma mogvcems: seifze? amoxsna: vinaidan am ori Zalis sidide da amocanis amoxsnis nimuSi yuTi Stormis dros molipul avtosadgomze = 1,17 m/w (pasuxi) 7.3  d = (–3  m) manZilze gadaadgildeba da masze qari F = (2 n) + (–6 n) ucvleli ZaliT moq a) ra muSaobas asrulebs qari yuTis gadaadgi medebs. 7.5 suraTze gamosaxulia mocemuli da viTareba da koordinatTa RerZebi. ar icvleba, muSaobis gamoTvlisTvis 7.7 lebis dros? amoxsna: vinaidan yuTi nivTieri wertilia qaris Zalis sidide da mimarTuleba 205 7.6. mizidulobis Zalis mier Sesrulebuli muSaoba kuri energia  d -s bolos? amoxsna: vinaidan Zala yuTze uaryofiT   sur. 7.5. d manZilze gadaadgilebisas F ucvleli Zala anelebs yuTis moZraobas.   (W = Fd cos φ) an 7.8 (W = F⋅ d ) formula SegviZlia gamoviyenoT. F da d erTeulovan- veqtorul CanawerSi gvaqvs mocemuli, amitom 7.8 formulas virCevT da vwerT:   W = F ⋅ d = [(2 n) + (–6 n) ] · [(–3 m) ]. · da k ⋅ k ar aris nuli toli (ix. E damateba). am SemTxvevaSi viRebT: W = (2 n)(–3 m) · + (–6 n)(–3 m) · = (–6 j)(1) + 0 = –6 j (pasuxi) asrulebs da yuTis kinetikur energias 6 j-s b) Tu rema 7.11 formulis saxiT gamoviyenoT da miviRebT: Kf = Ki + W = 10 j + (–6 j) = 4 j (pasuxi) kinetikuri energia 4 j-mde Semcirda da yuTic Senelda. saxulia oTxi mdgomareoba, romlebSic yuTze Zala moqmedebs da yuTi xaxunisgan Tavisufal zedapirze d manZils mar jvnisken gadis. Zalebis sidideebi Tana maSasadame, Zala yuTze uaryofiT 6 j muSaobas aklebs. mcirdeba. muSaoba-kinetikuri energiis Teo sakontrolo wertili 2. suraTze gamo SesaZlo skalaruli namravlebidan mxolod   · , muSaobas asrulebs, yuTis kinetikuri energia baria da maTi mimarTulebebic naCvenebia. daalageT mdgomareobebi gadaadgilebis dros yuTze Sesrulebuli muSaobis mi xedviT dididan mcireze.  d gadaadgilebis dasawyisSi yuTis kinetikuri energiaa 10 j, risi tolia kineti (a) (b) (g) (d) 7.6. mizidulobis Zalis mier Sesrulebuli muSaoba modiT, mizidulobis Zalis mier saganze Sesrulebuli muSaoba ganvixiloT. 7.6 suraTze naCvenebia m masis pomidori, romelic ν0 sawyisi siCqariT da, maSasadame, Ki=1 mν02 2 kinetikuri  energiiT maRla aisroles. maRla asvlisas pomidors Fg mizidulobis Zala anelebs anu mizidulobis Zala pomidorze muSaobas asrulebs da kinetikuri energia mcirdeba. vinaidan pomidori nivTier wertilad SegviZlia CavTvaloT,  d gadaadgilebis dros masze Sesrulebuli muSaoba 7.7  formuliT (W = Fd cos φ) SegviZlia CavweroT. Fg Zalis sididea mg da am Zalis mier Sesrulebuli Wg muSaobaa: Wg = mgd cosφ (mizidulobis Zalis mier Sesrulebuli muSaoba) (7.12) sur. 7.6. m masis maRla asrolil  pomidorze Fg mizidulobis  Zala moqmedebs da misi siCqare d ga daadgilebis dros 0-dan -mde mci rdeba. kinetikuri energiis sawyao kinetikuri energiis Ki(=1 mν02)-dan   2 maRla asrolili sagnis SemTxvevaSi Fg Zala d gadaadgilebis K (=1 mν2)-mde cvlilebas asaxavs. f 2 sapirispiro mimarTulebisaa, φ = 180° da gveqneba: Wg = mgd cos180° = mgd (–1) = –mgd (7.13) minus niSani gvauwyebs, rom maRla asvlis dros saganze moqmedi mizidulobis Zala sagnis kinetikur energias mgd raodenobas aklebs da, Sesabamisad, sagani neldeba. mas Semdeg rac sagani maqsimalur simaRles miaRwevs, is vardnas iwyebs da   Fg Zalasa da d gadaadgilebs Soris φ kuTxe nulis tolia: Wg= mgd cos 0° = mgd(+1) = +mgd (7.14) plius niSani gvauwyebs, rom mizidulobis Zala sagnis kinetikur energias mgd raodenobis energias umatebs. aqedan gamomdinare sagani Cqardeba. 206 Tavi 7 kinetikuri energia da muSaoba sagnis awevasa da dabla dawevaze Sesrulebuli muSaoba warmovidginoT, rom nivTieri wertilis magvar saganze  F vertikaluri ZaliT vmoqmedebT da mas maRla vwevT. gadaadgilebis dros Cveni Zala saganze Wa dadebiT muSaobas asrulebs, xolo mizidulobis Zala - uaryofiT Wg muSaobas. Cvens mier modebuli Zala sagans energias gadascems, xolo mizidulobis Zala sagnisTvis energiis warTmevas cdilobs. 7.10 formulidan gamomdinare, energiis gadacemebis Sedegad sagnis kinetikuri energiis ΔK cvlilebaa: ΔK = Kf – Ki = Wa + Wg , (7.15)  sur. 7.7. a) modebuli  F Zala sagans wevs. sagnis d gadaadgileba sadac Kf aris kinetikuri energia gadaadgilebis dasasruls, maRla  mizidulobis Zalisadmi f=180° F g xolo K – kinetikuri energia gadaadgilebis dasawyisSi. igive i kuTxes adgens. modebuli Zala formula sagnis dabla dawevis SemTxvevaSic gamosadegia, saganze dadebiT muSaobas asrulebs.  Tumca am SemTxvevaSi mizidulobis Zala sagans energias b) modebuli F Zala sagans dabla  gadascems, xolo Cvens mier modebuli Zala sagans energias wevs. sagnis d gadaadgileba Fg mizidulobis Zalisadmi f=0° kuTxes arTmevs. adgens. modebuli Zala saganze gavrcelebuli SemTxvevaa, rodesac awevamde da awevis uaryofiT muSaobas asrulebs. Semdeg sagani uZravia. magaliTad, warmovidginoT, rom wigns iatakidan Taroze debT. am SemTxvevaSi Kf da Ki kinetikuri energiebi nulis tolia da 7.15 formula Semdeg saxeze daiyvaneba: Wa + Wg = 0 anu Wa = –Wg (7.16) yuradReba miaqcieT, rom igive Sedegs miviRebT, Tu kinetikuri energiebi erTmaneTis tolia. orive SemTxvevaSi miRebuli Sedegi niSnavs, rom modebuli Zalis mier Sesrulebuli muSaoba mizidulobis Zalis mier Sesrulebuli muSaobis Sebrunebulia anu modebuli Zala sagans igive raodenobis energias gadascems, rasac mizidulobis Zala sagans arTmevs. 7.12 formulis gamoyenebiT 7.16 formula ase SeiZleba Caiweros: Wa = – mgd cosφ (awevasa da dawevaze Sesrulebuli muSaoba; Kf=Ki) (7.17)   sadac φ aris kuTxe Fg -sa da d -s Soris. Tu gadaadgileba vertikalurad zeviT aris mimarTuli (sur. 7.7a), maSin φ = 180° da modebuli Zalis mier Sesrulebuli muSaobaa mgd. Tu gadaadgileba vertikalurad qvemoT aris mimarTuli (sur. 7.7b), maSin φ = 0° da modebuli Zalis mier Sesru lebuli muSaoba –mgd-s utoldeba. 7.16 da 7.17 formulebi nebismier mdgomareobas Seesabameba, rodesac sagans maRla an dabla weven da awevamde da awevis Semdeg sagani uZravia. formulebi modebuli Zalis sidideze damokidebuli ar aris. magaliTad, Tu iatakidan Wiqas aiRebT da Tavs zemoT daiWerT, Tqveni Zala awevis dros mniSvnelovnad Seicvleba. miuxedavad amisa, awevamde da awevis Semdeg Wiqa uZravia da Tqveni Zalis mier Wiqaze Sesrulebuli muSaoba 7.16 da 7.17 formulebiT gamoiTvleba, sadac mg aris Wiqis wona, xolo d - awevis manZili. amocanis amoxsnis nimuSi 7.4 tvirTebis awevaSi legendad iqca (sur. 7.8). 1 sm simaRleze awia. platformaze mTlianad 27900 n wonis erT-erTi misi Sedegi dRemde rekordia. avtomobilis nawilebi da tyviiT savse seifi anedersoni xis platformis qveS moixara, iyo ganlagebuli. 50-ian wlebSi, pol andersoni uzarmazari xelebiT patara taburetebs daeyrdno da platforma daaxloebiT 7.6. mizidulobis Zalis mier Sesrulebuli muSaoba 207 a) tvirTis awevis dros ra muSaoba Seasrula  Fg mizidulobis Zalam tvirTze? amoxsna: tvirTi erT nivTier wertilad SegviZlia miviCnioT, vinaidan misi nawilebi erTdroulad moZraobdnen. amis Semdeg 7.12 formulis (Wg = mgd cosφ) saSualebiT  Fg Za lis mier tvirTze Sesrulebul Wg muSaobas gamovTvliT. qvemoT mimarTul mizidulobis Zalasa da zemoT mimarTul gadaadgilebas Soris φ kuTxea 180°. es da sxva monacemebi 7.12 formulaSi CavsvaT da miviRebT: Wg = mgd cos φ = (27900 n)(0,01 m)(cos180°) = –280 j (pasuxi) sur. 7.8. pol andersonma platforma da moxaliseebis jgufi zurgiT awia b) ra muSaoba Seasrula andersonis mier Sebrunebulia. 7.16 formula swored amgvar modebulma Zalam tvirTis awevaze? amoxsna: andersonis mier tvirTze modebuli Zalis gamosaxuleba gvqondes, misi Zala ar gvaqvs da namdvilad mdgomareobas asaxavs: WA = –Wg = +280 j romc ar iyo (pasuxi) miRebuli raodenobis muSaoba skolis gate mudmivi. aqedan gamomdinare, 7.7 formulaSi nili Zalis sididis Casma da muSaobis gamoTvla ki sakmarisia. maSin ratom iyo andersonis SeuZlebelia. miuxedavad amisa, viciT, rom Sedegi gasaocari? pasuxi is aris, rom muSaoba awevamde da awevis Semdeg tvirTi uZravia. da Zala sxvadasxva sidideebia. andersonis aqedan gamomdinare, andersonis Zalis mier  miRwevas ar sWirdeboda didi raodenobis Zalis Zala namdvilad aucilebeli iyo. Sesrulebuli WA muSaoba Fg mizidulobis mier Sesrulebuli muSaobis amocanis amoxsnis nimuSi zurgCanTis iatakidan aRebisTvisac energiis gadacema, samagierod, uzarmazari 7.5 15kg masis Tavdapirvelad uZrav yuTs Tokis saSualebiT xaxunisgan Tavisufal daxril zedapirze d = 5,7 m manZilze gadaaadgileben. h = 2,5 m simaRleze yuTi Cerdeba (sur. 7.9a). zedapiri  a) ra Wg muSaobas asrulebs yuTze Fg mizi dulobis Zala gadaadgilebisas? (a) amoxsna: Cven SegviZlia yuTi nivTier wer tilad miviCnioT da Fg Zalis mier masze Sesrulebuli Wg muSaobis gamoTvlisTvis 7.12 formula (Wg = mgd cos φ) gamoviyenoT miuxedavad imisa, rom  Fg -sa da d -s Soris φ kuTxe ar viciT. 7.9b suraTze gamosaxulia Tavisufali sxeulis grafiki yuTisTvis da Cans, rom φ = θ + 90°, sadac θ aris sibrtyis daxris kuTxe. 7.12 formula gvaZlevs: Wg = mgd cos (θ + 90°) = –mgd sin θ (7.18) (b) sur. 7.9. a) yuTi xaxunisgan Tavisufal daxril  sibrtyeze sibrtyisadmi paraleluri T Zalis moqme debiT gadaadgildeba. b)  Tavisufali sxeulis grafiki yuTisTvis. naCvenebia d gadaadgilebac. 208 kinetikuri energia da muSaoba Tavi 7 zemoT miRebuli formulis gamartivebisTvis trigonometriuli igiveoba gamoviyeneT. cvlileba nulis tolia. yuTze Sesrulebuli jamuri W muSaobis gamoTvlisTvis yuTze miRebuli Sedegi erTi SexedviT arafers moqmedi ar gvaZlevs, vinaidan θ ucnobia, magram 7.9 muSaoba unda SevkriboT. a)-dan ukve viciT, rom  suraTze Cans, rom d sin θ = h, sadac h CvenTvis cnobili sididea. amis Semdeg 7.18 formula mogvcems: Wg = –mgh (7.19) = –386 j (pasuxi) 7.19 formulis Tanaxmad mizidulobis Zalis mier Sesrulebuli Wg muSaoba vertikalur damokidebuli Zalis mier Sesrulebuli Fg mizidulobis Zalis mier Sesrulebuli Wg muSaoba –386j-ia, sibrtyidan yuTze moqmedi  FN drekadobis Zalis mier Sesrulebuli  WN muSaoba nulis tolia, vinaidan FN Zala gadaadgilebis = –(15 kg)(9,8 m/wm2)(2,5 m) gadaadgilebazea samive da perpendikularulia. Cven  mxolod T -s mier Sesrulebuli WT muSaoba dagvrCa. muSaoba-kinetikuri energiis Teo remis Tanaxmad: ΔK = WT + Wg + WN ara horizontalur gadaadgilebaze. b) ra WT muSaobas asrulebs yuTze Tokidan  moqmedi T Zala gadaadgilebis dros? amoxsna: 7.7 formulaSi (W = Fd cos φ) F-is nacvlad T-s ver CavsvamT, vinaidan T-s mniS vneloba ar viciT. samagierod SegviZlia yuTi nivTier wertilad miviCnioT da mas muSaobakinetikuri energiis Teorema (ΔK = W) mivu sadagoT. awevamde da awevis Semdeg yuTi uZravia, amitom misi kinetikuri energiis ΔK amocanis amoxsnis nimuSi anu 0 = WT – 368j + 0 da WT = 368 j (pasuxi) sakontrolo wertili 3. davuSvaT, yuTs igive h simaRlemde, oRond ufro grZel daxril sibrtyeze vamoZravebT. a) T Zalis mier Sesrulebuli muSaoba metia, naklebia Tu ar icvleba? b) yuTis moZraobisTvis  saWiro T Zalis sidide metia, naklebia Tu ar icvleba? 7.6 m=500kg masis lifti νi = 4m/wm siCqariT qvemoT   Cadis, gvarli fuWdeba da kabina a = g / 5 mudmivi aCqarebiT vardeba (sur. 7.10a). a) d = 12m manZilze vardnis ganmavlobaSi ra  Wg muSaobas asrulebs Fg mizidulobis Zala kabinaze? amoxsna: kabina SegviZlia nivTier wertilad miviCnioT da Fg Zalis mier Sesrulebuli Wg muSaoba 7.12 formuliT (Wg = mgd cos φ) gamovTvaloT.  7.10b suraTidan Cans, rom Zalasa da Fg d gadaadgilebas Soris kuTxe 0 -ia. 0 maSasadame, 7.12 formula gvaZlevs: Wg = mgd cos0° = (500 kg)(9,8 m/wm2)(12m)(1) = 5,88 × 104 j ≈ 59 kj (pasuxi) b) 12m-iani vardnis ganmavlobaSi ra WT muSao bas asrulebs kabinaze liftis  gvarlis gaswvriv zemoT mimarTuli T Zala? amoxsna: (W = Fd cosφ) 7.7 formulis sa (a) (b) sur. 7.10. liftis kabina νi siCqariT qvemoT eSveba da   aCqarebas iwyebs. a) kabina d manZils a = g/5 mudmivi aCqarebiT gadis. b) Tavisufali sxeulis grafiki kabinisTvis. naCvenebia gadaadgilebac. 209 7.7. drekadobis Zalis mier Sesrulebuli muSaoba SualebiT WT Tavdapirvelad muSaobas gamovTvliT, Tu T sididis gamosaxulebas Sesabamisad, kinetikuri energiac icvleba. aqedan gamomdinare, 7.16 formula am miviRebT. am gamosaxulebis miReba y RerZis SemTxvevaSi gamousadegaria. gaswvrivi komponentebisTvis niutonis meore g) ra jamuri W muSaoba sruldeba kabinaze kanonis CaweriT SeiZleba: T – Fg = ma es formula T-sTvis amovxsnaT, Fg-s nacvlad mg CavsvaT da miRebuli Sedegi 7.7 formulaSi gadavitanoT: WT = Td cos φ = m(a+g)d cos φ vardnis ganmavlobaSi? amoxsna: jamuri muSaoba kabinaze moqmedi Zalebis mier Sesrulebuli muSaobebis jamia: W = Wg + WT = 5,88×104 j – 4,7 × 104 j = 1,18×104 j ≈ 12 kj Semdeg qvemoT mimarTuli a aCqarebis nacvlad d) risi tolia kabinis kinetikuri energia kuTxis nacvlad 180°: amoxsna: kabinis kinetikuri energia icvleba,   –g/5 CavsvaT, xolo T da mg Zalebs Soris φ ( ) 4 g WT = m – + g d cos φ = mgd cos φ 5 5 4 = (500 kg)(9,8 m/w2)(12 m) cos 180° 5 = –4,7 × 104 j ≈ –47 kj (pasuxi) yuradReba miaqcieT, rom WT ar aris Wg-s Sebrunebuli. amis mizezi is aris, rom kabina vardnis dros Cqardeba, misi siCqare da, 12m-iani vardnis dasasruls? vinaidan masze jamuri muSaoba sruldeba. 7.1 formulidan gamomdinare, vardnis dasawyisSi 1 kinetikuri energiaa Ki= 2 mνi2. amis Semdeg 7.11 formula aseT saxes iRebs: 1 2 Kf = Ki + W = mνi2 + W 1 2 = (500 kg)(4 m/wm)2 + 1,18×104 j = 1,58 × 104 j ≈ 16 kj (pasuxi) 7.7. drekadobis Zalis mier Sesrulebuli muSaoba am TavSi cvladi Zalis garkveuli saxeobis, kerZod, drekadobis Zalis anu zambaridan moqmedi Zalis mier nivTieri wertilis magvar saganze Sesrulebul muSaobas ganvixilavT. bunebaSi arsebul mraval Zalas drekadobis Zalis magvari maTematikuri forma aqvs. aqedan gamomdinare, am Zalis Seswavlis dros mravali sxva Zalis bunebas gavigebT. drekadobis Zala 7.11 suraTze naCvenebia moSvebul mdgomareobaSi myofi zambara anu rodesac zambara SekumSuli an daWimuli ar aris. zambaris erTi bolo damagrebulia, xolo meore boloze nivTieri wertilis magvari sagani, magaliTad, blokia mibmuli. Tu bloks marjvniv gavwevT da zambaras davWimavT (7.11b suraTi), zambara bloks marcxnisken ubiZgebs (vinaidan drekadobis Zala zambaras pirvandel mdgomareobaSi abrunebs, mas aRmdgen Zalasac uwodeben). Tu bloks marcxnisken vubiZgebT da zambaras SevkumSavT (7.11g suraTi), zambara bloks marjvnisken ubiZgebs. zambaraze Catarebuli mravali eqsperimentiT gamarTlebulia daSveba, rom zambaridan moqmedi   Fs Zala zambaris Tavisufali bolos d gadaadgilebis proporciulia.  (7.20) Fs = - k d (hukis kanoni) ,  sadac Fs aris drekadobis Zala. es formula hukis kanonis saxeliTaa cnobili. robert huki ingliseli mecnieri iyo, romelic me-17 saukuneSi moRvaweobda. 7.20 formulaSi gamoyenebuli minus niSani gvauwyebs, rom drekadobis Zala yovelTvis zambaris Tavisufali bolos gadaadgilebis mimarTulebis sapirispiroa. k mudmivas zambaris sixiste ewodeba da igi zambaris elastiurobis maxasiaTebelia. rac ufro metia k, miT ufro araelastiuria zambara anu miT ufro meti Zalaa saWiro mocemuli gadaadgilebis misaRwevad. SI sistemaSi k-s erTeulia n/m. 210 Tavi 7 kinetikuri energia da muSaoba 7.11 suraTze x RerZi zambaris sigrZis paraleluradaa gavlebuli da misi aTvlis saTave (x=0) zambaris Tavisufali bolos mdebareobas emTxveva, rodesac zambara moSvebulia. amgvari ganlagebis SemTxvevaSi 7.20 formula Semdegi saxiT SeiZleba Caiweros: Fx = – kx (hukis kanoni). (7.21) wina formulisgan gansxvavebiT, amjerad indeqsia Secvlili. Tu x dadebiTia (zambara x RerZis marjvniskenaa daWimuli), Fx uaryofiTia (biZgi marcxniskenaa mimarTuli), xolo Tu x dadebiTia (zambara marcxniskenaa SekumSuli), maSin Fx dadebiTia (biZgi marjvniskenaa mimarTuli). gaiTvaliswineT, rom drekadobis Zala cvalebadia, vinaidan is zambaris Tavisufali bolos x mdebareobis funqciaa. aqedan gamomdinare, Fx Zala F(x) simboloTi SegviZlia aRvniSnoT. aseve yuradReba miaqcieT, rom hukis kanoni Fx-sa da x-s Soris sur. 7.11. a) zambara moSvebul mdgomareobaSi. x RerZis aTvlis saTave zambaris blokTan mierTebis  d bolos emTxveva. b) bloki manZilze gadaadgildeba, zambaris gaWimvis sidide x dadebiTia da  Fs drekadobis ZaliT moqmedebs. g) zambaris SekumSvis sidide x uaryofiTia da am SemTxvevaSic aRmdgeni Zala moqmedebs. wrfiv damokidebulebas asaxavs. drekadobis Zalis mier Sesrulebuli muSaoba 7.11a suraTze gamosaxuli blokis moZraobis dros drekadobis Zalis mier Sesrulebuli muSaobis gamoTvlisTvis ori mniSvnelovani daSvebis gakeTebaa saWiro: 1) zambaras masa ar gaaCnia anu misi masa blokis masasTan SedarebiT umniSvneloa da 2) zambara idealuria anu is hukis kanons zustad emorCileba. garda amisa, davuSvaT, rom bloksa da zedapirs Soris xaxuni ar arsebobs da bloki nivTieri wertilis magvari sagania. Tavdapirvelad bloks vubiZgoT marjvnisken da avamoZraoT. bloki marjvnisken gadaadgildeba, masze Fx drekadobis Zala muSaobas asrulebs da blokis kinetikuri energia (maSasadame siCqarec) mcirdeba. 7.7 formulis saSualebiT am muSaobis gamoTvla SeuZlebelia, vinaidan formulaSi mudmivi Zala unda Sediodes, drekadobis Zala ki cvalebadia. zambaris mier Sesrulebuli muSaobis dadgenas gamoTvlebi dasWirdeba. modiT blokis sawyisi mdebareoba xi simboloTi aRvniSnoT, xolo momdevno mdebareoba – xf simboloTi. Semdeg am or mdebareobas Soris manZili Zalian mcire Δx sigrZis segmentebad davyoT. es segmentebi xi-dan dawyebuli 1, 2 da a.S. ricxvebiT aRvniSnoT. blokis mier TiToeuli segmentis gavlis dros drekadobis Zala ar icvleba, vinaidan segmentis zomis simciris gamo x TiTqmis ucvlelia. aqedan gamomdinare, segmentSi moqmedi Zalis mniSvneloba mudmivad SegviZlia miviCnioT. Zalis mniSvneloba segmentebis mixedviT iyos Fx1, Fx2 da a.S. vinaidan TiToeul segmentSi Zala mudmivia, 7.7 formulis saSualebiT TiToeul segmentSi Sesrulebuli muSaobis gamoTvla SesaZlebeli xdeba. am SemTxvevaSi, φ = 180° da cos φ = –1. pirvel segmentSi Sesrulebuli muSaobaa –Fx1 Δx, meore segmentSi Sesrulebuli muSaoba _ –Fx2 Δx da a.S. zambaris mier Sesrulebuli jamuri Ws muSaoba xi-dan xf-mde am muSaobebis jamis tolia: , (7.22) sadac j aris segmentebis rigiTi nomeri. zRvarSi Δx nulisken miiswrafis da 7.22 formula aseT saxes iRebs: . (7.23) 211 7.7. drekadobis Zalis mier Sesrulebuli muSaoba 7.21 formulidan gamomdinare Fx Zalis sididea kx da gveqneba: x = (– 1k)[x2]xf = (– 1k)[x2]xi (– 1 k)(xf2 – xi2). 2 i 2 2 (7.24) gadamravlebis Semdeg miviRebT: Ws = 1 kxi2 – 1 kxf2 2 2 (drekadobis Zalis mier Sesrulebuli muSaoba) (7.25) drekadobis Zalis mier Sesrulebuli Ws muSaoba SeiZleba iyos dadebiTi an uaryofiTi, rac imazea damokidebuli blokidan gamodis energia Tu bloks gadaecema. gafrTxileba: xf saboloo mdebareoba 7.25 formulis marjvena mxaris meore wevria. aqedan gamomdinare, 7.25 formula gvauwyebs: Ws muSaoba dadebiTia, Tu bloki moSvebul mdebareobasTan (x = 0) ufro axlos Cerdeba, vidre Tavdapirvelad iyo. Ws muSaoba uaryofiTia, Tu bloki x = 0 mdebareobidan moSorebiT Cerdeba. Tu x = 0 mdebareobidan manZili ucvlelia, maSin Ws muSaoba nulis tolia. Tu xi = 0 da saboloo mdebareobaa x, maSin 7.25 formula gvaZlevs: Ws = – 1 kx2 (drekadobis Zalis mier Sesrulebuli muSaoba) 2 modebuli Zalis mier Sesrulebuli muSaoba amjerad warmoidgineT, rom bloki x RerZis gaswvriv gadaadgildeba da masze (7.26)  Fa Zala moqmedebas ganagrZobs. gadaadgilebis dros Cvens mier modebuli Zala blokze Wa muSaobas asrulebs, xolo drekadobis Zala - Ws muSaobas. energiis gadacemis Sedegad blokis kinetikuri energiis ΔK cvlilebaa: ΔK = Kf – Ki = Wa + Ws , (7.27) sadac Kf aris kinetikuri energia gadaadgilebis bolos, xolo Ki - kinetikuri energia gadaadgilebis dasawyisSi. Tu gadaadgilebamde da gadaadgilebis Semdeg bloki uZravia, maSin Kf da Ki nulis tolebia da 7.27 formula Semdeg saxeze daiyvaneba: Wa = –Ws (7.28) Tu zambaraze mimagrebuli bloki gadaadgilebamde da gadaadgilebis Semdeg uZravia, modebuli Zalis mier Sesrulebuli muSaoba drekadobis Zalis mier Sesrulebuli muSaobis Sebrunebulia. gafrTxileba: Tu bloki gadaadgilebamde da gadaadgilebis Semdeg moZraobs, maSin zemoT moyvanili mtkiceba marTebuli ar aris. sakontrolo wertili 4. 7.11 suraTze gamosaxuli blokis x RerZis gaswvriv sawyisi da saboloo mdebareobebia: a) –3sm, 2sm; b) 2sm, 3sm da g) –2sm, 2sm. TiToeul mdgomareobaSi drekadobis Zalis mier blokze Sesrulebuli muSaoba dadebiTia, uaryofiTia Tu nulis tolia? amocanis amoxsnis nimuSi 7.7 kanfetebiT savse paketi xaxunisgan Tavisufal amoxsna: magidaze devs da zambaris Tavisufal bo drekadobis Zala masze 7.25 an 7.26 formuliT loze 7.11a suraTis msgavsad aris mibmuli. mocemul muSaobas asrulebs. sawyisi xi koor x1 = 12 mm koordinatze paketis gaCerebisTvis marjvnisken moqmedi Fa= 4,9 n Zalaa saWiro. a) ra muSaobas asrulebs drekadobis Zala paketze, Tu paketi x = 0 koordinatidan x2 = 17 mm koordinatze gadaadgildeba? rodesac paketi gadaadgildeba, dinatia 0, xolo saboloo xf kordinatia 17 mm. zambaris k sixiste ucnobia. 7.21 formuliT k-s gamoTvla SesaZlebelia. Tu paketi x1 = 12 mm koordinatze uZravad aris gaCerebuli, drekadobis Zalam modebuli Zala 212 Tavi 7 kinetikuri energia da muSaoba unda gaawonasworos. maSasadame, Fx drekadobis Zala –4,9 n unda iyos. 7.21 formula gvaZlevs: F – 4,9 n k=– x =– = 408 n/m x1 12 × 10–3 m rodesac paketi x2=17mm kordinatzea, 7.26 formulidan gamomdinareobs: Ws = – 1 kx22 = – 1 (408 n/m)(17 × 10–3 m)2 2 = – 0,059 j 2 (pasuxi) b) Semdeg paketi marcxniv gadaadgildeba x3 = –12 mm koordinatze. ra muSaobas asrulebs drekadobis Zala paketze am gadaadgilebis dros? amocanis amoxsnis nimuSi amoxsna: amjerad xi = +17mm da xf = –12mm. 7.25 formula gvaZlevs: Ws = 1 kxi2 – 1 kxf2 = 1 k(xi2 – xf2 ) 2 2 2 = 1 (408 n/m)[(17 × 10–3 m)2 – (–12 × 10–3 m)2] 2 = 0,03 j = 30 mj (pasuxi) drekadobis Zalis mier Sesrulebuli muSaoba dadebiTia, vinaidan drekadobis Zala xi = +17 mm koordinatidan moSvebuli mdgomareobisken gadaadgilebis dros ufro met dadebiT muSaobas asrulebs, vidre moSvebuli mdgo mareobidan xf = –12 mm koordinatze gadaad gilebis dros uaryofiTi muSaoba sruldeba. 7.8 7.12 suraTze m = 0,4 kg masis yuTi xaxunisgan Tavisufal horizontalur zedapirze ν = 0,5 m/wm siCqariT misrialebs da k = 750 n/m sixistis zambaras kumSavs. rodesac zambara yuTs aCerebs, ra d manZiliT aris zambara SekumSuli? zedapiri sur. 7.12. m masis yuTi k sixistis zambarisken siCqariT moZraobs. amoxsna: 1. drekadobis Zalis mier yuTze Sesrulebuli Ws muSaoba da saWiro d manZili erTmaneTs 7.26 formuliT ukavSirdeba (Ws = – 1 kx2), 2 sadac d cvlis x-s. 2. Ws muSaoba da yuTis kinetikuri energia erTmaneTs 7.10 formuliT (Kf – Ki = W) ukav Sirdeba. 3. yuTis sawyisi kinetikuri energiaa 1 2 K= mν , xolo saboloo kinetikuri energia energiis Teorema Semdegi saxiT CavweroT: Kf – Ki = – 1 kd 2 2 Semdeg, mesame punqtSi moyvanili mniSvnelo bebi gaviTvaliswinoT da miviRebT: 0 – 1 mν2 = – 1 kd 2 2 2 gamartiveba, d-sTvis amoxsna da cnobili mniS vnelobebis Casma gvaZlevs: 2 (gaCerebis momentSi) - nuli. zemoT CamoTvlili pirveli ori punqti gava erTianoT da yuTisTvis muSaoba-kinetikuri = 1,2 × 10–2 m = 1,2 sm (pasuxi) 7.8. cvladi Zalis mier Sesrulebuli muSaoba erTganzomilebiani analizi modiT 7.2 suraTze gamosaxul mdgomareobas davubrundeT. amjerad Zala x RerZis dadebiTi mimarTulebisaa, xolo misi sidide x mdebareobasTan erTad icvleba. burTulis moZraobasTan erTad muSaobis Semsrulebeli Zalis F(x) sidide icvleba. xolo mimarTuleba ucvleli rCeba da nebismier mdebareobaSi Zalis sidide ar aris damokidebuli droze. 7.13a suraTze gamosaxulia erTganzomilebiani cvladi Zalis grafiki. Cven unda davadginoT am Zalis mier nivTier wertilze Sesrulebuli muSaobis gamosaxuleba, rodesac wertili xi sawyisi 7.8. cvladi Zalis mier Sesrulebuli muSaoba 213 mdebareobidan xf saboloo mdebareobisken gadaadgildeba. am miznis misaRwevad 7.7 formulas (W = Fd cos φ) ver gamo viyenebT, vinaidan is mxolod  F mudmivi ZalisTvis aris gamosadegi. modiT 7.13a suraTze gamosaxuli mrudis qveda areali Δx siganis viwro zolebad davyoT (sur. 7.13b). Δx sakmarisad mcirea, raTa am SualedSi F(x) Zala mudmivad miviCnioT. Fj,saS aris F(x)-is saSualo mniSvneloba romeliRac j rigis SualedSi da 7.13b suraTis SemTxvevaSi j rigis zolis (a) simaRlea. Tu Fj,saS mudmivia, Zalis mier j rigis SualedSi Sesrulebuli ΔWj muSaobis nazrdi 7.7 formulis saSualebiT miaxloebiT gamoiTvleba: ΔWj = Fj,saS Δx (7.29) Fj,saS 7.13b suraTze ΔWj aris j rigis marTkuTxedis farTobi. nivTieri wertilis xi-dan xf-mde gadaadgilebis dros Zalis mier Sesrulebuli jamuri W muSaobis gamoTvlisTvis xi-sa da xf-s Soris ganlagebuli yvelaze zolis farTobi SevkriboT: . (b) (7.30) 7.30 formula miaxloebiTia, vinaidan marTkuTxa zolebis mier moxazuli `konturi~ realur F(x) mruds mxolod uaxlovdeba. Sedegi realurTan ufro miaxloebuli iqneba Tu zolis Δx siganes SevamcirebT da raodenobas gavzrdiT (sur. 7.13g). zolis sigane zRvarSi nulisken miiswrafis, zolebis (g) raodenoba usasrulod didia da ufro zust Sedegs viRebT: . (7.31) es zRvari xi da xf zRvrebs Soris F(x) funqciis integrals niSnavs da 7.31 formula aseT saxes iRebs: (muSaoba, cvladi Zala). (7.32) Tu F(x) funqcia cnobilia, maSin mas 7.32 formulaSi CavsvamT, integrirebis Sesabamis zRvrebs SemovitanT da, integrirebiT muSaobas gamovTvliT. geometriuli TvalsazrisiT muSaoba aris F(x) mrudsa da x RerZs Soris arealis farTobis toli (xi da xf zRvrebs Soris, sur. 7.13d). samganzomilebiani analizi amjerad ganvixiloT nivTieri wertili, romelzec samganzo milebiani Zala moqmedebs: , (7.33) sadac Fx , Fy da Fz komponentebi wertilis mdebareobaze aris damokidebuli anu TiToeuli komponenti mdebareobis funqcia SeiZleba iyos. Cven sami gamartiveba Semogvaqvs: 1) Fx SeiZleba damokidebuli iyos x-ze, magram ara y-ze an z-ze; 2) Fy (d) sur. 7.13. a) erTganzomilebiani  F (x) Zalis x gadaadgilebaze da mokidebulebis grafiki. wertili xi-dan xf-mde gadaadgildeba. b) gra fikis qveda areali viwro zolebad aris dayofili. g) grafikis qveda areali kidev ufro viwro zolebad aris dayofili. d) zRvruli Sem Txveva. Zalis mier Sesrulebuli muSaoba 7.32 formuliT gamoiTvleba da x RerZsa da grafiks Soris ga muqebuli farTobis (xi-sa da xf-s Soris) tolia. 214 Tavi 7 kinetikuri energia da muSaoba SeiZleba damokidebuli iyos y-ze, magram ara x-ze an z-ze; 3) Fz SeiZleba damokidebuli iyos z-ze, magram ara x-ze an y-ze. am SemTxvevaSi wertilis gadaadgilebis nazrdi iqneba: (7.34) dW = F · dr = Fx dx + Fy dy + Fz dz (7.35)   dr gadaadgilebis dros wertilze F Zalis mier Sesrulebuli dW muSaobis nazrdi 7.8 formuliT gamoiTvleba:   ri sawyisi mdebareobidan  (xi, yi, zi koordinatebi) rf saboloo mdebareobisken (xf ,yf ,zf koordinatebi) gadaadgilebis dros F Zalis mier wertilze Sesrulebuli W muSaobaa: (7.36)  Tu F -s mxolod x komponenti aqvs, maSin 7.36 formulaSi y da z wevrebi nulis tolia da formula 7.32 formulamde daiyvaneba. muSaoba-kinetikuri energiis Teorema cvladi Zalis SemTxvevaSi cvladi Zalis mier nivTier wertilze Sesrulebuli muSaoba erTganzomilebian SemTxvevaSi 7.32 formuliT gamoiTvleba. modiT davrwmundeT, rom gamoTvlili muSaoba marTlac nivTieri wertilis kinetikuri energiis cvlilebis tolia, rogorc amas muSaoba-kinetikuri energiis Teorema amtkicebs. ganvixiloT m masis nivTieri wertili, romelic x RerZis gaswvriv moZraobs da masze RerZis gaswvriv mimarTuli jamuri F(x) Zala moqmedebs. am Zalis mier nivTier wertilze Sesrulebuli muSaoba xi sawyisi mdebareobidan xf saboloo mdebareobamde gadaadgilebis ganmavlobaSi 7.32 formuliT gamoiTvleba: , (7.37) sadac niutonis meore kanonis gamoyenebiT F(x) Zala ma-iT SevcvaleT. 7.37 formulaSi ma dx sidide ase SeiZleba Caiweros: dv dx dt (7.38) dv dv dx dv = = v , dt dx dt dx (7.39) dv v dx = mv dv, dx (7.40) ma dx = m gamoTvlis jaWvuri wesidan gamomdinare: 7.38 formula aseT saxes iRebs: ma dx = m 7.40 formula 7.37 formulaSi CavsvaT da miviRebT: = 1 mvf2 – 1 mvi2 2 2 (7.41) yuradReba miaqcieT, rom rodesac cvladi x-dan ν-ze SevcvaleT, aucilebeli gaxda integralis zRvrebis axali cvladiT gamosaxva. aseve gaiTvaliswineT, rom m mudmivia da misi integrals gareT gatana SesaZlebelia. 7.41 formulis marjvena mxares gamosaxuli wevrebi kinetikuri energiebia da viRebT: W = Kf – Ki = ΔK , es formula muSaoba-kinetikuri energiis Teoremas asaxavs. 215 7.8. cvladi Zalis mier Sesrulebuli muSaoba amocanis amoxsnis nimuSi epiduraluri proceduris dros, 7.9 eqimis mxridan nemsis SeRwevis SegrZneba cvladi xi = 0-dan xf = 0,3 m-mde gadaadgilebis Zalaa, romelmac nemsze unda imoqmedos, Cven raTa man qsovilebSi SeaRwios. 7.14a suraTze ganmavlobaSi Sesrulebuli muSaoba gvWir gamosaxulia Zalis F sididis nemsis wveris x deba. gadaadgilebaze damokidebulebis grafiki. integralis gamoTvla SesaZlebelia, Tu 7.14a rodesac x 0-dan zrdas iwyebs, kani nemss suraTze gamosaxuli grafikis qveda arealis ewinaaRmdegeba, magram x = 8 mm mdebareobaze farTobs gamovTvliT: Zala sakmarisad Zlieria da nemsi kanSi aRwevs. W= amis Semdeg saWiro Zala Semcirebas iwyebs. nemsi zurgis tvinis iogs gakveTs, rodesac x = 18 mm, xolo rodesac x = 30 mm, ( farTobi Zalis grafiksa da x RerZis Soris, xi-dan xf-mde ) Cveni grafiki wrfivi segmentebisgan Sedgeba, nemsi amitom SesaZlebelia, rom grafikis qveda SedarebiT myar myesSi SeaRwevs da Semdeg areali marTkuTxa da samkuTxa regionebad nemsis wveri epiduralur sivrceSi xvdeba. davyoT (sur. 7.14b). magaliTad, samkuTxa A swored am momentSi Zalis sidide mkveTrad regionis farTobia ecema. damwyebma eqima Zalis gadaadgilebaze farTobiA=1 (0,008 m)(12 n) = 0,048 n·m = 0,048 j damokidebulebis Tviseba kargad unda 2 Seiswavlos, raTa mixvdes rodis Sewyvitos mas Semdeg rac yvela dasaTaurebuli regionis nemsis wveris Seyvana. ra W muSaoba unda farTobs gavigebT, jamuri muSaobis gamoTvla Seasrulos nemsze modebulma Zalam, raTa ar gagviWirdeba: nemsis wveri x = 30 mm mdebareobaze anu epiduralur sivrceSi moxvdes? amoxsna: cvladi F(x) Zalis mier Sesrulebul muSaobas gamovTvliT, Tu gamoviyenebT 7.32 formulas: W = (A-K regionebis farTobebis jami) = 0,048 + 0,024 + 0,012 + 0,036 + 0,009 + 0,001 + 0,016 + 0,048 + 0,016 + 0,004 + 0,024 = 0,238 j (pasuxi) E (a) (b) sur. 7.14. a) epiduraluri proceduris dros nemsis x gadaadgilebaze Zalis F sididis damokidebulebis grafiki. b) grafiksa da gadaadgilebis RerZs Soris teritoria regionebad aris dayofili, raTa farTobis gamoTvla SesaZlebeli gaxdes. amocanis amoxsnis nimuSi  F = (3x2 n) + (4 n) Zala, sadac x metrebSia, 7.10 amoxsna: amocanaSi moqmedi Zala cvladia, nivTier wertilze moqmedebs da mis kine vinaidan misi Fx komponenti x-is mniSvnelo tikur energias cvlis. ra muSaoba srul bazea deba wertilze, rodesac is (2 m, 3 m) koor Sesrulebuli muSaobis gamoTvlisTvis 7.7 an dinatebidan (3 m, 0 m) koordinatebze gada 7.8 formulebs ver gamoviyenebT. samagierod adgildeba? wertilis siCqare izrdeba, mcir SegviZlia 7.36 formula gamoviyenoT: deba Tu ar icvleba? damokidebuli. aqedan gamomdinare, 216 Tavi 7 kinetikuri energia da muSaoba dadebiTi Sedegi niSnavs, rom = 3[ 1 x3]23 + 4[y]30 = [33 – 23] + 4[0 – 3] energias gadascems. aqedan gamomdinare, wer 3 =7j  F Zala wertils (pasuxi) tilis kinetikuri energia da siCqare izrdeba. 7.9. simZlavre muSam jalambaris saSualebiT agurebi Senobis saxuravze unda aitanos. Cven SegviZlia jalambaridan tvirTze moqmedi Zalis mier Sesrulebuli muSaoba gamovTvaloT, rac tvirTis awevisTvis aris saWiro. miuxedavad amisa, muSisTvis muSaobis Sesrulebis temps ufro aqvs mniSvneloba, vidre mis sidides. Cndeba SekiTxva: 5 wuTs waiRebs samuSao Tu mTel kviras? Zalis mier muSaobis Sesrulebis temps simZlavre ewodeba. Tu Zala Δt droSi W muSaobas asrulebs, saSualo simZlavrea: PsaS = W Dt (saSualo simZlavre) (7.42) P myisi simZlavre muSaobis Sesrulebis myisi tempia: W P= (myisi simZlavre) (7.43) Dt davuSvaT Zalis mier Sesrulebuli W(t) muSaobis droze damokidebulebis funqcia viciT. magaliTad, t = 3 wm droSi P myisi simZlavris gamoTvlisTvis W(t)-s pirveli rigis warmoebuli unda gavigoT da Semdeg t = 3 wm drosTvis gamovTvaloT. SI sistemaSi simZlavris erTeulia j/wm. es erTeuli Zalian xSirad gamoiyeneba da amitom mas vati (vt) ewodeba. jeims uati cnobili mecnieri iyo, romelmac orTqlis Zravebis muSaobis tempi Zalian gazarda da amiT gaiTqva saxeli. britanul sistemaSi simZlavris erTeulia futi•funti/wm. aseve xSirad gamoiyeneba cxenis Zalac. erTeulebs Soris Semdegi urTierTkavSiria: 1 vati = 1vt = 1 j/wm = 0,738 futi · funti/wm (7.44) sur. 7.15. satvirTo avtomobilis Zalis Sedegad miRebuli simZlavre tvirTze Zalis mier muSaobis Sesrulebis tempia da 1 cxenis Zala = 1cxZ = 550futi · funti/wm = 746vt (7.45) 7.42 formulaze dakvirveba cxadyofs, rom muSaoba droisa da simZlavris namravlia. aqedan gamomdinare: 1 kilovati · sT = 1kvt · sT = (103 vt)(3600 wm) = 3,6 × 106j = 3,6 mgj (7.46) vati da kvt · sT ZiriTadad eleqtroxelsawyoebTan asocirdeba, Tumca maTi gamoyeneba sxva xelsawyoebisTvisac SeiZleba. magaliTad, iatakidan wignis aRebisa da magidaze dadebis Semdeg SegiZliaT TqvaT, rom Tqven 4 × 10-6 kvt•sT muSaoba SeasruleT. Zalis mier nivTier wertilze muSaobis Sesrulebis tempi am Zalisa da nivTieri wertilis siCqaris saSualebiTac SeiZleba gamoisaxos. wertilisTvis, romelic wrfivad moZraobs da  masze am wrfisadmi raime φ kuTxiT mimarTuli mudmivi saxes iRebs: P= anu F Zala moqmedebs, 7.43 formula Semdeg ( ) dW F cos φ dx dx = = F cos φ dt dt dt P = Fν cos φ (7.47) 217 7.9. simZlavre 7.47 formulis marjvena mxare  F· skalaruli namravlis saxiTac SegviZlia warmovadginoT:  P= F · (7.48) (myisi simZlavre) 7.15 suraTze gamosaxuli satvirTo avtomobili drois romeliRac momentSi siCqariT moZrav    tvirTze F ZaliT moqmedebs. F -iT gamowveuli myisi simZlavre drois momentSi F -is mier Sesrulebuli muSaobis tempia da 7.47-7.48 formulebiT gamoiTvleba. xSirad misaRebia gamoTqma: `satvirTo avtomobilis simZlavre~, magram unda gvaxsovdes, rom simZlavre Zalis mier muSaobis Sesrulebis tempia. sakontrolo wertili 5. wris centrze damagrebuli Toki blokzea mibmuli da bloki arwriulad moZraobs. Tokidan blokze moqmedi Zalis Sedegad miRebuli simZlavre dadebiTia, uaryofiTia Tu nulis toli? amocanis amoxsnis nimuSi 7.16 suraTze naCvenebia yuTze moqmedi  F1 da 7.11  F2 Zala, rodesac yuTi xaxunisgan Tavisufal zedapirze Tqvengan marjvniv misrialebs. 2 n F1 Zala horizontaluria, xolo 4 n  sididis F2 Zala zedapiridan maRla 60°-iT sididis aris gadaxrili. garkveul momentSi yuTis ν siCqarea 3 m/wm. xaxunisgan Tavisufali zedapiri   sur. 7.16. yuTze moqmedebs ori F1 da F2 Zala, rodesac yuTi xaxunisgan Tavisufal zedapirze Tqvengan marjvniv misrialebs. yuTis siCqarea . a) risi tolia am momentSi yuTze moqmedi Zalebis Sedegad miRebuli simZlavreebi da jamuri simZlavre? icvleba Tu ara jamuri simZlavre am momentSi? nulis tolia. yuTis kinetikuri energia (K = 1 mν2) ar icvleba da misi siCqarec mu  2 F F dmivia 3 m/wm-ia. vinaidan 1 da 2 Zalebi simZlavre siCqaris msgavsad ar icvleba, 7.48 formulis da ara saSualo simZlavre drois raime Tanaxmad P1 da P2 simZlavreebic (maSasadame amoxsna: Cven gvWirdeba myisi SualedSi. garda amisa, viciT yuTis siCqare. Pj-ic) ucvlelia. modiT TiToeuli ZalisTvis 7.47 formula gamoviyenoT.  F1 Zala siCqarisadmi φ1=180° b) Tu  F2 Zalis sididea 6 n, risi tolia jamuri simZlavre da icvleba Tu ara is? kuTxes adgens da gveqneba: P1 = F1ν cos φ1 = (2 n)(3 m/wm)cos 180° = – 6 vt (pasuxi) miRebuli uaryofiTi Sedegi gvamcnobs, rom  F1 Zala yuTs energias 6 j/wm tempiT arTmevs. F2 Zala siCqarisadmi φ2 = 60° kuTxes adgens da gveqneba: P2 = F2ν cos φ2 = (4 n)(3 m/wm) cos 60° = 6 vt (pasuxi) amoxsna:  F2 Zalis Sedegad miRebuli simZlav rea: P2 = F2ν cos φ2 = (6 n)(3 m/wm) cos 60° = 9 vt  F1 ZaliT gamowveuli simZlavrea P1 = – 6 vt da jamuri simZlavrea: Pj = P1 + P2 = –6 vt + 9 vt = 3 vt (pasuxi)  dadebiTi Sedegi niSnavs, rom F2 Zala yuTs maSasadame, yuTisTvis energiis gadacemis energias 6 j/wm tempiT gadascems. kinetikuri energia da siCqare izrdeba. 7.48 jamuri simZlavre calkeuli simZlavreebis formulis Tanaxmad, siCqaris zrdasTan erTad P1 da P2 sidideebic icvleba. aqedan jamis tolia: Pt = P1 + P2 = –6 vt + 6 vt = 0 jamuri tempis mniSvneloba dadebiTia. yuTis (pasuxi) maSasadame, energiis gadacemis jamuri tempi gamomdinare, Pj = 3 vt Sedegi mxolod 3 m/wm siCqaris ganviTarebis momentSi fiqsirdeba. 218 kinetikuri energia da muSaoba Tavi 7 mimoxilva da Sejameba kinetikuri energia. m masis da ν siCqariT moZravi wertilis K kinetikuri energiaa: K=1 mν2 (kinetikuri energia) cvlilebas Semdegi formuliT ukavSirdeba: ΔK = Kf – Ki = Wa + Wg (7.15) (7.1) Tu moZraobis dawyebisa da dasrulebis Semdeg muSaoba. W muSaoba aris saganze Zalis kinetikuri energiebi Tanabaria, 7.15 formula 2 moqmedebis Sedegad sagnidan miRebuli an aseT saxemde daiyvaneba: sagnisTvis gadacemuli energia. sagnisTvis gadacemuli energia dadebiTi Wa = –Wg (7.16) muSaobaa, am formulis mixedviT, modebuli Zala igive xolo sagnidan miRebuli energia - uaryofiTi raodenobis energias gadascems sagans, rasac muSaoba. mizidulobis Zala arTmevs. mudmivi mu   Zalis mier Sesrulebuli Saoba. F mudmivi Zalis mier d gadaadgilebis drekadobis Zala. zambaridan moqmedi Fs dros wertilze Sesrulebuli muSaobaa:   W = Fd cos φ= F ⋅ d (muSaoba, mudmivi Zala) (7.7, 7.8)  φ aris mudmivi kuTxe F-sa da   d -s Soris. saganze muSaobas F Zalis d gadaa sadac dgilebisadmi gaswvrivi komponenti asru  drekadobis Zalaa:   Fs = – k d (7.20) (hukis kanoni) ,  sadac d aris zambaris Tavisufali bolos gadaadgileba moSvebuli mdgomareobidan (rodesac zambara SekumSuli an daWimuli ar aris), xolo k sixistis mudmivaa (zambaris lebs. rodesac saganze ori an meti Zala elastiurobis sazomi). Tu x RerZi zambaris moqmedebs, jamuri muSaoba Zalebis mier cal gaswvriv aris gavlebuli da misi aTvlis saTave calke Sesrulebuli muSaobebis jamia da Ft zambaris Tavisufali bolos mdebareobas tolqmedi Zalis mier saganze Sesrulebuli emTxveva (moSvebul mdgomareobaSi), maSin 7.20 muSaobis tolia. formula asec SeiZleba Caiweros: Fx = – kx (hukis kanoni) (7.21) muSaoba da kinetikuri energia. wertilis kinetikuri energiis ΔK cvlilebasa da wer tilze Sesrulebul jamur W muSaobas Soris maSasadame, drekadobis Zala cvladia. is Semdegi kavSiria: lebasTan erTad icvleba. ΔK = Kf –Ki =W (muSaoba-kinetikuri energiis Teorema) (7.10) sadac Ki aris wertilis sawyisi kinetikuri energia, xolo Kf - kinetikuri energia muSaobis zambaris Tavisufali bolos gadaadgi drekadobis Zalis mier Sesrulebuli muSaoba. rodesac sagani zambaris Tavisufal bolozea damagrebuli, drekadobis xi gvaZlevs: mdebareobisken gadaadgilebis ganmavlobaSi Kf = Ki + W (7.11) mizidulobis Zalis mier Sesrulebuli  d saganze muSaoba. m masis wertilis magvar  gadaadgilebis ganmavlobaSi Fg mizidulobis Zalis mier Sesrulebuli Wg muSaobaa: Wf = mgd cos φ   sadac φ aris kuTxe Fg -sa da d -s Soris. sawyisi sagnis awevasa da dabla dawevaze Sesru lebuli muSaoba. modebuli Zalis mier sagnis awevis an dawevis dros Sesrulebuli Wa muSaoba mizidulobis Zalis mier Sesrulebul Wg muSaobas da sagnis kinetikuri energiis ΔK saboloo saganze Ws muSaobas asrulebs: Ws= 1 xi2 – 1 kxf2 2 2 (7.25) Tu xi=0 da xf=x, 7.25 formula aseT saxes iRebs: (7.12) mdebareobidan xf Zala Sesrulebis Semdeg. 7.10 formulis gardaqmna Ws = – 1 kx2 2 (7.26) cvladi Zalis mier Sesrulebuli  muSaoba. rodesac saganze moqmedi F Zala sagnis mdebareobazea damokidebuli, mis mier saganze Sesrulebuli muSaoba ri sawyisi mdebareobidan (xi , yi , zi ) rf saboloo mdebareobamde (xf , yf , zf ) Zalis integrebis saSualebiT gamoiTvleba. Tu davuSvebT, rom 219 7.9. simZlavre Fx komponenti mxolod x-zea damokidebuli da ara y-ze an z-ze, Fy komponenti mxolod yzea damokidebuli da ara x-ze an z-ze, xolo Fz komponenti mxolod z-zea damokidebuli da ara x-ze an y-ze, Sesrulebuli muSaoba iqneba: W= (7.36)  Tu F -s mxolod x komponenti aqvs, maSin 7.36 tempia. Tu Δt drois SualedSi Zala W muSaobas asrulebs, drois simZlavrea: PsaS = am SualedSi saSualo W Dt (7.42) myisi simZlavre muSaobis Sesrulebis myisi tempia:  F P= W Dt (7.43) formula martivdeba: Tu lebisadmi φ kuTxes qmnis, myisi simZlavrea: (7.32) sagnis moZraobis  P = Fν cos φ = F · simZlavre. Zalis Sedegad miRebuli simZlavre saganze Zalis mier muSaobis Sesrulebis Zala sadac , mimarTu (7.47, 7.48) aris sagnis myisi siCqare SekiTxvebi 1. qvemoT moyvanili siCqareebi kinetikuri energiis mixedviT daalageT, romelic =4 +3 , = 3 –4 , siCqaris dros eqneba wertils: a) b) e) am kidebulebis sami grafiki. romeli grafiki Seesabameba Semdeg mdgomareobebs: a) F1 = F2, b) F1 > F2, g) F1 < F2? = –4 +3 , g) = –3 +4 , d) = 5 da v) ν = 5 m/wm da horizontalisadmi 30° kuTxes adgens.   2. F mudmivi Zalis mier wrfivi d gadaad gilebis dros wertilze Sesrulebuli muSa oba dadebiTia Tu uaryofiTia, rodesac a)   F d -s Soris kuTxea 30°, b) igive kuTxea   100° da g) F = 2iˆ -– 3 ˆj da d = –44iˆ? sur. 7.18. SekiTxva 4 -sa da 3. xaxunisgan Tavisufal yinulze mosriale Saibaze horizontaluri Zala moqmedebs da mis siCqares cvlis. 7.17 suraTze gamosaxulia Saibis νi sawyisi siCqare, νf sabolo siCqare da siCqaris veqtorebis mimarTuleba sam mdgomareobaSi. daalageT mdgomareobebi 5. 7.19 suraTze gamosaxul goWs miwaze Camosrialebis variantSi sami goWi arCevani xaxunisgan aqvs. samive Tavisufal zedapirze unda Camosrialdes. daalageT es variantebi mizidulobis Zalis mier goWze Sesrulebuli muSaobis mixedviT dididan mcireze. Zalis mier Saibaze Sesrulebuli muSaobis mixedviT dididan mcireze. sur. 7.17. SekiTxva 3 4. 7.18a suraTze naCvenebia sur. 7.19. SekiTxva 5 xaxunisgan Tavisufal zedapirze marjvnisken mosriale bloki, romelzec ori horizontaluri Zala moqmedebs. 7.18b suraTze gamosaxulia blokis K kinetikuri energiis t droze damo 6. 7.20 suraTze naCvenebia cvladi Zalis Fx komponentis (x RerZis gaswvrivi komponentis) wertilis x mdebareobaze damokidebulebis oTxi grafiki. daalageT grafikebi x = 0 da 220 Tavi 7 kinetikuri energia da muSaoba x = x1 mdebareobebs Soris wertilis gada fiki (K kinetikuri energiis t droze da adgilebis dros Zalis mier Sesrulebuli mu mokidebuleba) Saobis mixedviT dididan mcireze. romel grafiks Seesabameba? 7.21b suraTze gamosaxul 8. A zambara B zambaraze naklebad elastiuria (kA>kB). romeli zambaris drekadobis Zala Seasrulebs ufro met muSaobas Tu a) zambarebi erTi da igive manZiliT SeikumSa da b) zamba rebi erTi da igive Zalam SekumSa? 9. talaxis wveTi kldidan Camovarda an gadmoagdes. 7.22 suraTze gamosaxuli romeli grafiki (an grafikebi) SeiZleba asaxavdes talaxis kinetikuri energiis cvlilebas vardnis ganmavlobaSi? sur. 7.20. SekiTxva 6 7. 7.21a suraTze gamosaxulia oTxi mdgo mareoba, rodesac uZrav blokze horizon taluri Zala moqmedebs. ZalTa sidideebia F2 = F4 = 2F1 = 2F3. 7.21b suraTze naCvenebia igive oTxi mdgomareobisTvis blokis siCqa ris horizontaluri νx komponenti. a) 7.21b sur. 7.22. SekiTxva 9. suraTze gamosaxuli romeli grafiki Seesa 10. 7.23 suraTze gamosaxulia Zalis Fx bameba 7.21a suraTze gamosaxul Zalebs? b) 7.21g suraTze gamosaxuli romeli gra komponenti, romelic wertilze moqmedebs. Tu wertili moZraobas iwyebs x = 0 bareobidan, rodesac ra mas iqneba misi mde koordinati, a) udidesi kinetikuri (a) energia, b) udidesi siCqare da g) nu lovani eqneba? siCqare d) ra m i m a r T u l e b i T imoZravebs wertili (b) (g) sur. 7.23. SekiTxva 10. x = 6 m koordinatis gavlis Semdeg? sur. 7.21. SekiTxva 7. amocanebi SSM amoxsnebi mocemulia Student Solutions Manual-Si. www amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday ILW Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley. com/college/halliday. • – ••• wertilebis raodenoba amocanis sirTules gviCvenebs paragrafi 7.3. kinetikuri energia •1. Tu apolos kosmosur xomaldsa da masze damagrebul Saturn V (saturni V) raketas erToblivi masaa 2,9×105 kg da xomaldis siCqarea 11,2 km/wm, ra kinetikuri energia eqneba mas? 221 •2. 1972 wlis 10 agvistos, aSS-sa da kanadis CrdiloeTiT uzarmazarma meteorma gaifrina. cecxlovani burTi imdenad naTeli iyo, rom dRis sinaTlezec ki Canda da misi simkveTre Cveulebrivi meteoris mier datovebul kvals bevrad aRemateboda (sur. 7.24). meteoris masa daaxloebiT 4 × 106kg iyo da 15km/wm siCqariT moZraobda. atmosferoSi vertikalurad Se svlis SemTxvevaSi meteori dedamiwis ze dapirs igive siCqariT Seejaxeboda. a) gamo TvaleT SesaZlo vertikaluri sur. 7.24. amocana 2. meteoris kvali dRisiT Sejaxebis SemTxvevaSi meteoris kinetikuri energiis danakargi (joulebSi). b) miRebuli Sedegi 1 megatona trotilis afeTqebis energiis (4,2 × 1015 j) jeradis saxiT CawereT. g) xiro simaSi atomuri bombis afeTqebis energia daaxloebiT 13 kilotona trotilis afeTqebas udrida. ramdeni atomuri bombis afeTqebis eqvivalenturi iqneboda meteoris Sejaxeba? •3. protoni (m = 1,67×10–27kg) 3,6×1015m/wm2-iT wrfivad Cqardeba. Tu protonis sawyisi si Cqarea 2,4×107 m/wm da 3,5 sm manZils gadis, risi tolia misi a) saboloo siCqare da b) ki netikuri energiis nazrdi? SSM ••4. 1,8×10–2kg masis naxvretiani burTula x RerZis dadebiTi mimarTulebiT mavTulze moZraobs. rodesac t = 0, burTula x=0 mdebareobas 12 m/wm siCqariT gaivlis da masze mudmivi Zala moqmedebs. 7.25 suraTze naCvenebia burTulas mdebareoba, rodesac t = 0, 1, 2 da 3 wm. rodesac t = 3 wm, burTula wamierad Cerdeba. ra kinetikuri energia eqneba burTulas t = 10 wm dros? paragrafi 7.5. muSaoba da kinetikuri energia •6. 2 kg masis bidoni xy sibrtyeSi moZraobs da masze 5 n sididis erTaderTi Zala moqmedebs. bidonis sawyisi siCqarea 4 m/wm x RerZis dadebiTi mimarTulebiT, xolo garkveuli drois Semdeg - 6 m/wm y RerZis dadebiTi mimarTulebiT. ra muSaobas asrulebs bidonze 5 n Zala am drois ganmavlobaSi? •7. moneta xy koordinatTa sistemis aTvlis saTavidan (3 m, 4 m) koordinatebisken xaxu nisgan Tavisufal zedapirze misrialebs da masze mudmivi Zala moqmedebs. Zalis sididea 2 n da x RerZis dadebiTi mimarTulebisadmi 100° kuTxes adgens saaTis isris moZraobis sawinaaRmdego mimarTulebiT. ra muSaobas asrulebs Zala monetaze gadaadgilebis dros? •8. motivtive yinuls wylis dineba napiris  gaswvriv d = (15 m) – (12 m) manZilze gadaad gilebs. wyali yinulze F = (210 n) – (150 n) ZaliT moqmedebs. ra muSaobas asrulebs Zala yinulze gadaadgilebis dros? ••9. 3 kg masis sxeuli xaxunisgan Tavisufal, horizontalur sur. 7.25. amocana 4. ••5. mama da Svili erTmaneTs sirbilSi ejibrebian. mamis kinetikuri energia Svilis kinetikuri energiis naxevaria, xolo Svilis zedapirze uZravad da masze x RerZis dadebiTi mimarTulebis, mudmivi F horizontaluri Zala moqmedebs. 7.26 suraTze naCvenebia sxeulis mdebareobis stroboskopuli grafiki.  F Zala moqmedebas masa mamis masis naxevaria. rodesac mama 1m/wm siCqares ganaviTarebs, misi kinetikuri energia Svilisas utoldeba. ra sawyisi siCqareebi aqvT a) mamas da b) Svils? Zevs sur. 7.26. amocana 9. 222 kinetikuri energia da muSaoba Tavi 7 iwyebs, rodesac t = 0 da sxeulis mdebareoba a) ra jamur mu 0,5 wm-ian SualedebSi fiqsirdeba. ra muSaobas  F Zala sxeulze 0-dan 2wm-mde Saobas asrulebs sami Zala yuTze ga SualedSi? daadgilebis ••10. 8kg masis sagani x RerZis dadebiTi mavlobaSi? b) yuTis mimarTulebiT Zraobs. masze deba? SSM sur. 7.27. amocana 10 Cauvlis, moqmedebas. 7.27 kinetikuri mudmivi Zala iwyebs suraTze gamosaxulia x mdebareobaze energiis damokidebulebis grafiki x = 0 mdebareobebs gamosaxul ze ga mimarTuli sur. 7.29. amocana 13. ••14. 7.30 suraTze mde RerZis swvriv K izrdeba Tu mcir rodesac bareobas gan kinetikuri energia mo sxeuli x = 0 asrulebs Soris. amis da x = 5 m Semdeg Zala moqmedebas ganagrZobs. ra ν siCqare eqneba sagans, rodesac is x = –3 m mdebareobas ukan ubrundeba? ••11. 85 kg masis moTxilamure daRmarTidan swor trasaze 37 m/wm sawyisi siCqariT gadis. Tu Zala mas 2 m/wm2 aCqarebiT anelebs da Semdeg aCerebs, a) ra F sidide aqvs Zalas, b) ra d manZils gaivlis moTxilamure Senelebis ganmavlobaSi da g) ra W muSaobas asrulebs moTxilamureze Zala? risi tolia d) F, e) d da v) W, Tu Senelebisas aCqareba 4 m/wm2-ia? ILW ••12. WanWikebiT da qanCebiT savse yuTi sami yuT horizon taluri (F1, F2 da F3) Zala debs. yuTi moqme Tav dapirvelad uZravia sur. 7.30. amocana 14. da Semdeg xaxunisgan Tavisufal zedapirze gadaadgildeba. F1 = 3 n, F2 = 4 n da F3 = 10 n, xolo naCvenebi kuTxeebia θ2 = 50° da θ3 = 35°. ra jamur muSaobas asrulebs sami Zala yuTze pirveli 4 m-is gavlis dros? ••15. ucvleli mimarTulebis da 12 n sididis  ˆ (2i –- 44 ˆj++33kˆ) m gadaadgilebis dros Zala d = (2 wertilze muSaobas asrulebs. ra kuTxea Za lasa da gadaadgilebas Soris, Tu wertilis kinetikuri energiis cvlilebaa a) +30 j da b) –30 j? paragrafi 7.6. mizidulobis Zalis mier Sesrulebuli muSaoba manqanebis saxelos- •16. a) 1975 wels, monrealis velodromis 360kn wonis saxuravi 10 sm-iT maRla aswies da noSi, xaxunisgan Ta gaaswores. ra muSaoba Seasrules saxuravze visufal zedapirze, masze moqmedma Zalebma? b) 1960 wels, flo x RerZis gaswvriv 2 m manZilze gadaad gildeba. 7.28 ridis Statis qalaq tampaSi, dedam Svilze sur. 7.28. amocana 12. su Tu qalma 4000 n-is (manqanis mTliani wonis raTze naCvenebia mudmivi horizontaluri Zalis mier yuTze Sesrulebuli W muSaobis x mdebareobaze damokidebulebis grafiki. a) ra sididisaa es Zala? b) Tu yuTis sawyisi kinetikuri energiaa 3 j da x RerZis dadebiTi mimarTulebiT moZraobs, dacemuli manqanis erTi mxare maRla aswia. ra iqneba misi kinetikuri energia 2 m-is gavlis Semdeg? ••13. 7.29 suraTze gamosaxul yuTze sami Zala (F1, F2 da F3) moqmedebs da is xaxunisgan Tavisufal zedapirze marcxnisken 3 m manZils gadis. F1 = 5 n, F2 =9 n da F3 = 3 n, kuTxea θ = 60°. daaxloebiT 1/4-is) 5 sm simaRleze aweva SeZlo, ra muSaoba Seasrula misma Zalam manqanaze? •17. vertmfrens 72 kg masis adamiani okenidan saSualebiT 15 m simaRleze ayavs. adamianis aCqarebaa g/10. ra muSaobas asrulebs Tokis adamianze a) vertmfrenidan moqmedi Zala da b) mizidulobis Zala? vertmfrenSi Cajdomamde risi tolia adaminis g) kinetikuri energia da d) siCqare? SSM WWW  ••18. 7.31 suraTze, 20 n sididis Fa hori 223 zontaluri 3 kg Zala mi Tokis saSualebiT maRla swevs. samaSvelo wignze operacia sami stadiisgan Sedgeba da TiToeul moqmedebs da wigni stadiaze speleologi 10 m vertikalur man- xaxunisgan Zils gadis. samaSvelo stadiebi Semdegia: masis Tavi sufal a) daxril (θ = sibrtyeze uZravi speleologi 5 m/wm siCqares iZens; b) 5 m/wm mudmivi siCqariT d = 0,5 m man 30°) Tavdapirvelad sur. 7.31. amocana 18. speleogols maRla sweven; g) speleologis  jamur muSaobas asrulebs wignze Fa Zala, siCqare nuli xdeba. ra muSaoba sruldeba mizidulobis ganmavlobaSi? Zilze adis. a) ra Zala da drekadobis Zala gadaadgilebis dros? b) Tu gadaadgilebis dasawyisSi wignis kinetikuri energia nulis tolia, ra iqneba misi siCqare gadaadgilebis dasasruls? 80 kg masis speleologze TiToeuli stadiis •••23. 7.34 suraTze 0,25 kg masis bloki 900 kg- gamosaxuli iani liftis kabi ••19. M masis bloks Tokis saSualebiT g/4 aCqarebiT vertikalurad dabla weven. naSi Zevs. kabina sur. 7.34. amocana 23. Sesrulebuli muSaoba, b) mizidulobis Zalis d1 = 2,4 m da Semdeg d2 = 10,5 m manZilze gvarlis saSulebiT vertikalurad maRla adis. a) d1 gadaadgilebis mier blokze Sesrulebuli muSaoba, g) blokis dros iatakidan blokze moqmedi drekadobis kinetikuri energia da d) blokis siCqare, Zalaa FN = 3 n. ra muSaobas asrulebs kabinaze gamoTvaleT a) Tokis Zalis mier blokze rodesac bloki d manZils gaivlis. SSM  ••20. 7.32 suraTze, 82 n sididis Fa mudmivi Zala 3 kg masis yuTze φ = 53° kuTxiT moqmedebs da yuTi xaxunisgan Tavisufal daxril gvarlidan moqmedi Zala? b) d2 gadaadgilebis dros gvarlidan kabinaze moqmedi mudmivi Zala 92,61 kj muSaobas asrulebs. risi tolia FN ? paragrafi 7.7. drekadobis Zalis mier Sesrulebuli muSaoba sibrtyeze mudmivi siCqariT jer maRla adis. ra muSaobas •24. zafxulis semestris dros studenturi asrulebs saerTo sacxovreblis binadrebi erTmaneTs  Fa Zala yuTze, vid didi re yuTi h = 0,15 m vertikalur sur. 7.32. amocana 20 manZils gaivlis?  Fr ••21. 7.33 suraTze gamosaxuli bloki miliTaa Seqmnili. SeRebili wyliT savse buSti milTan dakavSirebul CanTaSia moTavsebuli, xolo mili oTaxis mTel sigrZezea gadaWimuli. davuSvaT mili hukis kanons emorCileba da moTavsebul buStze, vidre mili pirvandel ze maRla mimarTumoqmedebs. katapulta Seasrulebs milidan moqmedi Zala CanTaSi aldeba da muSa mas- ZaliT ebrZvian. damagrebuli Wimaven da Semdeg xels uSveben, ra muSaobas ril sibrtyeze sri  Fr CarCoze zambaris sixistea 100 n/m. Tu mils 5 m-iT θ=50° kuTxiT dax li 50 n sididis katapultebiT fanjris mdgomareobas daubrundeba? sur. 7.33. amocana 21. mdebareobaze uZravad gaCerebisTvis masze sibrtyeze d = 0,5 m manZilis gavlis Semdeg yuTis kinetikuri energia 80 j-iT izrdeba. ramdeniT ufro  meti kinetikuri energia eqneboda yuTs •25. 7.11 suraTze gamosaxuli blokis x = – 2 sm Fr Zalis ararsebobis pirobebSi? ••22. dazaralebul speleologs maSvelTa raz- 80n sididis ZaliT unda vimoqmedoT. Semdeg bloks nela gadavaadgilebT da zambarablokis sistema +4 j muSaobas asrulebs. amis Semdeg bloki kvlav uZravad Cerdeba. ra mdebareobazea bloki? (miniSneba: pasuxi oria). 224 kinetikuri energia da muSaoba Tavi 7 •26. zambara da bloki 7.11 suraTze gamo •••30. 7.11a suraTze gamosaxuli bloki xaxu saxulis msgavsad aris ganlagebuli. ro nisgan Tavisufal, horizontalur zedapirze x = +4 sm koordinatisken Zevs da zambaris sixistea 50 n/m. Tavdapirve- gamovwevT, mis gasaCereblad 360 n Zalaa lad zambara moSvebul mdgomareobaSia da desac bloks saWiro. Cven bloki x = 11 sm koordinatisken bloki x = 0 mdebareobaze imyofeba. amis Semdeg gamovwieT da xeli gavuSviT. ra muSaobas 3 n mudmivi sididis Zala bloks x RerZis dadebi asrulebs zambara blokze, rodesac bloki Ti mimarTulebiT ubiZgebs da zambara blokis xi gaCerebamde iWimeba. gaCerebis Semdeg a) ra = +5 sm koordinatidan b) x = –3 sm, x = –5 sm g) a) da d) x = +3 sm, x = –9 sm mdebareobazea bloki, b) ra muSaoba Seasrula koordinatisken gadaadgildeba? modebulma Zalam blokze da g) ra muSaoba ••27. 7.35 suraTze naCvenebia Fx dreka dobis Zalis x mde Seasrula drekadobis Zalam blokze? blokis bareobaze gadaadgilebis dros d) ra mdebareobazea blokis kinetikuri energia maqsimaluri da e) risi tolia kinetikuri energiis maqsimaluri damo kidebulebis mniSvneloba? gra fiki 7.11 suraTze gamosaxuli ganla sur. 7.35. amocana 27. gebisTvis. Cven bloki x = 12 sm koordinatisken paragrafi 7.8. cvladi Sesrulebuli muSaoba mier •31. gamovwieT da xeli gavuSviT. ra muSaobas 10 kg masis aguri x RerZis asrulebs zambara blokze, rodesac bloki gaswvriv moZraobs. xi = +8 sm sawyisi koordinatidan a) x = +5 sm, b) x = –5 sm, g) x = –8 sm da d) x = –10 sm Zalis 7.37 suraTze naCve nebia aguris aCqa koordinatisken gadaadgildeba? rebis aguris mde ••28. 7.11 suraTze bareobaze gamosaxuli debulebis grafiki. ra muSaobas asrulebs bloki sur. 7.37. amocana 31. damoki x = 0 koordinatidan x = +3 sm koordi koordinatidan natisken gadaadgilebis ganmavlobaSi? ILW agurze gadavaa dgileT da uZravad gavaCereT. 7.36 su raTze naCvenebia sur. 7.36. amocana 28. Saoba. Semdeg bloki x = +5 sm koordinatisken da xeli gamomwvevi x=8m Zala x=0 koordinatisken •32. 5 kg masis blo Cveni Zalis mier blokze Sesrulebuli mugavwieT aCqarebis gavuSviT. ra muSaobas Seasrulebs zambara blokze, rodesac bloki xi = +5 sm sawyisi koordinatidan a) x = +4 sm, b) x = –2 sm da g) x = –5 sm koordinatisken ki xaxunisgan Ta visufal, hori zontalur zeda pirze wrfivad moZraobs da masze Zala sur. 7.38. amocana 32. moqmedebs. 7.38 suraTze gamosaxulia am Zalis cvlileba blokis mdebareobis cvlilebasTan erTad. gadaadgildeba? ra muSaobas asrulebs Zala, rodesac bloki ••29. 2 kg masis sxeuli x RerZis dadebiTi aTvlis mimarTulebiT moZraobs da masze erTaderTi gadaadgildeba? Zala moqmedebs, romlis x komponentia Fx = –6x n, sadac x metrebSia mocemuli. x = 3m koordinatze sxeulis siCqarea 8 m/wm. a) ra iqneba sxeulis siCqare x = 4 m koordinatze? b) x-is ra dadebiTi mniSvnelobis dros iqneba sxeulis siCqare 5 m/wm? SSM ILW WWW saTavidan x = 8 m koordinatisken •33. wertilze moqmedi Zala x RerZis gaswvrivaa mimarTuli da F = F0(x/x0 – 1) formuliT aris mocemuli. gamoTvaleT Zalis mier wertilze Sesrulebuli muSaoba, rodesac wertili x = 0 koordinatidan x = 2x0 koordinatisken gadaadgildeba. amisTvis a) aageT F(x) grafiki 225 da muSaoba grafikis mixedviT gamoTvaleT da mibmuli Toki. yuTi b) moaxdineT F(x) integrireba. SSM WWW xaxunisgan ••34. sardinebiT savse bidoni x = 0,25 m koor dinatidan x = 1,25 m koordinatisken gadaad gildeba da masze F = exp(–4x2) Zala moqme debs, sadac x mocemulia metrebSi, xolo F – niutonebSi. ra muSaobas asrulebs bidonze sufal, horizon talur zedapirze Zala? ••35. 3 kg masis nivTieri wertilis magvar moZraobs, Tavi romlis gaswvriv x RerZia sur. 7.40. amocana 39. gavlebuli. Tokis marcxena bolo umniSvnelo masis da xaxunis mqone blokzea damagrebuli. bloki h = 1,2 m simaRlezea ganlagebuli da yuTi x1 = 3 m koordinatidan x2 = 1 m koor saganze erTaderTi Zala moqmedebs da sag dinatisken gadaadgildeba. moZraobis gan nis mavlobaSi Tokis daWimuloba ar icvleba mdebareobis droze damokidebulebis funqciaa x = 3t – 4t + t , sadac x mocemulia 2 3 metrebSi, xolo t – wamebSi. ra muSaobas Sea srulebs saganze Zala 0-dan 4 wm-mde drois SualedSi? ••36. 1,5 kg masis bloki xaxunisgan Tavisufal, horizontalur zedapirze uZravad aris ganlagebuli da masze x RerZis gaswvriv mimarTuli horizontaluri Zala iwyebs (x) = (2,5 – x2) n formuliT gamoisaxeba, sadac x mocemulia metrebSi, xolo blokis sawyisi mdebareobaa x = 0. a) ra kinetikuri energia eqneba bloks x = 2 m moqmedebas. Zala 25 n-s tolia. gamoTvaleT yuTis kinetikuri energiis cvlileba moZraobis ganmavlobaSi. paragrafi 7.9. simZlavre •40. liftis 3×103kg masis datvirTuli kabina 23 wamis ganmavlobaSi mudmivi siCqariT 210 miT maRla adis. ra saSualo tempiT asrulebs muSaobas gvarlidan moqmedi Zala kabinaze? •41. 100 kg masis bloki 5 m/wm mudmivi siCqariT horizontalur zedapirze gadaadgildeba da masze horizontisadmi 37° kuTxiT daxrili 122 n Zala moqmedebs. ra tempiT asrulebs koordinatis Cavlisas? b) ra maqsimaluri muSaobas Zala blokze? SSM ILW kinetikuri energia eqneba bloks x = 0 da x = 2 m •42. koordinatebs Soris? xaxunisgan Tavisufal aRmarTze adis. aRmarTi aRmarTis sac masze rode  Fa Za sur. 7.39. amocana 37. koordinatisken gadaaadgilebs. ra muSaoba Seasrula Zalam wertilze, rodesac wertili a) x = 4 m, b) x = 7 m da g) x = 9 m koordinats miaRwia? ra siCqare aqvs wertils da ra mimarTulebiT moZraobs, rodesac is d) x = 4 m, e) x = 7 m da v) x = 9 m koordinats miaRwevs?  ••38. F = (cx – 3x2) Zala wertilze moqmedebs da wertili x RerZis gaswvriv gadaadgildeba.  F Zala niutonebSia mocemuli, x metrebSia, xolo c mudmivaa. x = 0 koordinatze wertilis energiaa paralelurad aris gadaWimuli manZilis gavlis dros Tokidan moqmedi Zala la moqmedebs da x = 0 koordinatidan x = 9 m kinetikuri saSualebiT da 1 m/wm mudmivi siCqariT moZraobs. 8 m wertilis aCqareba, Tokis horizontisadmi 12º kuTxiTaa daxrili. Toki ••37. 7.39 suraTze naCvenebia 2 kg masis moTxilamure 20 j, xolo x=3m koordinatze _ 11 j. gamoTvaleT c. •••39. 7.40 suraTze naCvenebia yuTi da masze moTxilamureze 900 j muSaobas asrulebs. a) Tu Toki 2 m/wm mudmivi siCqariT imoZravebs, ra muSaobas Seasrulebs igive manZilis gavlis dros Tokidan moqmedi Zala moTxilamureze? ra tempiT asrulebs muSaobas Tokidan moqmedi Zala moTxilamureze, rodesac Toki b) 1 m/wm da g) 2 m/wm mudmivi siCqariT moZraobs? •43. 15 kg masis uZrav sxeulze 5 n Zala moq medebs. gamoTvaleT Zalis mier Sesrulebuli muSaoba a) pirvel wamSi, b) meore wamSi, g) mesame wamSi da gamoTvaleT d) mesame wamis dasasruls Zalis Sedegad miRebuli myisi simZlavre. SSM ••44. a) garkveul momentSi, nivTieri wertilis  magvar saganze F = (4 n) – (2 n) + (9 n) Za la moqmedebs da sagnis siCqarea = –(2 m/wm) + (4 m/wm) . ra myisi tempiT asrulebs muSaobas 226 Tavi 7 kinetikuri energia da muSaoba Zala saganze? b) raime sxva momentSi siCqares 7.41 suraTis mixe mxolod y komponenti aqvs. Tu Zala ar icvleba dviT da myisi simZlavrea –12 vt, ra siCqariT x = 0 koordinatze moZraobs am dros sagani? sxeulis ••45. satvirTo liftis nela moZravi kabinis masaa 1200 kg da 54 m manZils 3 wuTSi gadis, ris Semdegac Cerdeba. liftis sapirwonis masa mxolod 950 kg-ia, amitom lifts Zrava esaWiroeba. ra saSualo simZlavre unda uzrun velyos gvarlidan kabinaze moqmedma Zravis Zalam? SSM ••46. 0,3 kg masis sagani xaxunisgan Tavisufal, horizontalur erTi boloTi zedapirze srialebs horizontalur da zambarazea (k = 500n/m) mimagrebuli. zambaris meore bolo fiqsirebulia. (sadac wonasworul drekadobis Zala mdebareobaze nulis tolia) sagnis kinetikuri energiaa 10 j. a) ra tempiT asrulebs muSaobas zambara saganze, rodesac sagani wonasworul mdebareobas miaRwevs? b) ra tempiT asrulebs muSaobas zambara sagan ze, rodesac zambara 0,1 m-iT aris SekumSuli da sagani wonasworul mdebareobas Sordeba?  ••47. meqanizms tvirTi di = (0,5 m) + (0,75 m) + (0,2 m) sawyisi mdebareobidan (rodesac t = 0) d f = (7,5 m) + (12 m) + (7,2 m) saboloo mdebareobisken (rodesac t = 12 wm) gadaaqvs.  tvirTze modebuli Zalaa F = (2 n) + (4 n) + (6 n) . gamoTvaleT a) meqanizmis Zalis mier tvirTze Sesrulebuli muSaoba da b) meqanizmis Zalis saSualo simZlavre am gadaadgilebis dros.  •••48. 2kg masis moZrav saganze F = (3 n) + (7 n) + (7 n) Zala moqmedebs. sagani di = (3m) – (2 m) + (5 m) sawyisi mdebareobidan d f = –(5 m) + (4 m) + (7 m) saboloo mdebareo bisken 4 wamSi gadaadgildeba. gamoTva leT: a) Zalis mier saganze Sesrulebuli muSaoba Sedegad 4 wamis ganmavlobaSi, b) Zalis miRebuli saSualo simZlavre  drois igive SualedSi da g) kuTxe  d f -s Soris. icvleba. di -sa da damatebiTi amocanebi 49. 2 kg masis sxeuli x RerZis gaswvriv moZraobs da masze moqmedi erTaderTi Zala siCqarea 4 m/wm. a) ra kineti sur. 7.41. amocana 49. kuri energia aqvs sxeuls x = 3 m koordinatze? b) x-is ra dros eqneba sxeuls 8 j mniSvnelobis kinetikuri energia? g) ra maqsimaluri kinetikuri energia eqneba sxeuls x = 0 da x = 5 m koordinatebs Soris? 50. 250 g masis bloki k = 2,5 n/sm sixistis vertikalur zam baraze daeca (sur. 7.42). zambara 12 smiT SeikumSa da wa mierad gaCerda. zambaris SekumSvis sur. 7.42. amocana 50. ganmavlobaSi ra mu Saobas asrulebs blokze a) mizidulobis Zala da b) drekadobis Zala? g) risi tolia blokis siCqare zambaraze dacemis momentSi? (xaxuni ugulebelyaviT). d) Tu Sejaxebis siCqares gavaormagebT, ra maqsimalur manZilze SeikumSeba zambara? 51. 45 kg masis bloki xaxunisgan Tavisufal daxril sibrtyeze srialebs. sibrtyis sigrZea 1,5 m, xolo simaRle – 0,91 m. muSa bloks maRla awveba da bloki mudmivi siCqariT srialebs. a) gamoTvaleT muSis Zalis sidide. ra muSaobas asrulebs blokze b) muSa, g) mizidulobis Zala, d) zedapiridan blokze moqmedi drekadobis Zala da e) jamuri Zala? 52. 50 kg masis yuTis xaxunisgan Tavisufal, horizontalur zedapirze gadaadgilebisTvis muSa yuTze horizontisadmi 20° kuTxiT daxrili 210 n sididis ZaliT moqmedebs. 3 m-is gavlis ganmavlobaSi ra muSaobas asrulebs yuTze a) muSa, b) mizidulobis Zala da g) zedapiridan yuTze moqmedi drekadobis Zala? d) ra mTliani muSaoba sruldeba yuTze?  53. gaxvretil burTulaze Fa Zala moqmedebs da burTula swor mavTulze +5 sm manZils  gadis. Fa Zalas garkveuli fiqsirebuli 227  mniSvneloba xolo aqvs,  Fa -sa da burTulis 56. F = (4 n) + c Zala wertilze moqmedebs da wertili d = (3 m) – (2 m) gadaadgilebas gada asrulebs (wertilze  sxva Zalebic moqmedebs). adgilebas Soris φ risi tolia c, Tu kuTxis arCeva Sesa Zlebelia. 7.43 su raTze gamosaxulia F Zala wertilze a) 0, b) 17 j da g) –18 j muSaobas asrulebs? sur. 7.43. amocana 53. 57. 7.46 suraTze naCvenebia xaxunisgan  Fa Zalis mier burTulaze Sesrulebuli W Tavisufal zedapirze marjvnisken mosriale Fa , rodesac φ aris a) 64° Zala moqmedebs. ori Zala horizontaluria da muSaobis φ-ze damokidebulebis grafiki. ra muSaobas asrulebs da b) 147°? 54. 7.44a suraTze gamosaxuli 4 kg masis bloki, romelzec 2 n Zala moqmedebs. bloki xaxunisgan Tavisufal zedapirze marjvniv 1 m manZils gadis, xolo Zala horizontisadmi θ kuTxes adgens. daadgineT gadaadgilebis bolos blokis νf siCqaris gamosaxuleba, Tu blokis sawyisi siCqarea a) 0 da b) 1 m/wm. g) 7.44b yuTi. yuTi d = 20 sm manZils gadis da masze sami maTi sidideebia F1 = 5 n da F2 = 1 n. mesame Zala θ = 600 kuTxiT dablaa daxrili da misi sididea F3 = 4 n. a) ra jamur muSaobas asrulebs yuTze mizidulobis Zala, drekadobis Zala da samive modebuli Zala 20 sm gadaadgilebis ganmavlobaSi? b) Tu yuTis masaa 2 kg da sawyisi kinetikuri energiaa 0, ra iqneba misi siCqare gadaadgilebis dasasruls? suraTze naCvenebia analogiuri mdgomareoba, oRond amjerad Zala mimarTulia marcxnisken. daadgineT 1m-iani gadaadgilebis bolos blokis νf siCqaris gamosaxuleba. d) aageT νf-s grafikebi θ kuTxis 0°-90° arealisTvis. sur. 7.46. amocana 57.  58. x RerZis dadebiTi mimarTulebis F Zala RerZis gaswvriv moZrav saganze moqmedebs. Zalis sididea F = 10e –x/2 n, sadac x mocemulia sur. 7.44. amocana 54. metrebSi. gamoTvaleT  F Zalis mier 55. 2 kg masis yuTi xaxunisgan Tavisufal zedapirze x RerZis dadebiTi mimarTulebiT moZraobs. t = 0 droidan dawyebuli, qari x Sesrulebuli muSaoba x = 0 koordinatidan RerZis uaryofiTi mimarTulebiT uberavs. RerZebis SemosazRvruli arealis farTobi 7.45 suraTze naCvenebia yuTis x mdebareobis gamoTvaleT. t droze damokidebulebis grafiki. grafikze moaxdineT integrireba. dayrdnobiT gamoTvaleT yuTis kinetikuri 59. muSa 25kg masis yuTs xaxunisgan Tavisufal daxril zedapirze 209 n ZaliT ubiZgebs. daxris kuTxe 25°-ia. yuTi 1,5 m manZils gadis. energia, rodesac a) t = 1 wm da b) t = 5 wm. g) ra muSaobas asrulebs yuTze qaris Zala t = 1 wmdan da t = 5 wm-mde SualedSi? x=2m koordinatamde. a) amisTvis F(x) grafiki aageT da grafikiTa da sakoordinato b) muSaobis gamoTvlisTvis ra muSaobas asrulebs am dros yuTze a) muSis Zala, b) mizidulobis Zala da g) zedapiridan moqmedi drekadobis Zala? d) ra jamuri muSaoba sruldeba yuTze? 60. = (3 + 7 + 7 ) n Zala 2 kg masis saganze   (3 – 2 + 5 ) m sawyisi mdebareobidan d f = (-5iˆ - 4 ˆj + 7 kˆ) (–5 + 4 + 7 ) m saboloo mdebareobisken gada moqmedebs, romelic 4 wamis ganmavlobaSi d i = (3iˆ - 2 ˆj + 5kˆ) sur. 7.45. amocana 55. 228 kinetikuri energia da muSaoba Tavi 7 adgildeba. gamoTvaleT a) Zalis mier saganze Tokidan blikze Tokis daWimulobaze orjer Sesrulebuli muSaoba drois am SualedSi, b) meti jamuri Zala moqmedebs). Zalis Sedegad miRebuli saSualo simZlavre 64. 2 kg masis sagani Tavdapirvelad uZravia, Semdeg Tanabrad Cqardeba da 3 wamis gan mavlobaSi 10 m/wm siCqares anviTarebs. a) ra drois igive SualedSi da g) kuTxe   di da d f veqtorebs Soris. 61. 7.11a suraTze, m masis bloki xa muSaobas asrulebs saganze misi amaCqarebeli xunisgan Zala 3 wamis ganmavlobaSi? risi tolia am Tavisu Zalis Sedegad miRebuli myisi simZlavre fal, horizon b) Sualedis dasasruls da g) Sualedis pirveli talur zedapir sur. 7.47. amocana 61. ze Zevs da hori zontaluri zambaris (k sixistis) erT bo lozea damagrebuli. zambaris meore bolo fiqsirebulia. bloki Tavdapirvelad uZravia (x = 0), xolo Semdeg x RerZis dadebiTi  F naxevris dasasruls?  65. ra muSaobas asrulebs F = (2x n) + (3 n)  Zala (x metrebSia), romelic sagans ri =  (2m) +(3m) mdebareobidan r f = – (4 m) – (3m) mdebareobaze gadaadgilebs? moqmedebas ganicdis. 7.47 suraTze naCvenebia 66. lifts 660 n wonis 100 mgzavri 60 wamis ganmavlobaSi 150 m simaRleze mudmivi siCqa blokis kinetikuri energiis x mdebareobaze riT ahyavs. ra saSualo simZlavre unda uz mimarTulebis horizontaluri Zalis  damokidebulebis grafiki. a) ra sididisaa F ? b) risi tolia k? 62. 10 n Zala xy sibrtyeSi moZrav 2 kg masis saganze moqmedebs da x RerZis dadebiT runvelyos wevis Zalam? 67. yuTebi sawyobSi eskalatoris saSualebiT gadaadgildeba, romelic 0,5 m/wm mudmivi siCqariT moZraobs. erT-erT adgilze eska 150° kuTxes adgens. ra 2 m sigrZis da horizontisadmi 10°-iT daxril aRmarTze adis, Semdeg 2 m-s muSaobas asrulebs Zala saganze, rodesac horizontalurad moZraobs da sabolood mimarTulebasTan (saaTis isris moZraobis sawinaaRmdegod) latori sagani aTvlis saTavidan im wertilisken 2 m sigrZis da horizontisadmi aseve 10°- gadaadgildeba, romlis mdebareobis veqtoria iT (2 m) – (4 m) ? eskalatoriT 2 kg masis yuTi gadaadgildeba g) daRmarTze Cadis? 68. erT yinulmWreli blokze m = 20 kg ma gayinul sis sagania Camokide qari xaxunisgan uZravad qari dgas, Tavisufal rodesac aRmosavleTisken 200 n ZaliT moqmedebs. qari yinulmWrels 8 m manZilze Tavisufal boloze  F Zala moqmedebs.  iyos F Zala, raTa tbaze dauberavs. uberavs buli, xolo Tokis a) ra sididis unda davuSvaT marTze adis, b) horizontalurad moZraobs da masis armqone blokebzea damagrebuli. Cadis. eskalatoris Zala yuTze, rodesac yuTi a) aR xaxunisgan Tavisufal, daRmarTze da ar srialebs. ra tempiT asrulebs muSaobas 63. 7.48 suraTze, Toki daxril da yinulmWrelze wrfivad gadaadgilebs da es wrfe Crdilosur. 7.48. amocana 63. aRmosavleTiT 20° kuTxiTaa mimarTuli. ra kinetikuri energia eqneba yinulmWrels am sagani mudmivi siCqariT maRla aiwios? b) manZilis gavlis Semdeg? sagnis 2 sm-iT maRla asawevad ra manZilze unda 69. bavSvi sasrialoze erToba da, rom gamoiwios Tokis Tavisufali bolo? maRla ar SeSindes, deda iWers. dedidan Svilze awevis dros ra muSaobas asrulebs saganze 100 n Zala moqmedebs, romelic srialis g) Tqveni Zala da d) mizidulobis Zala? sawinaaRmdegod (miniSneba: rodesac Toki blokzea Semoxveuli, 1,8 m manZilze srialdeba da misi kinetikuri aris mimarTuli. bavSvi 229 energia 30 j-iT izrdeba. a) ra muSaobas asrulebs bavSvze mizidulobis Zala am drois ganmavlobaSi? b) Tu deda gaiweva, ramdeniT gaizrdeba bavSvis kinetikuri energia igive 72. wertili wrfivad  gadaadgilda  d = (8 m) + c da masze F = (2 n) – (4 n) Zalis moqmedebiT (wertilze sxva Zalebic  moqmedebs). risi tolia c, Tu wertilze F manZilze Camosrialebis dros? Zalis mier Sesrulebuli muSaoba a) nulia, 70. b) dadebiTia da g) uaryofiTia? grZis yuTs 73. liftis 4500 kg masis kabina maqsimum 1800 kg masis tvirTs uZlebs. Tu srulad datvirTuli kabina 3,8 m/wm siCqariT maRla ZaliT adis, ra simZlavre unda uzrunvelyos kabinis 230 kg masis yuTi L = 12 m si dia. Tokze Tqven cvladi  F ki ubiZgeT da d = 4 m mamoZravebelma manZiliT SenarCunebisTvis? gverdze gadaadgileT (sur. 7.49). a) ra sidide aqvs F Zalas, ro Zalam amgvari siCqaris 74. wertili x RerZis gaswvriv moZraobs da sur. 7.49. amocana 70. desac yuTi gadaad gilebis saboloo wertils miaRwevs? yuTis gadaadgilebis ganmavlobaSi b) ra jamuri muSaoba sruldeba masze, g) ra muSaobas asrulebs yuTze mizidulobis Zala da d) ra muSaobas asrulebs yuTze Tokidan moqmedi Zala? e) gadaadgilebamde da gadaadgilebis Semdeg yuTi uZravia. gamoiyeneT b), g) da  d) pasuxebi da Tqveni F Zalis mier yuTze masze RerZis dadebiTi mimarTulebis Zala moqmedebs. 7.51 suraTze naCvenebia F Zalis wertilis x mdebareobaze damokidebulebis grafiki. F=a/x2, sadac a = 9 n · m2. gamoTvaleT Zalis mier wertilze Sesrulebuli muSaoba, rodesac x=3m wertili x=1m koordinatamde koordinatidan gadaadgildeba. a) muSaoba grafikis saSualebiT gamoTvaleT da b) muSaoba Zalis funqciis integrebis saSualebiT gamoTvaleT. Sesrulebuli muSaoba gamoTvaleT. v) ratom ar aris Tqveni Zalis mier Sesrulebuli muSaoba horizontaluri gadaadgilebis da a)-s pasuxis namravlis toli? 71. maCvenebliani zambara milimetrebSi da graduirebuli skalis gverdiT aris Camo kidebuli. 7.50 suraTze naCvenebia zambaraze Camokidebuli sami sxvadasxva yuTi. a) skalis romel niSnuls miuTiTebs maCvenebeli, ro desac zambaraze yuTi ar kidia? b) risi tolia mesame yuTis W wona? sur. 7.51. amocana 74. 75. sagani x RerZis dadebiTi mimarTulebiT  misrialebs da masze Fa Zala moqmedebs. Zala x RerZis gaswvrivaa mimarTuli da misi x mm mm 0 mm 0 komponentia Fax = 9x – 3x2, sadac x mocemulia 0 metrebSi, xolo Fax – niutonebSi. sagani 30 garkveuli drois mere Cerdeba. a) aageT saganze moZraobas x = 0 koordinatidan iwyebs da 40 moqmedi 60 110 n W Fa Zalis x-ze damokidebulebis grafiki. b) romel koordinatzea Sesrulebuli muSaoba maqsimaluri da g) risi tolia muSaobis maqsimaluri mniSvneloba? d) romel 240 n sur. 7.50. amocana 71. koordinatze xdeba muSaoba nulis toli? e) romel koordinatze Cerdeba sagani? 230 Tavi 7 kinetikuri energia da muSaoba 76. ricxviTi integreba. yuTi x RerZis gaswvriv x = 0,15 m koordinatidan x = 1,2 m koordinatamde gadaadgildeba da masze qari mociguraves dasavleTisken ubiZgebs. 7.52 suraTze naCvenebia mociguravis x mdebareobis t droze damokidebulebis grafiki, sadac t = 0 qaris amovardnis droa, xolo x RerZi F = exp(–2x2) Zala moqmedebs, sadac x moce mulia metrebSi, xolo F – niutonebSi. ra aRmosavleTisken muSaobas asrulebs Zala yuTze? kinetikuri energia aqvs mociguraves, rodesac 77. cxeni urems 40 funti ZaliT eqaCeba. Zala horizontidan maRla, 30°-iT aris gadaxrili mociguraveze 0-dan 3wmde drois SualedSi? da uremi 6mili/sT siCqariT moZraobs. a) ra 79. dedamiwis doneze momxdarma afeTqebam muSobas asrulebs Zala 10 wuTis ganmavlobaSi? Rrmuli datova, romlis diametri afeTqebis b) risi tolia Zalis saSualo simZlavre (cxenis energiis ZalebSi)? megatona trotilis afeTqeba 1 km diametris 78. 50 kg masis mocigurave xaxunisgan Tavi Rrmuls tovebs. miCiganSi, huronis tbis sufal gayinul tbaze mudmivi siCqariT aRmosavleTisken srialebs. uecrad monaberi aris mimarTuli. a) ra t = 0? b) ra muSaobas asrulebs qaris Zala 1 3 xarisxis proporciulia. 1 fskerze 50km diametris Rrmulia, romelic meteoris Sejaxebis Sedegad aris darCenili. risi tolia amgvari Sejaxebis kinetikuri energia? pasuxi a) megatona trotilis (1 megatona 4,2 × 1015 joulia) da b) xirosimaSi afeTqebuli bombis (daaxloebiT 13 kilotona trotilis) energiis saSualebiT gamosaxeT? (meteoris an kometis Sejaxebam SesaZloa mniSvnelovnad Secvala dedamiwis klimaturi pirobebi da, arc is aris gamoricxuli, rom dinozavrebis da sicocxlis sxva formebis gaqroba gamoiwvia). 80. 1200 kg masis manqana trasaze 120 km/sT siCqariT sur. 7.52. amocana 78. moZraobs. risi tolia manqanis kinetikuri energia trasis ganapira mdgomi adamianis TvalTaxedviT? 8 potenciuuri energia da energiis Senaxva kldeze cocvis dros lideri (win mimavali mTamsvleli) meore mTamsvlelze damagrebuli TokiTaa dazRveuli. Toki erT an met gorgolaWSi (liTonis rgolSi) gadis, romelsac lideri kedelze amagrebs. gorgolaWebis upiratesoba is aris, rom Secdomis SemTxvevaSi mTamsvleli yvelaze maRla damagrebuli gorgolaWidan mxolod garkveul manZilze Gamovardeba. Tavis mxriv Tokic safrTxes qmnis, vinaidan vardnis dros misi gawyveta sulac ar aris gamoricxuli. bevri damwyebi mTamsvleli Tvlis, rom riskis xarisxi qveda gorgolaWidan lideris asvlis simaRlezea damokidebuli. rac ufro metia es simaRle, miT ufro Zlierad iWimeba Toki da TiTqos safrTxec metia. amgvari msjeloba swori ar aris. ratom? pasuxs am TavSi SeityobT 8.1. ra aris fizika? fizikis erT-erTi amocana sxvadasxva energiis Seswavlaa. energiis erT-erTi saxeoba U potenciuri energiaa. saxis zogadi teqnikuri TvalsazrisiT, potenciuri energia saganTa sistemis mdgomareobasTan (ganlagebasTan) dakavSirebuli energiaa, rodesac sagnebi erTmaneTze ZalebiT moqmedeben. zemoT motanili gansazRvreba yvelasTvis kargad nacnobi movlenis formaluri daxasiaTebaa. kargi ma– galiTia elastiuri gvarliT xtoma (sur. 8.1). saganTa sistema dedamiwisa da mxtomelisgan Sedgeba. saganTa Soris moqmedi Zala mizidulobis Zalaa. sistemis mdgomareoba icvleba (dedamiwasa da mxtomels Soris manZili mcirdeba). mxtomelis moZraobisa da kinetikuri energiis zrdis SefasebisTvis U mizidulobis potenciur energias ganvsazRvravT. es energia saganTa mdgomareobasTan aris dakav Sirebuli, romlebic erTmaneTs mizidulobis Zalis saSualebiT izidaven. am SemTxvevaSi, sistema mxtomeli da dedamiwaa. sur. 8.1. elastiuri gvarliT mxtomelis rodesac mxtomeli emzadeba, saganTa sistema gvar- kinetikuri energia Tavisufali vardnis dros lisa da mxtomelisgan Sedgeba. saganTa Soris izrdeba. Semdeg gvarli iWimeba da mxtomelis drekadobis Zala moqmedebs. Semdeg sistemis siCqare mcirdeba. 231 232 potenciuuri energia da energiis Senaxva Tavi 8 mdgomareoba icvleba (gvarli iWimeba). gvarlis sigrZis zrdisa da mxtomelis kinetikuri energiis Semcirebis SefasebisTvis U elastiur potenciur energias ganvsazRvravT. es energia drekadi sagnis (am SemTxvevaSi gvarlis) SekumSvasTan an gaWimvasTan aris dakavSirebuli. fizika adgens, Tu rogor unda gamoiTvalos sistemis potenciuri energia, raTa energiis Senaxva da gamoyeneba SesaZlebeli gaxdes. magaliTad, vidre elastiuri gvarliT mxtomeli triuks ganaxorcielebs, vinmem mizidulobis da elastiuri potenciuri energiebi unda daTvalos, raTa gvarli sworad SeirCes. zusti gamoTvlebis SemTxvevaSi naxtomi saSiSi ar iqneba. 8.2. muSaoba da potenciuri energia me-7 TavSi muSaobasa da kinetikuri energiis cvlilebas Soris kavSiri ganvixileT. amjerad muSaobasa da potenciuri energiis cvlilebas Soris damokidebulebas SeviswavliT. modiT, pomidori maRla avisroloT (sur. 8.2). ukve viciT, rom maRla asvlis dros mizidulobis Zalis mier pomidorze Sesrulebuli Wg muSaoba uaryofiTia, vinaidan Zala pomidors kinetikur energias arTmevs. gravitaciuli Zala uaryofiT muSaobas asrulebs gravitaciuli Zala dadebiTi muSaobas asrulebs Semdeg mizidulobis Zala am energias dedamiwa-pomidvris sistemis mizidulobis potenciur energias umatebs. mizidulobis Zalis moqmedebis Sedegad pomidori neldeba, Cerdeba da vardnas iwyebs. vardnis dros energiis gadacema piriqiT xdeba. mizidulobis Zalis mier pomidorze Sesrulebuli Wg energia dadebiTia da Zalas pomidori-dedamiwis sistemis mizidulobis potenciuri energia pomidvris kinetikur energiaSi gadaaqvs. orive SemTxvevaSi, mizidulobis potenciuri energiis ΔU cvlileba mizidulobis Zalis mier pomidorze Sesrulebuli muSaobis Sebrunebulia: ΔU = –W (8.1) sur. 8.2. pomidori maRla aisroles. maRla asvlis dros mizidulobis Zala pomidorze uaryofiT muSaobas asrulebs da misi kinetikuri energia mcirdeba. Camovardnis dros mizidulobis Zala dadebiT muSaobas asrulebs da pomid vris kinetikuri energia izrdeba igive formula 8.3 suraTze gamosaxul bloki-zambaris sistemas miesadageba. Tu bloks marjvniv vubiZgebT, drekadobis Zala marcxnisken aris mimarTuli, blokze uaryofiT muSaobas asrulebs da blokis kinetikuri energia zambara-blokis elastiur potenciur energiaSi gadadis. bloki neldeba, wamierad Cerdeba da marcxnisken moZraobas iwyebs, vinaidan drekadobis Zala kvlav (a) marcxniskenaa mimarTuli. am dros energiis gadacema piriqiT xdeba - energia zambara-blokis potenciuri energiidan blokis kinetikur energiaSi gadadis. Semnaxveli da araSemnaxveli Zalebi modiT, zemoT naxseneb or SemTxvevasTan dakavSirebuli ZiriTadi faqtebi CamovweroT: 1. 2. sistema ori an meti sagnisgan Sedgeba. Zala nivTieri wertilis magvar sagans (pomidors an bloks) da sistemis danarCen nawils Soris moqmedebs. (b) sur. 8.3. zambaraze mimagrebuli bloki x = 0 koordinatidan marjvniv iwyebs moZraobas. a) marjvniv gada adgilebis dros drekadobis Zala blokze uaryofiT muSaobas asrulebs. b) x = 0 koordinatisken dabrunebis dros drekadobis Zala blokze dadebiT muSaobas asrulebs. 8.3. Semnaxveli Zalebis traeqtoriisgan damoukidebloba 3. 233 rodesac sistemis mdgomareoba icvleba, Zala nivTieri wertilis magvar saganze muSaobas (vuwodoT mas W1) asrulebs da energia sagnis K kinetikur energiasa da sistemis raime sxva saxis energias Soris gadaaqvs. 4. rodesac sistemis mdgomareoba sapirispirod icvleba, energiis gadacema piriqiT xdeba da Zala W2 muSaobas asrulebs. rodesac W1 = –W2 , sxva saxis energia potenciuri energiaa da Zalas Semnaxveli (konservatiuli) ewodeba. albaT mixvdiT, rom mizidulobis Zala da drekadobis Zala Semnaxvelia (sxva SemTxvevaSi mizidulobis da elastiurobis potenciur energiaze ver visaubrebdiT). sxva saxis Zalebs araSemnaxveli ewodeba. araSemnaxveli Zalis magaliTebia kinetikuri xaxunis Zala da wevis Zala. modiT, bloki xaxunis mqone zedapirze gavasrialoT. srialis dros zedapiridan moqmedi kinetikuri xaxunis Zala blokze uaryofiT muSaobas asrulebs da kinetikuri energia siTbur energiaSi gadaaqvs. eqsperimentis Sedegad dgindeba, rom energiis amgvari gadacemis Sebruneba SeuZlebelia (atomebisa da molekulebis moZraobasTan dakavSirebuli siTburi energia blokis kinetikur energiaSi ver gadava). aqedan gamomdinare, saxezea sistema (blokisa da zedapirisgan Semdgari), sistemis nawilebs Soris moqmedi Zala da Zalis mier energiis gadacema, magram Zala Semnaxveli ar aris. maSasadame, siTburi energia potenciuri energia ar aris. rodesac nivTieri wertilis magvar saganze mxolod Semnaxveli Zala moqmedebs, sagnis moZraobasTan dakavSirebuli rTuli amocanebi Zalian martivdeba. momdevno paragrafSi saubari Semnaxveli Zalebis dadgenaze gveqneba da amgvari amocanebis gamartivebis erT saSualebasac ganvixilavT. 8.3. Semnaxveli Zalebis traeqtoriisgan damoukidebloba Zalis Semnaxvelobis an araSemnaxvelobis gansazRvrisTvis Semdegi testi arsebobs. davuSvaT Zala Caketil traeqtoriaze moZrav wertilze moqmedebs, rodesac wertili erTi mdebareobidan iwyebs moZraobas da sabolood igive mdebareobas ubrundeba. Zala mxolod im SemTxvevaSi aris Semnaxveli, Tu mTeli moZraobis ganmavlobaSi nebismier Caketil traeqtoriaze moZravi wertilisTvis gadacemuli energia nulis tolia: Semnaxveli Zalis mier nebismier Caketil traeqtoriaze moZrav wertilze Sesrulebuli jamuri muSaoba nulis tolia. mizidulobis Zala Caketili traeqtoriis tests abarebs. magaliTi 8.2 suraTze gamosaxuli pomidoria. pomidori moZraobas ν0 sawyisi siCqariT da 1 mν02 kinetikuri energiiT iwyebs. masze 2 moqmedi mizidulobis Zala pomidors anelebs, aCerebs da Semdeg mis vardnas iwvevs. rodesac pomidori asrolis wertils ubrundeba, misi siCqare kvlav ν0-ia, xolo kinetikuri energia isev 1 mν02 xdeba. maSasadame, mizidulobis Zala igive raodenobis energias arTmevs pomidors maRla 2 asvlis dros, rasac dabla Camovardnis ganmavlobaSi awvdis. mizidulobis Zalis mier pomidorze Sesrulebuli jamuri muSaoba nulia. Caketili traeqtoriis testi mniSvnelovani daskvnis gakeTebis saSualebas iZleva: or wertils Soris moZrav nivTier wertilze Semnaxveli sur. 8.4. a) nivTier wertilze Zalis mier Sesrulebuli damokidebuli ar aris. muSaoba traeqtoriaze Semnaxveli Zala moqmedebs da is a da b wertils Soris pirveli an 8.4 suraTze nivTieri wertili a da b wertils Soris pirveli an meore traeqtoriiT moZraobs. Tu wertilze mxolod Semnaxveli Zala moqmedebs, maSin masze Sesrulebuli muSaoba meore traeqtoriiT gadaadgildeba. b) nivTieri wertili a wertilidan b wertilisken pirveli traeqtoriiT gadaadgildeba da Semdeg a wertils meore traeqtoriiT ubrundeba. 234 Tavi 8 potenciuuri energia da energiis Senaxva orive traeqtoriis SemTxvevaSi Tanabaria. modiT zemoT miRebuli daskvna simboloebiT gamovsaxoT: Wab,1=Wab,2 , (8.2) sadac ab simbolo sawyis da saboloo wertilebs asaxavs, xolo 1 da 2 indeqsebi traeqtoriaze mianiSnebs. miRebuli daskvna Semnaxveli Zalis Semcveli rTuli amocanebis gamartivebis saSualebas iZleva. warmoidgineT, rom Semnaxveli Zalis mier Sesrulebuli muSaoba mocemul traeqtoriaze unda gamovTvaloT, rodesac damatebiTi informaciis gareSe gamoTvla Zalian rTulia an sulac SeuZlebelia. am SemTxvevaSi, raime sxva traeqtorias gamoiyenebT, rac muSaobis gamoTvlas bevrad ufro martivs an zogadad SesaZlebels gaxdis. 8.1 amocanis amoxsnis nimuSSi swored amgvari magaliTia ganxiluli, magram Tavdapirvelad 8.2 formulas damtkiceba esaWiroeba. 8.2 formulis damtkiceba 8.4b suraTze naCvenebia Caketil traeqtoriaze moZravi nivTieri wertili, romelzec erTaderTi Zala moqmedebs. nivTieri wertili a sawyisi wertilidan b saboloo wertilisken pirveli traeqtoriiT gadaadgildeba da Semdeg a wertils meore traeqtoriiT ubrundeba. TiToeul traeqtoriaze Zala nivTier wertilze muSaobas asrulebs. pirveli traeqtoriiT a-dan b-sken moZrav nivTier wertilze Sesrulebuli muSaoba Wab,1 simboloTi gamovsaxoT, xolo b-dan a-sken meore traeqtoriiT moZrav nivTier wertilze Sesrulebuli muSaoba Wba,2 iyos. Tu Zala Semnaxvelia, mTeli traeqtoriis ganmavlobaSi Sesrulebuli jamuri muSaoba nulis tolia: Wab,1 + Wba,2 = 0 aqedan gamomdinare: Wab,1 = –Wba,2 (8.3) modiT amjerad Wab,2 muSaoba ganvixiloT, romelsac Zala maSin asrulebs, rodesac nivTieri wertili a-dan b-sken meore traeqtoriiT gadaadgildeba (sur. 8.4a). Tu Zala Semnaxvelia, es muSaoba Wba,2 muSaobis Sebrunebulia: Wab,2 = –Wba,2 (8.4) 8.3 formulaSi -Wba,2-is nacvlad Wab,2 CavsvaT da miviRebT: Wab,1= Wab,2 formula damtkicebulia. sakontrolo wertili 1. suraTze naCvenebia a da b wertilebis damakavSirebeli sami traeqtoria. TiToeuli traeqtoriis  gaswvriv moZrav F Zala moqmedebs, romlis mimarTuleba naCvenebia. aris Tu ara F Zala Semnaxveli nivTier wertilze erTaderTi TiToeul SemTxvevaSi? amocanis amoxsnis nimuSi 8.1  Fg mizidulobis Zalis mier ab 8.5a suraTze naCvenebia 2kg masis yvelis naWeri, amoxsna: romelic xaxunisgan Tavisufal zedapirze wirze Sesrulebul muSaobas 7.12 formulis a-dan b wertilisken gadaadgildeba. yveli mTlianad 2 m manZils gadis, xolo jamuri vertikaluri gadaadgileba 0,8 m-ia. ra muSa (Wg = mgd cos φ) saSualebiT ver gamovTvliT,  obas asrulebs yvelze mizidulobis Zala? vinaidan  Fg -sa da d gadaadgilebas Soris f kuTxe icvleba. 8.4. potenciuri energiis mniSvnelobebis gansazRvra vinaidan Soris da 235  Fg Zala Semnaxvelia, a da b wertils raime sxva gamoTvlebi traeqtoria gavamartivoT. avirCioT modiT es traeqtoria 8.5b suraTze gamosaxuli wyvetili xazi iyos. horizontaluri segmentis gaswvriv φ kuTxe mudmivad 90°-ia. miuxedavad imisa, rom am segmentis gaswvrivi gadaadgileba ar viciT, 7.12 formulis Tanaxmad masze Sesrulebuli Wh muSaobaa: Wh = mgd cos 90° = 0 sur. 8.5. a) yvelis naWeri a wertilidan b wertilisken gadaadgildeba. b) mizidulobis Zalis mier yvelze Sesrulebuli muSaobis gamoTvla ufro advilia, Tu yvelis namdvil traeqtorias wyvetili traeqtoriiT SevcvliT.  adgilebaa 0,8 m, Fg da d mimarTulia qvemoT Wν = mgd cos 0° = (2 kg)(9,8 m/wm2)(0,8 m)(1) = 15,7 j a-dan b-mde wyvetili traeqtoriiT gada gamomdinare, adgilebisas vertikaluri segmentis  gaswvriv d gada da φ kuTxe mudmivad 0°-ia. 7.12 formulidan wyvetili traeqtoriis ver tikaluri nawilis gaswvriv Sesrulebuli Wν muSaobaa:  Fg Zalis mier yvelze Se srulebuli jamuri muSaobaa: W = Wh + Wν = 0 + 15,7 j ≈ 16 j (pasuxi) 8.4. potenciuri energiis mniSvnelobebis gansazRvra mocemul paragrafSi miviRebT formulebs, romelTa saSualebiT mizidulobis da elastiurobis potenciuri energiis mniSvnelobis gamoTvla SeiZleba. Tavdapirvelad Semnaxvel Zalasa da masTan dakavSirebul potenciur energias Soris kavSiri davadginoT. ganvixiloT nivTieri wertilis magvari sagani, romelic sistemis nawilia. sistemaSi  F Semnaxveli Zala moqmedebs. rodesac Zala saganze W muSaobas asrulebs, sistemasTan dakavSirebuli potenciuri energiis ∆U cvlileba Sesrulebuli muSaobis Sebrunebulia. es faqti 8.1 formulaSi (ΔU = –W) aisaxeba. rodesac Zala mdebareobasTan erTad icvleba, Sesrulebuli W muSaoba 7.32 formulis msgavsad SeiZleba Caiweros: (8.5) am formulis saSualebiT Zalis mier Sesrulebuli muSaoba gamoiTvleba, rodesac sagani xi wertilidan xf wertilisken gadaadgileba da sistemis mdgomareoba icvleba. 8.1 formulaSi 8.5 formula CavsvaT da mdgomareobis cvlilebis Sedegad potenciuri energiis cvlilebas gamovTvliT: (8.6) 8.6 formula Semnaxvel Zalasa da masTan dakavSirebul potenciur energias Soris zogadi kavSiria. modiT misi gamoyeneba vcadoT. mizidulobis potenciuri energia Tavdapirvelad m masis nivTieri wertili ganvixiloT, romelic y RerZis gaswvriv vertikalurad moZraobs. yi wertilidan yf wertilamde gadaadgilebis dros nivTier wertilze  Fg mizidulobis Zala muSaobas asrulebs. nivTieri wertili-dedamiwis sistemis mizidulobis potenciuri energiis Sesabamisi cvlilebis gamoTvlisTvis 8.6 formulas viyenebT, oRond saWiroa ori cvlileba: 1) radgan mizidulobis Zala vertikalurad moqmedebs, integreba y RerZis gaswvriv xdeba da ara x RerZis gaswvriv; 2) F-is nacvlad –mg-s vsvamT, vinaidan  Fg Zalis sididea mg da 236 potenciuuri energia da energiis Senaxva Tavi 8 mimarTulia y RerZis sapirispirod. amgvari cvlilebebis Semdeg miviRebT: saidanac gamomdinareobs: ΔU = mg(yf – yi) = mg Δy (8.7) mizidulobis potenciuri energiis mxolod ΔU cvlilebas aqvs fizikuri mniSvneloba. miuxedavad amisa, simartivisTvis nivTieri wertili-dedamiwis sistemas zogjer raime U mniSvnelobas vukavSirebT, rodesac nivTieri wertili garkveul y simaRlezea. amisTvis 8.7 formula Semdegi saxiT gadavweroT: U – Ui = mg(y – yi) (8.8) amis Semdeg Ui-s sistemis mizidulobis potenciur energiad miviCnevT, rodesac sistema aTvlis mdgomareobaSia anu nivTieri wertili yi aTvlis wertilze imyofeba. Cveulebriv miviCnevT, rom Ui=0 da yi=0. 8.8 formulaSi am monacemebis Casmis Sedegad miviRebT: U(y)=mgy (mizidulobis potenciuri energia) (8.9) am formulis Tanaxmad: nivTieri wertili-dedamiwis sistemasTan dakavSirebuli mizidulobis potenciuri energia mxolod nivTieri wertilis y=0 aTvlis mdebareobisadmi y vertikalur mdebareobazea damokidebuli da ara horizontalur mdebareobaze. elastiuri potenciuri energia ganvixiloT 8.3 suraTze naCvenebi bloki-zambaris sistema, rodesac bloki k sixistis mqone zambarazea damagrebuli. bloki xi wertilidan xf wertilisken gadaadgildeba da Fx=–kx drekadobis Zala blokze muSaobas asrulebs. bloki-zambaris sistemis elastiuri potenciuri energiis Sesabamisi cvlilebis gamoTvlisTvis 8.6 formulaSi F(x)-is nacvlad –kx CavsvaT: ΔU = 1 kxf2 – 1 kxi2 anu 2 2 (8.10) x mdebareobaze myof blokTan potenciuri energiis U mniSvnelobis dakavSirebisTvis aTvlis mdgomareobad virCevT iseT ganlagebas, rodesac zambara moSvebulia da bloki xi=0 mdebareobaze imyofeba. am SemTxvevaSi, Ui elastiuri potenciuri energia nulis tolia da 8.10 formula aseT saxes iZens: U – 0 = 12 kx2 – 0 rac gvaZlevs: U(x)= 12 kx2 (elastiuri potenciuri energia) sakontrolo wertili 2. (8.11) nivTieri wertili x = 0 koordinatidan x1 koor dinatisken gadaadgildeba da masze x RerZis gaswvriv mimarTuli Semnaxveli Zala moqmedebs. suraTze sami mdgomareoba, naCvenebia rodesac Zalis x komponenti x-Tan erTad icvleba. samive mdgomareobaSi Zalis maqsimaluri F1 sidide Tanabaria. daalageT mdgomareobebi nivTieri wertilis moZraobis dros potenciuri energiis cvlilebis mixedviT dididan mcireze. 8.5. meqanikuri energiis Senaxva 237 amocanis amoxsnis taqtika taqtika 1: misaRebia, magram yovelTvis unda gaxsovdeT, rom sinamdvileSi potenciuri energia mTels sistemas, am SemTxvevaSi ki vaSli-dedamiwis sistemas ukavSirdeba. aseve unda gaxsovdeT, rom potenciuri energiis raime mniSvnelobis sagnisTvis an sistemisTvis miniWebas mxolod maSin aqvs azri, rodesac potenciuri energiis termin `potenciuri ener giis~ gamoyeneba potenciuri energia mTlianad sistemas ukavSirdeba. miuxedavad amisa, SesaZloa SegxvdeT amocanebi, rodesac potenciuri energia mxolod sistemis nawilTan aris dakavSirebuli. magaliTad: `xeze Camo kidebuli vaSlis mizidulobis potenciuri energiaa 30 j~. msgavsi winadadeba xSirad amocanis amoxsnis nimuSi aTvlis mniSvneloba cnobilia. 8.2 2 kg masis zarmaca miwidan 5 m simaRleze xis totzea Camokidebuli (sur. 8.6). a) risi tolia zarmaca-dedamiwis sistemis U mizidulobis potenciuri energia, Tu y = 0 aTvlis wertili 1) miwazea, 2) miwidan 3m simaRlis aivanzea, 3) xis totzea da 4) totidan 1 m simaRlezea? y = 0 wertilSi mizidulobis potenciuri energia nulis tolia. amoxsna: mas Semdeg rac aTvlis wertilad y=0 miviCnieT, SesaZlebeli gaxda sistemis U mizidulobis potenciuri energiis am aTvlis wertilis mimarT gamoTvla (8.9 formulis saSualebiT). pirveli variantis SemTxvevaSi zarmacas koordinatia y = 5 m da gveqneba: U = mgy =(2 kg)(9,8 m/wm2)(5 m) = 98j (pasuxi) sxva variantebis SemTxvevaSi U-s mniSvnelo bebia: 2) U = mgy = mg(2 m) = 39 j; 3) U = mgy = mg(0) = 0 j; 4) U = mgy = mg(–1 m) = –19,6 j ≈ –20 j (pasuxi) b) zarmaca miwaze Camovarda. aTvlis wertilis sur. 8.6. y = 0 aTvlis wertilis mdebareobis oTxi arCevani. TiToeuli y RerZi metrebSia dagraduirebuli. aTvlis wertilis mdebareoba zarmaca-dedamiwis sistemis U potenciuri energiis mniSvnelobaze axdens gavlenas, xolo sistemis ΔU cvlileba masze potenciuri energiis damokidebuli ar aris. TiToeuli arCevanis SemTxvevaSi ra iqneba zarmaca-dedamiwis sistemis potenciuri energiis ΔU cvlileba? amoxsna: aTvlis potenciuri wertilis damokidebuli ar aris da mxolod simaRlis Δy cvlilebas eqvemdebareba. oTxive SemTxvevaSi energiis mdebareobis cvlileba arCevaze Δy = –5m. 8.7 formulidan gamomdinare: ΔU= mgΔy = (2kg)(9,8m/wm2)(–5m) = –98j (pasuxi) 8.5. meqanikuri energiis Senaxva sistemis Emeq meqanikuri energia misi U potenciuri energiis da Semadgeneli sagnebis K kinetikuri energiis jamis tolia: Emeq = K + U (meqanikuri energia) (8.12) am paragrafSi ganvixilavT ra mosdis meqanikur energias, rodesac sistemis SigniT energiis 238 Tavi 8 potenciuuri energia da energiis Senaxva gadacema mxolod Semnaxveli Zalebis saSualebiT xdeba anu sistemis Semadgenel sagnebze xaxunis an wevis Zalebi ar moqmedebs. garda amisa vuSvebT, rom sistema garemosgan izolirebulia anu sistemis gareT mdebare sagnidan moqmedi gare Zala sistemis SigniT energiis cvlilebas ar iwvevs. rodesac Semnaxveli Zala sistemis saganze W muSaobas asrulebs, energia sagnis K kinetikur energiasa da sistemis U potenciur energias Soris mimoiqceva. 7.10 formulidan gamomdinare, kinetikuri energiis ΔK cvlilebaa: ΔK = W (8.13) 8.1 formulidan gamomdinare, potenciuri energiis ΔU cvlilebaa: ΔU = –W (8.14) 8.13 da 8.14 formulebis gaerTianeba gvaZlevs: ΔK = –ΔU (8.15) sityvebiT rom vTqvaT, erTi saxis energia imdeniT izrdeba, ramdeniTac meore saxis energia mcirdeba. 8.15 formula aseTi saxiTac SeiZleba Caiweros: K2 – K1 = – (U2 – U1) (8.16) indeqsebi or sxvadasxva moments anu sistemaSi sagnebis or sxvadasxva ganlagebas asaxavs. 8.16 formulis gardaqmna gvaZlevs: K2 + U2 = K1 + U1 (meqanikuri energiis Senaxva) 8.17 formula Semdeg rames gvauwyebs: sistemis nebismier mdgomareobaSi K-s da U-s jami ( ) ( = sistemis nebismier sxva mdgomareobaSi K-s da U-s jami (8.17) ) es formula marTebulia, rodesac sistema izolirebulia da sagnebze mxolod Semnaxveli Zalebi moqmedebs. izolirebul sistemaSi, sadac energiis cvlilebas mxolod Semnaxveli Zalebi iwvevs, kinetikuri da potenciuri energia SeiZleba Seicvalos, magram maTi jami - sistemis Emeq meqanikuri energia, ar icvleba am wess meqanikuri energiis Senaxvis principi ewodeba (albaT mixvdiT, saidan modis saxelwodeba Semnaxveli Zala). 8.15 formulis saSualebiT es principi sxva formiTac SeiZleba Caiweros: ΔEmeq = ΔK + ΔU = 0 (8.18) meqanikuri energiis Senaxvis principi sakmaod rTul amocanebs amartivebs, romelTa amoxsna mxolod niutonis kanonebis gamoyenebiT TiTqmis SeuZlebeli iqneboda. rodesac sistemis meqanikuri energia inaxeba, garkveul momentSi kinetikuri da potenciuri energiebis jami raime sxva momentSi energiebis jams SegviZlia davukavSiroT. Sualeduri moZraobis da Zalebis mier Sesrulebuli muSaobis ganxilva saWiro ar aris. 8.7 suraTze naCvenebia magaliTi, rodesac meqanikuri energiis Senaxvis principis gamoyeneba SeiZleba. qanqaris moZraobis dros dedamiwa-qanqaris sistemis energia K kinetikur energiasa da U mizidulobis potenciur energias Soris mimoiqceva, K+U jami ki ar icvleba. Tu mizidulobis potenciuri energia viciT, rodesac qanqaris balansiri umaRles wertilzea (sur. 8.7g), maSin 8.5. meqanikuri energiis Senaxva 239 sur. 8.7. qanqaris masa ba lansirSia koncentrirebuli da qanqara win da ukan irxeva. suraTze naCvenebia moZraobis erTi sruli cikli. qanqaradedamiwis sistemis potenciuri da kinetikuri energiebi balan siris rxevasTan erTad ciklis ganmavlobaSi icvleba, magram sistemis Emeq meqanikuri energia mudmivia. Emeq energia kinetikur da potenciur energiebs Soris mimoiqceva. a) da e) etapebze mTliani energia kinetikuria. am dros balansiri udables wertilzea da udidesi siCqare aqvs. g) da z) etapebze mTliani energia potenciuria. am dros balansiri umaRles wertilzea da nulovani siCqare aqvs. b, d), v) da T) etapebze naxevari kinetikuri energiaa, xolo meore naxevari - potenciuri energia. Tu qanqaris mimagrebis adgilze xaxunis Zala moqmedebs an haeris wevis Zala arsebobs, maSin Emeq energia ar Seinaxeba da qanqara sabolood gaCerdeba. T) a) b) g) z) d) v) e) 8.17 formulis saSualebiT balansiris kinetikur energias gamovTvliT, rodesac is udables wertilzea (sur. 8.7e). modiT, udablesi wertili aTvlis wertilad miviCnioT, sadac mizidulobis potenciuri energia U2 = 0. aseve davuSvaT, rom aTvlis wertilis mimarT umaRles wertilSi potenciuri energia U1 = 20 j. vinaidan balansiri umaRles wertilSi wamierad Cerdeba, misi kinetikuri energia K1 = 0. es mniSvnelobebi 8.17 formulaSi CavsvaT da udables wertilSi K2 kinetikur energias gamovTvliT: K2 + 0 = 0 + 20 j anu K2 = 20 j yuradReba miaqcieT, rom es Sedegi umaRles da udables wertilebs Soris moZraobis ganxilvis da Zalebis mier Sesrulebuli muSaobis gamoTvlis gareSe miviReT. sakontrolo wertili 3. suraTze naCvenebia oTxi mdgomareoba. erT-erTi maTganis dros uZravi bloki qvemoT vardeba, xolo danarCen SemTxvevaSi bloki xaxunisgan Tavisufal daxril sibrtyeze curdeba. a) daalageT mdgomareobebi B wertilSi blokis kinetikuri energiis mixedviT dididan mcireze. b) daalageT mdgomareobebi B wertilSi blokis siCqaris mixedviT dididan mcireze. 240 potenciuuri energia da energiis Senaxva Tavi 8 amocanis amoxsnis nimuSi 8.8 suraTze gamosaxuli m 8.3 masis bavSvi h = 8,5 simaRlis wylis sasrialoze erToba. davuSvaT sasrialos zedapiri xaxunisgan Tavisufalia. ra siCqare eqneba bavSvs wylis auzSi Sesvlisas? amoxsna: bavSvis siCqares aCqarebis saSua lebiT ver gamovTvliT, vinaidan wina Tavebisgan gansxvavebiT, sasrialos daxris kuTxe ar viciT. miuxedavad amisa, vinaidan siCqare kinetikur energias sur. 8.8. bavSvi h simaRlis sasrialo ze curdeba ukavSirdeba, siCqaris gamoTvlisTvis meqanikuri energiis principidan gamomdinare: Emeq,b = Emeq,t Senaxvis principi SegviZlia gamoviyenoT. am SemTxvevaSi daxris kuTxe saWiro ar iqneba. gavSaloT es formula: Kb + Ub = Kt + Ut sistemaSi meqanikuri energia inaxeba, Tu sistema izolirebulia da energiis gadacemas anu SevamowmoT, sruldeba Tu ara es pirobebi. mgyb = 12 mνt2 + mgyt am formulis m-ze gayofa da gardaqmna Zalebi: bavSvze ori Zala moqmedebs. mizi mogvcems: mxolod Semnaxveli Zalebi iwvevs. modiT, dulobis Zala Semnaxvelia da bavSvze muSaobas asrulebs. sasrialodan bavSvze moqmedi 1 mνb2 + 2 νb2= νt2+ 2g(yt – yb) CavsvaT νt = 0 da yt – yb = h da miviRebT: drekadobis Zala muSaobas ar asrulebs, vinaidan misi mimarTuleba nebismier wer tilSi bavSvis moZraobis mimarTulebis per pendikularulia. sistema: vinaidan bavSvze muSaobas mxolod mizidulobis Zala asrulebs, Cveni sistema bavSvisa da dedamiwisgan Sedgeba da izo lirebulia. yovelive saxezea zemoTqmulidan izolirebul gamomdinare, sistemaSi moqmedi Semnaxveli Zala, romelic muSaobas asrulebs da meqanikuri energiis Senaxvis principis gamoyeneba dasaSvebia. davuSvaT Emeq,t iyos meqanikuri energia, rodesac bavSvi srials iwyebs, xolo Emeq,b - meqanikuri energia, rodesac bavSvi auzSi Sedis. energiis Senaxvis amocanis amoxsnis nimuSi = 13 m/w (pasuxi) igive siCqare eqneboda bavSvs 8,5 m simaRlidan vertikalurad Camovardnis SemTxvevaSi. sinamdvileSi, bavSvze xaxunis Zalebi imoqme debda da ase swrafad Camosrialeba Seu Zlebeli iqneboda. am amocanis niutonis kanonebis saSualebiT amoxsna sakmaod rTulia, energiis Senaxvis principis gamoyeneba ki viTarebas amartivebs. miuxedavad amisa, gamoTvlas SevZlebT, Camosrialebis energiebis amisTvis drois saSualebiT sasrialos ver formis codna gaxdeba saWiro da amocanac sakmaod garTuldeba. 8.4 cocvis qvemoT H manZilze Camovardeba, Toki daWimvas magaliTi, rodesac win mimavali mcocavi iwyebs. daWimuloba maqsimaluri maSin xdeba, (lideri) (damagrebuli rodesac lideri kidev d manZils gaivlis da liTonis rgolidan, romelSic Toki gadis) gaCerdeba. am dros Tokze Fmax maqsimaluri 8.9 suraTze naCvenebia gorgolaWidan kldeze aTvlili H simaRlidan vardeba, xolo ukan Zala moqmedebs. vardnis es monakveTi Zalian mimavali mcocavi Toks eWideba, raTa vardna saxifaToa, vinaidan Fmax Zalam SesaZloa Toki Seasustos. rodesac lideri gorgolaWidan gawyvitos. 241 8.5. meqanikuri energiis Senaxva Fmax Zalis gamoTvlisTvis vuSvebT, lideri rom Toki hukis kanonis mixedviT (7.21 vardna formula, Fx = –kx) zambaris msgavsad gorgolaWi iWimeba. nebismieri Tokis SemTxvevaSi k sixistis mudmiva Tokis L sigrZeze da nivTierebis eToki elastiurobazea damokidebuli: k= Toki eToki (8.19) L Tokis tipiuri elastiurobaa eToki = 20 kn, xolo lideris masaa m = 80 kg. gamoTvaleT Fmax Zala 8.9a suraTze gamosaxul did manZilze vardnisTvis da 8.9g suraTze gamosaxul mcire manZilze vardnisTvis. romel Sem TxvevaSi ufro gawydeba Toki? a) meore mcocavi b) g) sur. 8.9. a) meore mcocavisgan L manZiliT daSorebuli lideri kldeze mimagrebuli liTonis gorgolaWidan H manZilze imyofeba. b) lideri maqsimalur manZilze vardeba da Toki d manZiliT iWimeba. g) igive SemTxveva ufro mcire L da H manZilebisTvis amoxsna: vinaidan Toki zambaris msgavsad cvlileba. 8.20 formulaSi mocemuli samive iWimeba, masSi Semnaxveli drekadobis ZaliT cvlileba gamowveuli elastiuri potenciuri energia mdebareobebs Soris unda gamoiTvalos. lideris da saboloo inaxeba. meqanikuri energiis Senaxvis principi kinetikuri lideris sawyis mdebareobasa da saboloo saboloo mdebareobas (Tokis maqsimaluri daWimvisas) Cerdeba, ΔK = 0. Soris gamoviyenoT. Cvens sistemaSi (lideri- mizidulobis potenciuri energia: 8.7 for dedamiwa-Toki) meqanikuri energia inaxeba, mulidan gamomdinare ΔUg = mgΔy, sadac Δy Tu sistema izolirebulia da masSi energiis aris lideris mdebareobis cvlileba vardnis gadacemas mxolod Semnaxveli Zalebi iwveven. ganmavlobaSi. lideri jer H manZilze ecema modiT, SevamowmoT, sruldeba Tu ara es (gorgolaWamde), Semdeg kidev H manZils gadis pirobebi. (rodesac Toki daWimvas iwyebs) da daWimvis Zalebi: lidersa da dedamiwas Soris moqmed ganmavlobaSi kidev d manZils faravs (sur. mizidulobis Zalas energia mizidulobis 8.9b). aqedan gamomdinare: potenciuri energiidan kinetikur energiaSi gadaaqvs. rodesac Toki iWimeba, drekadobis Zala Tokis elastiur potenciur energias zrdis. mizidulobis Zala da drekadobis Zala Semnaxvelebia. sistema: orive Zala sistemaSi Sedis da energiis gadacemac masSi xdeba, amitom sistema elastiuri meqanikuri energiis Senaxvis principi am sistemisTvis SegviZlia gamoviyenoT. 8.18 potenciuri (8.20) sadac ΔK aris lideris kinetikuri energiis cvlileba, ΔUg – sistemis mizidulobis po tenciuri energiis cvlileba, xolo ΔUToki – sistemis elastiuri potenciuri energiis energia: vardnis 2 2 elastiuri potenciuri energiis cvlilebaa: ΔUToki = 1 kd 2 – 1 k(0)2 = 1 kd 2 2 2 energiis miRebuli cvlilebebi 8.20 formulaSi CavsvaT da miviRebT: anu ΔK + ΔUg + ΔUToki = 0 , da wamierad vardnis bolo d manZiliT iWimeba. 8.10 formulidan (ΔU= 1 kxf2 – 1 kxi2) gamomdinare, formulis Tanaxmad: sawyis lideri dasawyisSi Toki daWimuli ar aris, xolo 2 gamomdinare, vinaidan mdebareobaze ΔUg = mg Δu = mg(–2H – d) izolirebulia. zemoTqmulidan energia: sawyis miviReT 0+mg(–2H – d)+ 12 kd 2=0 1 kd 2 – mgd – 2mgH = 0 2 kvadratuli zogadi amonaxsnia: gantoleba, romlis 242 potenciuuri energia da energiis Senaxva Tavi 8 8.9g suraTze gamosaxuli vardnis SemTxvevaSi lideri mxolod 2H = 2 m manZilze vardeba, daWimvis d manZilis dadebiT mniSvnelobas mxolod dadebiTi fesvi gvaZlevs. 7.21 formulidan (Fx = –kx) gamomdinare, Tokze moqmedi Zalaa: nacvlad gamomdinare, Toki gawyvetis Sansi erTi SexedviT naklebia. miuxedavad amisa, rode sac 8.21 formulaSi 2H = 2 m da L = 3 m mniS vnelobebs CavsvamT, miviRebT: Fmax = kd = mg + k-s ris Semdegac Toki daWimvas iwyebs. aqedan Fmax = 5,4 × 103 n 8.19 formula CavsvaT da miviRebT: (pasuxi) maSasadame, mcire manZilze vardnis dros safrTxe ufro metia. gamocdilma kldeze mcocavebma kargad ician, rom Tokis gawyvetis saSiSroeba mxolod gorgolaWidan lideris H dacilebaze da (8.21) 2H sidide aris manZili, romelsac lideri gadis Tokis gamosaxuli daWimvamde. vardnis 8.9a SemTxvevaSi suraTze 2H = 6 m mokidebuli ar aris. safrTxe ufro 2H/L Sefardebazea damokidebuli, romelsac vardnis koeficienti ewodeba. saSiSroebis gansazRvrisTvis kldeze mcocavi vardnis koeficients gamoTvlis. rac ufro metia da Tokis sigrZea L = 13 m. lideris wonaa misi mniSvneloba, miT ufro maRalia To- mg = (80 kg)(9,8 m/wm ) = 784 n da Tokis niv Tierebis elastiurobaa eToki = 20000 n. kis gawyvetis albaToba. rodesac safrTxe es monacemebi 8.21 formulaSi CavsvaT da gorgolaWebs miviRebT: gamosaxuli 2 amagrebs. vardnis 8.9a suraTze SemTxvevaSi vardnis koeficientia 6/13=0,46, xolo 8.9g suraTze Fmax = 784 n + = 4,7 × 103 n Zalian realuria, kldeze mcocavi damatebiT (pasuxi) gamosaxuli vardnis SemTxvevaSi koeficienti 2/3=0,67-ia. igive maSasadame, 8.9g suraTze Camovardnis safrTxe ufro metia. amocanis amoxsnis taqtika taqtika 1: meqanikuri energiis Senaxva meqanikuri energiis Senaxvis principis Semcveli amocanebis amoxsnisas Semdegi SekiTxvebi unda dasvaT: ra sistemisTvis inaxeba meqanikuri energia? Tqven unda SegeZloT sistemis garemosgan gamoyofa da imis gansazRvra, Tu ra aris sistemis SigniT da ra mdebareobs garemoSi. 8.3 amocanis amoxsnis nimuSSi sistema bavSvi+dedamiwaa, 8.4 amocanis amoxsnis nimuSSi - lideri+dedamiwa+Toki. arsebobs Tu ara xaxuni an weva? Tu xaxuni an weva arsebobs, maSin meqanikuri energia ar inaxeba. aris Tu ara sistema izolirebuli? meqani kuri energiis Senaxvis principi mxolod izolirebul sistemebs miesadageba. es niSnavs, rom sistemis SigniT mdebare sagnebze gare Zalebma muSaoba ar unda Seasrulos. ra sawyis da saboloo mdgomareobaSia Tqveni sistema? sistema mdgomareobas icvlis. meqanikuri energiis Senaxvis principidan gamomdinare Emeq energia orive mdgomareobaSi erTi da igivea. yuradRebiT gansazRvreT, Tu romelia es ori mdgomareoba. 8.6. potenciuri energiis grafikis wakiTxva am SemTxvevaSic sistemis Semadgenel nivTier wertils ganvixilavT, rodesac sistemaSi Semnaxveli Zala moqmedebs. amjerad davuSvaT, rom nivTieri wertili x RerZis gaswvriv moZraobs da Semnaxveli Zala masze muSaobas asrulebs. sistemis potenciuri energiis U(x) grafikis saSualebiT nivTieri wertilis moZraobaze didi raodenobiT informaciis mopoveba SeiZleba. vidre amgvari grafikebis ganxilvaze gadavalT, kidev erTi formulis miReba gvesaWiroeba. 8.6. potenciuri energiis grafikis wakiTxva 243 Zalis gamoTvla analitikurad 8.6 formulis saSualebiT erTganzomilebian mdgomareobaSi or wertils Soris potenciuri energiis ΔU cvlilebas gamovTvliT, Tu F(x) Zala viciT. modiT sawinaaRmdego SemTxveva ganvixiloT, rodesac potenciuri energiis U(x) funqcia viciT da Zalis gamoTvla gvinda. erTganzomilebiani moZraobis dros Δx manZilis gavlisas nivTier wertilze Zalis mier Sesrulebuli W muSaobaa F(x)Δx. aqedan gamomdinare, 8.1 formula Semdegi saxiT SegviZlia CavweroT: ΔU(x) = –W = –F(x)Δx F(x)-sTvis amoxsna da diferencialur zRvarze gadasvla mogvcems: F(x) = – dU(x) dx (erTganzomilebiani moZraoba) (8.22) es swored is formulaa, romlis miRebac gvindoda. Sedegis Sesamowmeblad drekadobis Zalis nacvlad elastiuri potenciuri energiis U(x)= 12 kx2 funqcia CavsvaT. amis Semdeg 8.22 formula mogvcems F(x) = –kx, rac hukis kanonia. nivTieri wertili – dedamiwis sistemisTvis mizidulobis potenciuri energiis U(x) = mgx funqciac SegviZlia CavsaT, sadac m nivTieri wertilis masaa, romelic dedamiwis zedapiridan x simaRlezea ganlagebuli. am SemTxvevaSi 8.22 formula mogvcems F = –mg, rac nivTier wertilze moqmedi mizidulobis Zalaa. potenciuri energiis grafiki 8.10a suraTze gamosaxulia potenciuri energiis U(x) funqciis grafiki. grafiki agebulia sistemisTvis, romelSic nivTieri wertili erT ganzomilebaSi moZraobs da F(x) Semnaxveli Zala masze muSaobas asrulebs. sxvadasxva wertilebSi U(x) grafikis daxrilobis saSualebiT F(x) Zalas iolad gamovTvliT (8.22 formulis Tanaxmad, F(x) Zala U(x) grafikis daxrilobis Sebrunebulia). 8.10b suraTze am gziT miRebuli F(x)-is grafikia naCvenebia. sur. 8.10. a) potenciuri energiis U(x) funqciis grafiki sistemisTvis, romelSic nivTieri wertili x RerZis gaswvriv moZraobs. xaxuni ar gvaqvs, amitom meqanikuri energia inaxeba. b) nivTier wertilze moqmedi F(x) Zalis grafiki potenciuri energiis grafikis saSualebiT aris miRebuli. g) U(x) grafiki Emeq-is sam SesaZlo mniSvnelobasTan erTad. (a) (b) (g) 244 Tavi 8 potenciuuri energia da energiis Senaxva Semobrunebis wertilebi araSemnaxveli Zalis ararsebobis dros sistemis E meqanikuri energiis mniSvneloba mudmivia: U(x) + K(x) = Emeq (8.23) am formulaSi K(x) aris sistemaSi arsebuli nivTieri wertilis kinetikuri energiis funqcia (K(x) kinetikuri energiis nivTieri wertilis x mdebareobaze damokidebulebis funqciaa). 8.23 formula asec SeiZleba Caiweros: K(x) = Emeq – U(x) (8.24) davuSvaT Emeq meqanikuri energia 5 joulia. 8.10a suraTze amgvari energia horizontaluri wrfiT gamoisaxeba, romelic energiis RerZis 5j mniSvnelobaze gadis. 8.24 formula gvauwyebs Tu rogor gamovTvaloT K kinetikuri energia nivTieri wertilis nebismieri x mdebareobisTvis. U(x) grafikze x-is am mdebareobisTvis U gamoTvaleT da Semdeg Emeq- s gamoakeliT. magaliTad, Tu nivTieri wertili x5-is marjvniv nebismier wertilze mdebareobs, maSin K = 1 j. K-s mniSvneloba udidesia (5 j), rodesac nivTieri wertili x2 koordinatze mdebareobs da umciresia (0 j), rodesac nivTieri wertili x1 koordinatzea. K ar SeiZleba uaryofiTi iyos (vinaidan ν2 yovelTvis dadebiTia) da maSasadame nivTieri wertili x1-dan marcxniv verasdros gadaadgildeba, sadac Emeq – U uaryofiTia. rodesac nivTieri wertili x2-dan x1-sken gadaadgildeba, K mcirdeba da x1 koordinatze K = 0 (nivTieri wertili Cerdeba). yuradReba miaqcieT, rom rodesac nivTieri wertili x1-s miaRwevs, 8.22 formuliT mocemuli masze moqmedi Zala dadebiTia (vinaidan dU/dx daxriloba uaryofiTia). amgvarad nivTieri wertili x1-ze ar rCeba da marjvnisken moZraobas iwyebs. aqedan gamomdinare x1 Semobrunebis wertilia, sadac K=0 da nivTieri wertili moZraobis mimarTulebas icvlis. grafikis marjvena mxareze Semobrunebis wertili (sadac K=0) ar arsebobs da marjvnisken moZraoba usasrulod grZeldeba. wonasworobis wertilebi 8.10g suraTze naCvenebia 8.10a suraTze gamosaxuli potenciuri energiis U(x) funqciis grafikze aRebuli Emeq-is sami sxvadasxva mniSvneloba. vnaxoT, rogor icvleba is. Tu Emeq = 4 j (iisferi), Semobrunebis wertili x1-dan x1-sa da x2-s Soris inacvlebs. garda amisa, x5-is marjvniv mdebare nebismier wertilze sistemis meqanikuri energia misi potenciuri energiis tolia anu nivTier wertils kinetikuri energia ar gaaCnia, masze Zala ar moqmedebs da, maSasadame, uZravia. amgvar mdebareobaze myof nivTier wertils neitralur wonasworobaSi myofi ewodeba. Tu Emeq = 3 j (vardisferi wrfe), Semobrunebis wertili oria. erTi maTgani x1-sa da x2-s Soris aris ganlagebuli, xolo meore maTgani - x4-sa da x5-s Soris. garda amisa, x3 koordinatze K = 0. Tu nivTieri wertili swored x3 koordinatzea ganlagebuli, masze moqmedi Zala nulis tolia da nivTieri wertili ar moZraobs. rodesac nivTieri wertili am koordinatis romelime mxares aris ganlagebuli, Zala mas igive mimarTulebiT ubiZgebs da nivTieri wertili moZraobas ganagrZobs. amgvar mdgomareobaSi myofi nivTieri wertili arastabilur wonasworobaSia. amjerad ganvixiloT nivTieri wertilis moqmedeba, rodesac Emeq = 1 j (mwvane wrfe). Tu nivTier wertils x4-ze movaTavsebT, igi iqve darCeba. nivTieri wertili marcxniv an marjvniv ar gadaadgildeba, vinaidan amisTvis uaryofiTi kinetikuri energiaa saWiro. Tu nivTier wertils marcxnisken an marjvnisken vubiZgebT, aRmdgeni Zala mas kvlav x4-ze abrunebs. amgvar mdgomareobaSi myofi nivTieri wertili stabilur wonasworobaSia. Tu nivTier wertils Wiqis magvar potenciur ormoSi movaTavsebT, romlis centri x2-ia, nivTieri wertili or Semobrunebis wertils Soris aRmoCndeba. mas moZraoba mxolod x1-sken an x3-sken SeeZleba. 245 8.7. gare Zalis mier sistemaze Sesrulebuli muSaoba sakontrolo wertili 4. suraTze naCvenebia potenciuri energiis U(x) funqcia sistemisTvis, romelSic nivTieri wertili erT ganzomilebaSi moZraobs. a) daalageT AB, BC da CD regionebi nivTier wertilze moqmedi Zalis mixedviT x dididan mcireze. b) ra mimarTuleba aqvs Zalas, rodesac nivTieri wertili AB regionSi imyofeba? 8.7. gare Zalis mier sistemaze Sesrulebuli muSaoba sistema me-7 TavSi muSaoba ganvsazRvreT, rogorc saganze moqmedi Zalis mier sagnisTvis energiis gadacema an warTmeva. amjerad igive gansazRvrebas saganTa sistemaze moqmedi gare ZalisTvis ganvavrcobT. dadebiTi W muSaoba aris sistemaze moqmedi gare Zalis mier sistema sistemisTvis energiis gadacema an warTmeva. 8.11a suraTze naCvenebia dadebiTi muSaoba (sistemisTvis energiis gadacema), xolo 8.11b suraTze asaxulia uaryofiTi uaryofiTi W muSaoba (sistemis energiis warTmeva). rodesac sistemaze erTze meti Zala moqmedebs, maTi jamuri muSaoba sistemisTvis gadacemuli an warTmeuli energiaa. energiis gadacema bankis erTi angariSidan meoreze fulis gadatanas waagavs. Tu sistema erTi nivTieri wertilis an nivTieri wertilis magvari sagnisgan Sedgeba, Zalis mier sur. 8.11. a) sistemaze Sesrulebuli dadebiTi W muSaoba niSnavs, rom sistemas energia gadaeca. b) siste maze Sesrulebuli uaryofiTi W muSaoba niSnavs, rom sistemas energia waerTva. sistemaze Sesrulebuli muSaoba mxolod sistemis kinetikur energias cvlis. energiis amgvari gadacemisTvis muSaoba-kinetikuri energiis Teorema arsebobs (7.10 formula, ΔK=W) anu nivTier wertils energiis mxolod erTi angariSi - kinetikuri energia gaaCnia. gare Zalebs am angariSze energiis damateba an gamokleba SeuZliaT. Tu sistema ufro rTulia, gare Zala energiis sxva formebsac cvlis anu rTul sistemas energiis ramdenime angariSi gaaCnia. modiT, amgvari sistemebisTvis, garkveuli mtkicebebi CamovayaliboT, rodesac xaxuni arsebobs an ar arsebobs. xaxunis ararsebobis SemTxveva boulingis TamaSis dros moTamaSe Tavdapirvelad Caimuxleba da iatakze ganlagebul burTs xels mokidebs. Semdeg moTamaSe swordeba da burTs daaxloebiT saxis donidan isvris. am dros moTamaSis Zala burTze muSaobas asrulebs anu gare Zala energias gadascems romeliRac sistemas, magram romelia es sistema? pasuxis gasacemad SevamowmoT sur. 8.12. boulingis burTisa da de romeli energiebi damiwisgan Semdgar sistemaze W dadebiTi icvleba. burTis kinetikuri energia icvleba da vinaidan muSaoba sruldeba, rac sistemis meqa nikuri energiis ΔEmeq cvlilebas, burTis dedamiwasa da burTs Soris dacileba izrdeba, burTi- kinetikuri energiis ΔK cvlilebas da sis dedamiwis sistemis mizidulobis potenciuri energiac temis mizidulobis potenciuri energiis icvleba. moTamaSis Zala gare Zalaa da sistemaze mis mier ΔU cvlilebas iwvevs. 246 Tavi 8 potenciuuri energia da energiis Senaxva Sesrulebuli muSaobaa: W = ΔK + ΔU anu W=ΔEmeq (sistemaze Sesrulebuli muSoba xaxunis ararsebobis dros), (8.25) (8.26) sadac ΔEmeq aris sistemis meqanikuri energiis cvlileba. zemoT miRebuli ori formula 8.12 suraTzea asaxuli da erTmaneTis eqvivalenturia, rodesac gare Zala sistemaze muSaobas asrulebs da xaxuni ar arsebobs. xaxunis arsebobis SemTxveva modiT, 8.13a suraTze gamosaxuli magaliTi ganvixiloT.  F Zala bloks x RerZis gaswvriv   d manZilze gadaaadgilebs da blokis siCqare ν 0 -dan ν -mde horizontaluri mudmivi izrdeba. moZraobis ganmavlobaSi zedapiridan blokze  a) f k kinetikuri xaxunis Zala moqmedebs. Tavdapirvelad bloki – zedapiris sistema bloks sistemad miviCnevT da mas niutonis meore kanons mivusadagebT. x RerZis gaswvrivi komponentebisTvis niutonis meore kanoni aseTi saxiT Caiwereba: radgan Zalebi F – fk = ma  a mudmivebia, aCqarebac (8.27) ν2 = ν02 + 2ad es gantoleba a-sTvis amovxsnaT, miRebuli Sedegi 8.27 formulaSi CavsvaT da gardavqmnaT: Fd = 12 mν2 – 12 mν02+ fkd (8.28) vinaidan blokisTvis 1 mν2 – 1 mν02=ΔK, gveqneba: 2 2 Fd = ΔK + fkd (8.29) sxva SemTxvevaSi (magaliTad rodesac bloki daxril sibrtyeze moZraobs) potenciuri energia  sur. 8.13. a) bloks  zedapirze F Zala maSasadame, 2.16 formulis gamoyenebiT vwerT: b) mudmivia. ubiZgebs, xolo f k kinetikuri xaxunis  Zala moZraobas ewinaaRmdegeba. d gadaadgilebis dasawyisSi blokis  siCqarea ν 0 , xolo gadaadgilebis  dasasruls blokis siCqare ν xdeba.  b) F Zala bloki-zedapiris sistemaze W dadebiT muSaobas asrulebs, rac blokis meqanikuri energiis ΔEmeq cvlilebas da blokisa da zedapiris siTburi energiis ΔEsiT cvlilebas iwvevs. SeiZleba Seicvalos. am cvlilebis mxedvelobaSi misaRebad, 8.29 formula zogadi saxiT CavweroT: Fd = ΔEmeq + fkd (8.30) eqsperimentis Sedegad dgindeba, rom bloki da zedapiris nawili moZraobis dros xurdeba. me-18 TavSi ganvixilavT, rom sagnis temperatura sagnis EsiT siTbur energias (atomebisa da molekulebis moZraobasTan dakavSirebuli energia) ukavSirdeba. am SemTxvevaSi blokisa da zedapiris siTburi energia izrdeba, vinaidan maT Soris xaxuni arsebobs da bloki zedapirze misrialebs. gavixsenoT, rom xaxuni zedapirebs Soris civi SeduRebiTaa gamowveuli. rodesac bloki zedapirze misrialebs, bloksa da zedapirs Soris SeduRebis wertilebi Cndeba da orive maTgani xurdeba. aqedan gamomdinare, sriali blokisa da zedapiris EsiT siTbur energias zrdis. eqsperimentis Sedegad aseve dgindeba, rom siTburi energiis ΔEsiT zrda fk da d sidideebis namravlis tolia: ΔEsiT = fk d (siTburi energiis zrda srialis dros) (8.31) 8.30 formula aseTi saxiT SegviZlia CavweroT: Fd = ΔEmeq + ΔEsiT (8.32)  Fd aris gare F Zalis mier Sesrulebuli W muSaoba, magram romel sistemaze sruldeba muSaoba? 247 8.7. gare Zalis mier sistemaze Sesrulebuli muSaoba pasuxis gasacemad SevamowmoT Tu romeli energiebi icvleba. blokis meqanikurienergia icvleba da blokisa da zedapiris siTburi energiebic icvleba. aqedan gamomdinare, F Zala muSaobas bloki-zedapiris sistemaze asrulebs. es muSaobaa: W = ΔEmeq + ΔEsiT (sistemaze Sesrulebuli muSaoba xaxunis pirobebSi) (8.33) es formula 8.13b suraTzea naCvenebi da gare Zalis mier sistemaze Sesrulebul muSaobas asaxavs, rodesac xaxuni arsebobs. sakontrolo wertili 5. horizontaluri Zala bloks xaxunisgan Tavisufal zedapirze sam SemTxvevaSi ubiZgebs. modebuli Zalis F sidideebi da blokis siCqaris cvlilebebi cxrilSia mocemuli. samive SemTxvevaSi bloki Tanabar d manZils gadis. daalageT SemTxvevebi blokisa da zedapiris siTburi energiis cvlilebis mixedviT dididan mcireze. SemTxveva F blokis siCqaris cvlileba a 5n mcirdeba b 7n ucvleli rCeba g 8n izrdeba 8.5 amocanis amoxsnis nimuSi aRdgomis kunZulze mcxovrebma preistoriuli xanis adamianebma aTasobiT qvis uzarmazari qandakeba daamzades da mTels kunZulze mimofantes (sur. 8.14). rogor moaxerxes adamianebma yovelgvari damxmare meqanizmebis gareSe qandakebebis gadatana? 10 km-mde savaraudod, manZilze qandakebebs xis marxilebze awyobdnen da ise gadahqondaT. Tanamedrove gamoTvlebiT, 25 adamiani 9000 kg masis qandakebas 2 wuTis ganmavlobaSi 45 m manZilze gadaaadgilebs. sur. 8.14. qvis qandakeba aRdgomis kunZulze a) gamoTvaleT adamianebidan moqmedi jamuri Zalis mier Sesrulebuli muSaoba 45m manZilze gadaadgilebis procesSi da gansazRvreT = (50)(80 kg)(9,8 m/wm2)(45 m) cos 0° = 1,8 × 106 j ≈ 2 mgj (pasuxi) sistema, romelzec muSaoba sruldeba. sistemis gansazRvrisTvis SevamowmoT romeli amoxsna: Sesrulebuli muSaobis gamoTvla energia icvleba. qandakeba gadaadgildeba 7.7 formuliT SeiZleba (W = Fd cos φ). da, amitom, misi kinetikuri energia icvleba. am SemTxvevaSi d manZili aris 45 m, F aris 25 aseve adamianidan qandakebaze moqmedi jamuri Zalis da miwas Soris mniSvnelovani kinetikuri sidide, xolo φ = 0°. modiT SevTanxmdeT, rom xaxuni arsebobs, rac maTi siTburi energiis TiToeuli adamiani sakuTar wonaze orjer cvlilebas iwvevs. meti sididis ZaliT moqmedebs da am Zalis romelzec muSaoba sididea 2mg. marxilisa da miwisgan Sedgeba. aqedan gamomdinare, jamuri iolad misaxvedria, rom marxilsa maSasadame, sruldeba, sistema, qandakebis, Zalis sididea F = (25)(2mg) = 50mg. adamianis b) ramdeniT gaizarda sistemis siTburi energia masa daaxloebiT 80kg-ia da 7.7 formula aseT 45m manZilze gadaadgilebis procesSi? saxes miiRebs: W = Fd cos φ = 50 mgd cos φ amoxsna: ΔEsiT energiis cvlileba Zalis mier Sesrulebul W muSaobas 8.33 formuliT 248 potenciuuri energia da energiis Senaxva Tavi 8 ukavSirdeba (xaxunis arsebobis pirobebSi): amocanis a) W = ΔEmeq + ΔEsiT nawilidan W-s cvlileba nulis tolia, vinaidan gadaadgilebamde da gadaadgilebis Semdeg ΔEsiT = W = 1,8 × 106 j ≈ 2 mgj (pasuxi) g) gamoTvaleT, ra muSaoba unda Seesrulebina 25 adamians, raTa qandakeba 10 km manZilze amocanis amoxsnis nimuSi kombostoTi savse  masis yuTs qandakeba- amoxsna: W muSaobas amocanis a) nawilis msgavsad gamovTvliT, oRond d-s nacvlad amjerad 1×104 m gvaqvs: W = ΔEsiT = 3,9×108j ≈ 400 mgj gamomdinare: m = 14 kg gamoTvaleT cvlileba. qandakeba uZravia. yovelive zemoTqmulidan mtvirTavi aseve marxili-miwis sistemis siTburi energiis ΔEsiT mniSvneloba ukve viciT. qandakebis meqanikuri energiis ΔEmeq gadaetana? yvavilovani 30 n sididis horizontaluri F ZaliT betonis zedapirze miacurebs. d = 0,5 m manZilze gadaadgilebis dros yuTis siCqare 0,6 m/wm-dan 0,2 m/wm-mde mcirdeba. (pasuxi) 25 adamiani qandakebis gadatanas 10km manZilze SeZlebda da araviTari zesaidumlo saSualeba saWiro ar iqneboda. 8.6 xdeba. ese igi xaxuni mainc arsebobs da yuTisa da zedapiris siTburi energiac icvleba. yovelive zemoTqmulidan gamomdinare, Cveni sistema yuTisa da zedapirisgan Sedgeba. b) ramdeniT izrdeba yuTisa da zedapiris  a) ra muSaobas asrulebs F Zala da romel siTburi energia? sistemaze sruldeba muSaoba? Zalis mier Sesrulebul W muSaobasTan 8.33 amoxsna: siTburi energiis ΔEsiT cvlileba  F  gamosadegia da F Zalis mier Sesrulebuli W formuliT SegviZlia davakavSiroT, rodesac muSaobaa: amoxsna: am SemTxvevaSi 7.7 formula W = Fd cos φ = (40 n)(0,5 m)cos 0° = 20 j (pasuxi) sistemaSi xaxuni arsebobs: W = ΔEmeq + ΔEsiT (8.34) sistemis dasadgenad vnaxoT, Tu romeli W-s mniSvneloba amocanis a) nawilidan viciT. yuTis meqanikuri energiis ΔEmeq cvlileba energia misi kinetikuri energiis cvlilebaa, vinaidan icvleba. vinaidan yuTis siCqare icvleba, misi kinetikuri energiac auci leblad icvleba. arsebobs Tu ara xaxuni yuTsa da zedapirs Soris anu icvleba Tu ara siTburi energia? yuradReba miaqcieT,  rom F Zalas da yuTis siCqares erTi da igive mimarTuleba aqvs. maSasadame, xaxuni  ar arsebobs, magram maSin F Zala yuTs unda aCqarebdes, Cvens SemTxvevaSi ki piriqiT potenciuri energia ar icvleba: ΔEmeq = ΔK = 1 mν2 – 1 mν02 2 2 es formula 8.34 formulaSi CavsvaT da ΔEsiTsTvis amovxsnaT: ΔEsiT = W – (2mν2 – 1 mν02) = W – 1 m(ν2 – ν02) 2 2 = 20 j – 12 (14 kg)[(0,2 m/wm)2 – (0,6 m/wm)2] = 22,2 j ≈ 22 j (pasuxi) 8.8 energiis Senaxva Cven ukve ganvixileT SemTxvevebi, rodesac energia sagnebs an sistemebs gadaecema iseve, rogorc fuli gadairicxeba angariSebs Soris. TiToeul SemTxvevaSi vuSvebdiT, rom mTliani energia inaxeba anu mas gaCena an gaqroba ar SeuZlia. formaluri eniT rom vTqvaT, energia Senaxvis kanons emorCileba da es kanoni sistemis E mTlian energias exeba. mTliani energia sistemis meqanikuri energiis, siTburi energiis da nebismieri sxva saxis Sida energiis jamis tolia. energiis Senaxvis kanonis Tanaxmad: 8.8 energiis Senaxva 249 sistemis E mTliani energia mxolod im raodenobiT icvleba, ra raodenobis energiac sistemas gadaecema an waerTmeva. Cven energiis gadacemis mxolod erTi saxeoba - sistemaze Sesrulebuli W muSaoba ganvixileT. maSasadame, Cvens mier am etapze miRebuli codnis energiis Senaxvis kanoni ase SeiZleba Camoyalibdes: W = ΔE = ΔEmeq + ΔEsiT + ΔES (8.35) sadac ΔEmeq aris sistemis meqanikuri energiis cvlileba, ΔEsiT – sistemis siTburi energiis cvlileba, xolo ΔES – sistemis nebismieri sxva saxis Siga energiis cvlileba. ΔEmeq meqanikuri energiis cvlilebaSi kinetikuri energiis ΔK cvlileba da sur. 8.15. potenciuri energiis ΔU cvlileba Sedis. energiis Senaxvis kanoni fizikis ZiriTadi principebidan ar miiReba. es kanoni didi raodenobis eqsperimentebzea dafuZnebuli. mecnierebs am kanonis gamonaklisi SemTxveva jerjerobiT ar SexvedriaT. izolirebuli sistema kldeze mcocavma qvemoT Casasvlelad sistemas mizidulobis potenciuri energia unda waarTvas. sistema mcocavisgan, misi aRWur vilobisgan da dedamiwisgan Se dgeba. Toki liTonis rgolebzea Semotarebuli da rgolebsa da Toks Soris xaxuni arsebobs. swored amis Sedegad, gadacemuli energiis didi nawili Tokisa da rgolebis siTbur energiaSi gadadis da ara mcocavis kinetikur energiaSi. garemosgan izolirebul sistemaSi energiis garedan gadacema ar xdeba. am SemTxvevaSi energiis Senaxvis kanoni amtkicebs: izolirebuli sistemis E mTliani energia ar icvleba. energiis gadacema SesaZlebelia izolirebuli sistemis SigniT xdebodes, magaliTad, kinetikur energiasa da potenciur energias an kinetikur energiasa da siTbur energias Soris. miuxedavad amisa, sistemis yvela saxis energiis jami ar Seicvleba. magaliTad 8.15 suraTze gamosaxuli kldeze mcocavi SegviZlia moviyvanoT. izolirebuli sistema mcocavisgan, misi aRWurvilobisgan da dedamiwisgan Sedgeba. rodesac mcocavi kldeze Camodis, sistemis mdgomareoba icvleba da, maSasadame, sistemis mizidulobis potenciuri energiidan energiis gadatana xdeba. am energiis nawili kldeze mcocavis kinetikur energiaSi gadadis, Tumca es sasurveli ar aris, vinaidan mcocavis siCqare izrdeba. energiis arasasurveli gadacemis Tavidan asacileblad, kldeze mcocavi Toks liTonis rgolebs axvevs da Toksa da rgolebs Soris xaxuni Cndeba. xaxunis Sedegad sistemis mizidulobis potenciuri energia rgolebisa da Tokis siTbur energiaSi gadadis. mcocavi-aRWurviloba-dedamiwis sistemis mTliani energia (mizidulobis potenciuri energia, kinetikuri energia da siTburi energia) ar icvleba. energiis Senaxvis kanoni izolirebuli sistemisTvis orgvarad SeiZleba Caiweros. Tavdapirvelad 8.35 formulaSi W = 0 CavsvaT da miviRebT: ΔEmeq + ΔEsiT + ΔES = 0 (izolirebuli sistema) (8.36) aseve SegviZlia miviCnioT, rom ΔEmeq = Emeq,2 – Emeq,1, sadac 1 da 2 indeqsebi or sxvadasxva moments, magaliTad raime procesis dasawyiss da dasasruls mianiSneben. am SemTxvevaSi 8.36 formula aseT saxes iRebs: Emeq,2 = Emeq,1 – ΔEsiT – ΔES (8.37) 250 potenciuuri energia da energiis Senaxva Tavi 8 8.37 formula gvauwyebs: izolirebuli sistemis SemTxvevaSi, erT momentSi arsebuli mTliani energia meore momentSi arsebul mTlian energiasTan SegviZlia davakavSiroT da Sualedur droSi energiebis gaTvaliswineba saWiro ar aris. es faqti izolirebul sistemebTan dakavSirebuli rTuli amocanebis amoxsnas bevrad amartivebs. 8.5 paragrafSi gansakuTrebuli SemTxveva ganvixileT, rodesac izolirebul sistemaSi araSemnaxveli Zalebi (magaliTad kinetikuri xaxunis Zala) ar moqmedebs. am dros ΔEsiT da ΔES nulis tolia da 8.37 formula 8.18 formulaze daiyvaneba. sxva sityvebiT rom vTqvaT, izolirebuli sistemis meqanikuri energia inaxeba, rodesac araSemnaxveli Zalebi masSi ar moqmedeben. gare Zalebi da energiis Siga gadacema gare Zalas kinetikuri an potenciuri energiis Secvla saganze muSaobis Sesrulebis gareSe anu sagnisTvis energiis gadacemis gareSe SeuZlia. Tumca am Zalis mier aris gamowveuli sagnis Siga energiis erTi saxeobidan meoreSi gadasvla. 8.16 suraTze naCvenebia magaliTi. Tavdapirvelad uZravi mocigurave moajirs ubiZgebs da yinulze srialdeba (sur. 8.16a da b). mociguravis kinetikuri energia izrdeba, vinaidan  moajiridan masze F gare Zala moqmedebs. am Zalas moajiridan mociguraveze energia ar gadaaqvs anu muSaobas ar asrulebs. yinuli (a) kinetikuri energiis zrda mociguravis kunTebSi arsebuli bioqimiuri energiis Siga gadacemebs ukavSirdeba. 8.17 suraTze naCvenebia kidev erTi magaliTi. Zrava manqanis siCqares oTxive borblis saSualebiT zrdis (Zrava oTxive borbals abrunebs). aCqarebis ganmavlobaSi saburavebi gzis safars ukan ubiZgeben da am biZgis Sedegad win mimarTuli  f xaxunis  Zala oTxive saburavze moqmedebs. gzidan moqmedi F jamuri gare Zala xaxunis Zalebis jamia da sworedes Zala aCqarebs manqanas anu mis kinetikur energias zrdis. F Zalas (b) gzidan manqanaze energia ar gadaaqvs da, maSasadame, manqanaze muSaobas ar asrulebs. manqanis kinetikuri energia sawvavSi Senaxuli energiis Siga gadacemebis Sedegad izrdeba. zemoT moyvanili magaliTebis msgavs SemTxvevebSi,  F gare Zala sagnis meqanikuri energiis cvlilebas SegviZlia davukavSiroT, Tu mdgomareobas gavamartivebT. modiT, kvlav mociguravis magaliTs davubrundeT. 8.16g suraTze d manZiliT gadaadgile bis dros vuSvebT, rom mociguravis aCqareba mudmivia da misi siCqare ν0 = 0-dan ν-mde icvleba (anu F Zalis sidide da φ kuTxe ar icvleba). gadaadgilebis Semdeg mociguraves nivTier wertilad miviCnevT da ugulebelvyofT im faqts, rom kunTebis daWimvis Sedegad Sida siTburi energia da sxva fiziologiuri maxasiaTeblebi Seicvala. amgvari daSvebebis Sedegad 7.5 formulis ( 1 mν2 – 1 mν02 = Fxd ) gamoyenebiT davweroT: 2 2 (g) sur. 8.16. a) mocigurave sakuTar Tavs moajiridan  ubiZgebs da moajiridan masze F Zala moqme debs. b) moZraobis dasawyisSi mociguravis siCqarea . g) moci guraveze horizontaluri x Rer Zisadmi f kuTxis  mdgeneli F Zala d moqmedebs. manZilis gavlis  Semdeg F Zalis horizontaluri komponenti mociguravis siCqares (=0)-dan -mde cvlis. 0 251 8.8 energiis Senaxva K – K0 = (Fcos φ) d ΔK = Fd cos φ anu (8.38) Tu yvelaferTan erTad sagani maRlac iweva, maSin mizidu lobis potenciuri energiis ΔU cvlilebac Semogvaqvs: sur. 8.17. manqana oTxive borblis saSualebiT Cqardeba. gzis safari saburavebze oTxi xaxunis ZaliT (ori maTgani naCvenebia) moqmedebs. es  Zalebi manqanaze moqmed jamur F gare Zalas warmoadgens. ΔU + ΔK = Fd cos φ (8.39) formulis marjvena mxares Cawerili Zala saganze muSaobas ar asrulebs, magram formulis marcxena mxareze naCveneb energiis cvlilebaze pasuxismgebeli mainc aris. simZlavre Cven ukve vnaxeT rogor gadadis energia erTi saxidan meoreSi da 7.9 paragrafSi mocemuli simZlavris gansazRvreba SegviZlia ganvavrcoT. gaixseneT, rom simZlavre ganvsazRvreT, rogorc Zalis mier muSaobis Sesrulebis tempi. ufro zogadad, P simZlavre aris Zalis mier energiis erTi saxeobidan meoreSi gadatanis tempi. Tu Δt droSi ΔE odenobis energiis gadacema xdeba, Zalis Sedegad miRebuli saSualo simZlavrea: ΔE PsaS = Δt . (8.40) Zalis Sedegad miRebuli myisi simZlavrea: dE P = dt . amocanis amoxsnis nimuSi 8.18 suraTze gamosaxuli 2 kg masis yuTi (8.41) 8.7 vinaidan zedapirsa da yuTs Soris xaxuni zedapirze ν1 = 4 m/wm siCqariT misrialebs, arsebobs, maTi siTburi energia izrdeba. Semdeg zambaras ejaxeba, kumSavs mas da sistema: yvela Zala yuTi-zambara-zedapiri- wamierad kedlis Cerdeba. zambarisken moZraobis sistemaSi moqmedebs da energiis dros xaxuni ar arsebobs, magram zambarasTan gadacema izolirebul sistemaSi xdeba. radgan Sejaxebis Semdeg zedapiridan yuTze 15 n sistema izolirebulia, misi mTliani energia sididis kinetikuri xaxunis Zala moqmedebs. ar icvleba. aqedan gamomdinare, energiis Tu k = 10000 n/m, ra d manZiliT SeikumSeba Senaxvis kanoni 8.37 formulis saxiT Cvens zambara? sistemas mivusadagoT: amoxsna: Tavdapirvelad viswavloT da yvela davadginoT, Zala sistema Se izo lirebulia Tu gare Zala masze muSaobas asrulebs. Emeq,2 = Emeq,1 – ΔEsiT (8.42) indeqsi 1 yuTis sawyis mdgomareobas See sabameba, xolo indeqsi 2 im moments asaxavs, Zalebi: zedapiridan yuTze moqmedi drekado bis Zala yuTze muSaobas ar asrulebs, vinaidan am Zalis mimarTuleba yuTis ga daadgilebisadmi perpendikularulia. igive mizezis gamo arc mizidulobis Zala asrulebs yuTze muSaobas. zambaris SekumSvis Semdeg, drekadobis Zala yuTze muSaobas asrulebs da zambaris elastiur potenciur energias zrdis. drekadobis Zala myar kedelzec moqmedebs. sur. 8.18. yuTi xaxunisgan Tavisufal zedapirze 1 siCqariT zambarisken misrialebs. zambaris sixistea k. rodesac yuTi zambaramde miaRwevs, zedapiridan masze xaxunis Zala iwyebs moqmedebas. 252 potenciuuri energia da energiis Senaxva Tavi 8 Emeq,2 = K2 + U2 = 0 + 12 kd 2 rodesac yuTi wamierad Cerdeba da zambara d manZiliT ikumSeba. orive mdgomareobaSi sistemis meqanikuri energia yuTis kinetikuri energiis (K = 1 mν2) da zambaris potenciuri 2 energiis (U = 1 kx2) jamis tolia. pirvel 2 mdgomareobaSi U = 0 (vinaidan zambara Se kumSuli ar aris) da yuTis siCqarea ν1. aqedan gamomdinare: saboloo jamSi, 8.31 formulis saSualebiT yuTisa da zedapiris siTburi energiis ΔEsiT cvlileba fk d-iT SevcvaloT da 8.42 formula Semdegi saxiT CavweroT: 1 2 1 kd = 2 mν12 2 gardaqmna da cnobili – fkd sidideebis Casma mogvcems: Emeq,1 = K1 + U = 12 mν12+ 0 meore mdgomareobaSi K = 0 (vinaidan yuTi Cerdeba) da SekumSvis manZilia d. 5000d 2 + 15d – 16 = 0 kvadratuli gantolebis amonaxsnia: aqedan d = 0,055 m = 5,5 sm (pasuxi) gamomdinare: amocanis amoxsnis nimuSi 8.8 8.19 suraTze gamosaxulia sacirko warmodgena. m = 6 kg masis ZaRli oRroCoRro zedapirze y0 = 8,5 m simaRlidan ν0 = 7,8 m/wm siCqariT iwyebs moZraobas, marjvnisken srialdeba da wamierad Cerdeba, rodesac y = 11,1 m simaRles miaRwevs. zedapiri xaxunisgan Tavisufali ar aris. ramdeniT izrdeba ZaRlis da zedapiris siTburi energia? amoxsna: modiT, ZaRlze moqmedi yvela Zala ganvixiloT da davadginoT sistema izolirebulia Tu masze gare Zala muSaobas asrulebs. Zalebi: zedapiridan ZaRlze moqmedi drekadobis Zala muSaobas ar asrulebs, vinaidan misi mimarTuleba yovelTvis ZaRlis gadaadgilebisadmi perpendikularulia. sur. 8.19. ZaRli oRroCoRro zedapirze srialdeba. cxoveli moZraobas y0 simaRlidan ν0 siCqariT iwyebs da y simaRleze wamierad Cerdeba. ZaRlze moqmedi mizidulobis Zala muSaobas asrulebs, daSoreba vinaidan icvleba. dedamiwidan ZaRlsa da ZaRlis zedapiris garda amisa, ΔEmeq cvlileba ZaRlis kine Soris xaxunis Sedegad maTi siTburi energia tikuri energiis ΔK cvlilebisa da sistemis izrdeba. mizidulobis potenciuri energiis ΔU cvli sistema: yvela Zala ZaRli-zedapiri-dedami lebis jamis tolia, sadac: ΔK = 0 – 1 mν02 wis sistemaSi moqmedebs da energiis gadacema erT izolirebul sistemaSi xdeba. radgan sistema izolirebulia, misi mTliani energia ar icvleba. modiT energiis Senaxvis kanoni am sistemisTvis 8.36 formulis saxiT CavweroT: ΔEmeq + ΔEsiT = 0 (8.43) energia ZaRlis sawyis mdgomareobasa da wamieri gaCerebis moments Soris icvleba. 2 da ΔU = mgy – mgy0 es formulebi 8.43 formulaSi CavsvaT da ΔEsiT-sTvis amovxsnaT: ΔEsiT = 1 mν02 – mg(y – y0) = 2 1 (6kg)(7,8m/wm)2–(6kg)(9,8m/wm2)(11,1m–8,5m) 2 ≈ 30j (pasuxi) 253 mimoxilva da Sejameba Semnaxveli Zalebi. Zala Semnaxvelia, elastiuri potenciuri energiaa: rodesac nebismier Caketil traeqtoriaze moZrav nivTier wertilze mis U(x) = 12 kx2 mier Sesrulebuli jamuri muSaoba nulis tolia. aseve, Zala Semnaxvelia, Tu or wertils Soris moZrav nivTier wertilze mis mier (8.11) aTvlis mdgomareobaSi zambara moSvebulia, x = 0 da U = 0. meqanikuri energia. sistemis Emeq meqanikuri Sesrulebuli jamuri muSaoba traeqtoriaze energia misi K kinetikuri energiis da U damokidebuli ar aris. mizidulobis Zala potenciuri energiis jamis tolia: da drekadobis kinetikuri Zala xaxunis Semnaxveli Zala Zalebia. araSemnaxveli Emeq = K + U (8.12) sistemas izolirebuli ewodeba, Tu gare Zala Zalaa. masSi energiis cvlilebas ar iwvevs. rodesac potenciuri energia. potenciuri energia izolirebul sistemaSi mxolod Semnaxveli sistemis imgvar mdgomareobas ukavSirdeba, rodesac sistemaSi Semnaxveli Zala moqmedebs. rodesac Semnaxveli Zala sistemis Sida nivTier wertilze W muSaobas asrulebs, sistemis potenciuri energiis ΔU cvlilebaa: ΔU = –W Zalebi asruleben muSaobas, sistemis Emeq meqanikuri energia ar icvleba. meqanikuri energiis Senaxvis principi ase SeiZleba Caiweros: K2 + U2 = K1 + U1 (8.17) (8.1) indeqsebi energiis gadacemis procesis or Tu nivTieri wertili xi-dan xf wertilisken sxvadasxva moments asaxaven. energiis Senaxvis gadaadgildeba, sistemis potenciuri energiis principi asec SeiZleba gamoisaxos: cvlilebaa: (8.6) mizidulobis potenciuri energia. ΔEmeq = ΔK + ΔU = 0 potenciuri rodesac energiis sistemaSi (8.18) grafikebi. nivTier wertilze axlomdebare erTganzomilebiani F(x) Zala moqmedebs da nivTieri wertilisgan Semdgar sistemasTan am sistemisTvis potenciuri energiis U(x) dakavSirebul funqcia viciT, Zalas Semdegi formuliT dedamiwisa da masTan potenciur energias mizidulobis potenciuri energia ewodeba. gamovTvliT: Tu nivTieri wertili yi-dan yf simaRleze gadaadgildeba, nivTieri wertili-dedamiwis sistemis mizidulobis potenciuri energiis cvlileba: ΔU = mg(yf – yi) = mg Δy (8.7) F(x) = – dU(x) dx Tu mocemulia U(x)-is grafiki, maSin x-is nebismieri mniSvnelobis dros F(x) Zala am grafikis mrudis Sebrunebulia da nivTieri Tu nivTieri wertilisTvis aTvlis wertilad wertilis kinetikuri energiaa: yi = 0 aris miCneuli da sistemis Sesabamisi mizidulobis potenciuri energiaa Ui = 0, maSin nivTieri wertilis y simaRleze mdebareobis dros U mizidulobis potenciuri energiaa: U(y) = mgy elastiuri elastiuri (8.9) potenciuri potenciuri energia. energia aris drekadi sagnis SekumSvasTan an gaWimvasTan dakavSirebuli romelic energia. Tavisufali bolos zambarisTvis, x manZilze gadaadgilebis dros F=-kx ZaliT moqmedebs, (8.22) K(x) = Emeq – U(x) (8.24) sadac Emeq aris sistemis meqanikuri energia. Semobrunebis wertili aris raime x wertili, sadac nivTieri wertili moZraobis mimarTulebas icvlis (am wertilze K = 0). nivTieri wertili wonasworobaSia im wertilebze, sadac U(x) grafikis daxriloba nulia (am wertilebze F(x) = 0). gare Zalis mier sistemaze Sesrulebuli muSaoba. W muSaoba aris sistemaze moqmedi gare Zalis saSualebiT sistemisTvis an 254 potenciuuri energia da energiis Senaxva Tavi 8 sistemidan gadacemuli rodesac jami) mxolod im raodenobiT icvleba, ra moqmedebs, raodenobiTac energia sistemas miewodeba an gadacemuli energia maT mier Sesrulebuli waerTmeva. es faqti eqsperimentis Sedegad sistemaze erTze energia. meti Zala jamuri muSaobaa. rodesac xaxuni ar arsebobs, aris dadgenili da mas energiis Senaxvis sistemaze Sesrulebuli muSaoba da sistemis kanoni ewodeba. Tu sistemaze W muSaoba meqanikuri energiis ΔEmeq cvlileba erTmane sruldeba, maSin: W = ΔE = ΔEmeq + ΔEsiT + ΔES Tis tolia: W = ΔEmeq = ΔK + ΔU rodesac Zala sistemaSi moqmedebs, (8.26, 8.25) kinetikuri siTburi energia icvleba. am SemTxvevaSi, sistemaze Sesrulebuli muSaobaa: W = ΔEmeq + ΔEsiT (8.33) ΔEsiT cvlileba fk xaxunis Zalas da gare Zalis Sedegad gamowveul d gadaadgilebas Semdegi formuliT ukavSirdeba: ΔEsiT = fk d energia (8.31) Senaxva. sistemis E mTliani energiis (meqanikuri da Sida Tu sistema izolirebulia (W=0): xaxunis EsiT sistemis (8.35) da ΔEmeq + ΔEsiT + ΔES = 0 (8.36) Emeq,2 = Emeq,1 – ΔEsiT – ΔES (8.37) 1 da 2 indeqsebi or sxvadasxva moments asaxavs. simZlavre. Zalis Sedegad miRebuli simZlavre Zalis mier energiis gadacemis tempia. Tu Δt droSi Zalas ΔE raodenobis energia gadaaqvs, Zalis saSualo simZlavrea: ΔE PsaS = Δt (8.40) Zalis Sedegad miRebuli myisi simZlavrea: dE P = dt energiebis (8.41) SekiTxvebi 1. 8.20 suraTze naCvenebia i wer tilidan f wertilamde siCqaris da b) finiSis xazamde misasvleli drois mixedviT dididan mcireze. gavle buli erTi wrfi sur. 8.20. SekiTxva 1. vi da oTxi ara wrfivi toria. traeqwrfivi arawrfivi traeqtoriiT traeqtoriiT da moZrav sami saganze mxolod Fc Semnaxveli Zala moqmedebs. meoTxe arawrfivi traeqtoriiT moZrav saganze Fc sur. 8.21. SekiTxva 2. 3. 8.22 suraTze gamosaxuli, Tavdapirvelad Zala da Fnc araSemnaxveli Zalebi moqmedeben. uZravi TiToeuli swor moZraobas. gorakebis simaRleebi naCvenebia, ΔEmeq maT identuri gluvi mwvervalebi aqvT da sagnis meqanikuri energiis ΔEmeq cvlileba arawrfivi monakveTzea traeqtoriis naCvenebi. risi tolia bloki uswormasworo, Tavisufal zedapirze 3m simaRlidan iwyebs a) wrfiv traeqtoriaze da b) rogor icvleba meqanikuri energia Fnc-is Sedegad? 2. 8.21 suraTze gamosaxul horizontalurad moZrav bloks finiSis konturul xazamde misasvlelad zedapirze xaxunisgan SeuZlia Tavisufal imoZraos, sam romlebic erTmaneTisgan mxolod simaRliT gansxvavdeba. daalageT zedapirebi a) finiSis xazze blokis xaxunisgan sur. 8.22. SekiTxva 3. 255 bloki arc erTi gorakidan ar vardeba. a) romel xaxunis Sedegad Cerdeba. a) h-is Semcirebis goraks ver gadaivlis bloki? b) ra moxdeba amis SemTxvevaSi D manZili Semcirdeba, gaizrdeba Semdeg? romeli gorakis mwvervalze iqneba Tu igive darCeba? b) blokis masis gazrdis g) blokis centriskenuli aCqareba udidesi da SemTxvevaSi D manZili Semcirdeba, gaizrdeba d) blokze moqmedi drekadobis Zala umciresi? Tu igive darCeba? 4. 8.23 suraTze naCvenebia sami mdgomareoba, rodesac bloki sibrtyeze srialdeba da maT Soris xaxuni arsebobs. samive mdgomareobaSi blokis da sawyisi manamde siCqare ganagrZobs erTi da igivea moZraobas, vidre kinetikuri xaxunis Zala gaaCerebs. daalageT mdgomareobebi srialis Sedegad siTburi energiis zrdis mixedviT dididan mcireze. sur. 8.25. SekiTxva 6. 7. 8.26a suraTze, damagrebul vertikalur cilindrze Zelakze mibmul Toks maRla weven. cilindri Zelakze mWidrodaa damagrebuli da srials xaxuni ewinaaRmdegeba. cilindri-Zelaki-dedamiwis sistemaze Zala W = +100 j muSaobas asrulebs (sur. 8.26b). 8.26g suraTze gamosaxulia sistemis energiis sur. 8.23. SekiTxva 4. 5. 8.24 suraTze naCvenebia nivTieri wertilis potenciuri energiis funqcia. a) daalageT AB, BC, CD da DE ubnebi nivTier wertilze moqmedi Zalis sididis mixedviT dididan cvlilebebi: K kinetikuri energia 50 j-iT izrdeba da Ug mizidulobis potenciuri ener gia 20 j-iT izrdeba. sistemaSi EsiT siTburi energiac icvleba. risi tolia es cvlileba? mcireze. ra mniSvnelobas ar unda gadaaWarbos nivTieri wertilis Emeq meqanikurma energiam, raTa b) nivTieri wertili potenciur ormoSi aRmoCndes datyvevebuli da d) nivTierma wertilma or potenciur ormos Soris ΔK = +50 j imoZraos, magram H wertilis marjvniv ar ΔUg = +20 j gadaadgildes? d) SemTxvevaSi BC, DE da FG ΔEsiT = ? (a) ubnebidan romelSi eqneba nivTier wertils (g) (b) e) udidesi kinetikuri energia da v) umciresi siCqare? sur. 8.26. SekiTxva 7. 8. 8.27 suraTze gamosaxuli ganlageba me-7 SekiTxvis mWidrod 6. 8.25 suraTze simaRlidan gzis daqanebul daRmarTi rodesac gamosaxuli bloki gzaze xaxunisgan gzis swor bloki h srialebs. Tavisufalia. monakveTze Camodis, garkveuli D manZilis gavlis Semdeg damagrebul Toks qvemoT Cebian. garda Camosvlis cilindri meore saSualebiT qmedebs da Zelakze cilindrze mibmul eqa dros kis amjerad amisa, qvemoT blokze sur. 8.24. SekiTxva 5. analogiuria. To mo bloki laboratoriis magidaze gadaa dgildeba. am SemTx vevaSic 8.26b suraTze sur. 8.27. SekiTxva 8. 256 potenciuuri energia da energiis Senaxva Tavi 8 gamosaxulis dedamiwis msgavsi sistema cilindri-Zelaki- ganvixiloT. sistemaze 200 j muSaoba sruldeba, Tavad sistema ki blokze 60 j muSaobas asrulebs. kinetikuri energia 130 j-iT izrdeba, xolo mizidulobis potenciuri energia 20 j-iT mcirdeba. sadac a) AB masze xaxunis Zala monakveTze, b) BC moqmedebs. monakveTze da g) CD monakveTze blokis kinetikuri energia izrdeba, mcirdeba Tu ar icvleba? d) blokis meqanikuri energia am monakveTebze izrdeba, mcirdeba Tu ar icvleba? a) sistemis energiis cvlilebebi 8.26g suraTis msgavsad gamosaxeT. b) risi tolia siTburi energiis cvlileba sistemis SigniT? 9. 8.28 suraTze gamosaxuli bloki xaxunisgan Tavisufal uswormasworo zedapirze A-dan C wertilamde srialdeba da Semdeg CD horizontalur monakveTze gadaadgildeba, sur. 8.28. SekiTxva 9. amocanebi g) wignis vardnis dawyebisas da d) wignis xelSi SSM amoxsnebi mocemulia Student Solutions Manual-Si. daWerisas? Tu dedamiwis doneze U = 100 j, www amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday da T) U wignis xelSi daWerisas? ILW Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley. com/college/halliday. • – ••• wertilebis raodenoba amocanis sirTules gviCvenebs risi tolia e) Wg , v) ΔU, z) U vardnis dawyebisas •3. 8.30 suraTze 2 g r = 22 sm radiusis naxevarsferos for mis WiqaSi Caagdes. paragrafi 8.4. potenciuri energiis mniSvnelobebis gansazRvra •1. risi tolia zambaris sixiste, romelic yinulsa 7,5sm-iT Saobas asrulebs mi SekumSvis dros 25j yinulis natexi masis yinulis natexi elastiur Soris da Wiqas xaxuni ar arsebobs. a) ra mu sur. 8.30. amocanebi 3 da 9 potenciur energias inaxavs? SSM zidulobis Zala yi •2. Tqven 2 kg masis wigni D = 10 m simaRlidan nulze, vidre yinuli Wiqis fskerze daeSveba? megobars CaugdeT. gawvdili xelebi Tu Tqveni megobris dedamiwidan d = 1,5 m simaRlezea (sur. 8.29), a) ra Wg muSaobas asrulebs mizidu b) rogor Seicvleba am drois ganmavlobaSi yinuli-dedamiwis sistemis potenciuri energia? g) Tu Wiqis fskerze potenciuri energia nulis tolia, ra iqneba potenciuri lobis Zala wignze energia vardnis dros? d) Tu potenciuri energia yinulis WiqaSi b) risi tolia wigni- Cavardnisas nulis tolia, ra iqneba is Wiqis dedamiwis fskerze? e) yinulis natexis masis gaormagebis sistemis yinulis WiqaSi zemoT Cavardnisas? mizidulobis pote SemTxvevaSi nciuri energiis ΔU gaizrdeba, Semcirdeba Tu ar Seicvleba? SSM miRebuli pasuxebi cvlileba vardnis •4. 8.31 suraTze naCvenebia L = 0,425 m sigrZis dros? Tu dedamiwis da umniSvnelo masis Zelakze damagrebuli doneze sistemis U m = 0,341 kg masis burTi. Zelakis meore mizidulobis poten bolo fiqsirebulia da burTi vertikalur ciuri energis nulis sur. 8.29. amocanebi 2 tolia, ra iqneba U da 10 wreze moZraobs. Zelaki Tavdapirvelad horizontalurad aris gaCerebuli da Semdeg 257 qvemoT iseTi ZaliT energia burTis frenis ganmavlobaSi? g) Tu burTi kldis simaRleze mizidulobis potenciuri arasrul wres Semowers da energia nulis tolia, ra iqneba igive energia naCveneb adgilze Cerdeba. burTis miwaze dacemisas? ra muSaobas Seasrulebs ••7. 8.33 suraTze burTze mizidulobis gamosaxuli sur. 8.31. Zala sawyisi wertilidan m = 0,032 kg masis amocanebi 4 da 12 a) udables wertilamde, patara ubiZgeben, rom bloki b) umaRles wertilamde da g) sawyisi wertilis xaxunisgan mopirdapire wertilamde? Tu sawyis wertilze sufal burTi-dedamiwis srialdeba. sistemis mizidulobis Tavi rgolze rgo potenciuri energia nulis tolia, ra iqneba lis igive energia, rodesac burTi d) udables R = 12 sm. wertils, e) umaRles wertils da v) sawyisi moZraobas wertilis mopirdapire wertils miaRwevs? lis zedapiridan h = 5R simaRlidan iwyebs (P T) davuSvaT Zelaks ufro Zlierad ubiZges wertili). ra muSaobas asrulebs mizidulobis radiusia sur. 8.33. amocanebi 7 da 21 bloki rgo da burTma umaRlesi wertili aranulovani Zala blokze P wertilidan a) Q wertilamde siCqariT da gaiara. udablesi wertilidan b) rgolis wveramde gadaadgilebis umaRles wertilamde ΔUg cvlileba ufro ganmavlobaSi? Tu rgolis zedapirze bloki- meti iqneba, naklebi iqneba Tu ar Seicvleba? dedamiwis sistemis mizidulobis potenciuri •5. 8.32 suraTze gamosaxuli m = 825 kg masis manqana pirveli gorakis h = 42 m simaRlis mwvervalze ν0 = 17 m/wm siCqariT adis. ra energia nulis tolia, ra iqneba igive energia muSaobas asrulebs mizidulobis Zala manqanaze am wertilidan a) A wertilamde, g) P wertilze, d) Q wertilze da e) rgolis wverze? v) Tu bloks raime sawyisi siCqare mianiWes, rogor Seicvleba a)-e) amocanis pasuxebi? b) B wertilamde da g) C wertilamde? Tu ••8. C naCvenebia wertilze manqana-dedamiwis sistemis 8.34 suraTze L=2m nulis sigrZis da umniS tolia, ra iqneba igive energia d) B wertilze vnelo masis Zelaki, da e) A wertilze? v) manqanis masis gaormagebis romlis erTi bolo SemTxvevaSi, A wertilsa da B wertils Soris fiqsirebulia, sistemis mizidulobis potenciuri energia lo gaizrdeba, Semcirdeba Tu igive darCeba? vertikalur wreze mizidulobis potenciuri energia meore brunavs. masis ros m = 5 kg burTi am xo bolo Re bolozea sur. 8.34. amocanebi 8, 20 da 26 mimagrebuli. Rero θ = 30° kuTxiT gadaxares da man 0 = 0 sawyisi siCqariT daiwyo moZraoba. rodesac burTi udables wertilamde Camova, a) ra muSaobas sur. 8.32. amocanebi 5 da 11 •6. 1,5 kg masis burTi 12,5 m Seasrulebs masze mizidulobis Zala da b) simaRlis rogor Seicvleba burTi-dedamiwis sistemis kldidan isroles. burTis sawyisi siCqarea mizidulobis potenciuri energia? g) Tu 14 m/wm da horizontalidan zeviT 41°-iT udables wertilze mizidulobis potenciuri aris gadaxrili. a) ra muSaobas Seasrulebs energia nulis tolia, ra iqneba igive energia mizidulobis Zala burTze, vidre burTi burTis gaSvebis momentSi? d) θ kuTxis gazrdis miwaze daecema? b) ramdeniT Seicvleba burTi- SemTxvevaSi rogor Seicvleba pasuxebi a)-g) dedamiwis sistemis mizidulobis potenciuri kiTxvebze? 258 Tavi 8 paragrafi Senaxva 8.5. potenciuuri energia da energiis Senaxva meqanikuri energiis Tu b) gasrolis kuTxe horizontalidan qvemoT 41°-ia da g) burTis masaa 2,5 kg? d •9. a) risi tolia yinulis natexis siCqare Wiqis •15. fskerze me-3 amocanaSi? b) Tu WiqaSi orjer manZili a) 8.9a da b) 8.9g suraTze gamosaxul meti masis yinulis meore natexs CavagdebT, SemTxvevebSi? ra siCqare eqneba Camovardnil ra fskerze? kldeze mcocavs, rodesac Toki g) 8.9a suraTze g) Tu yinulis natexs raime sawyis siCqares da d) 8.9 g suraTze daWimvas iwyebs? e) 8.9a mivaniWebT, a)-s pasuxi gaizrdeba, Semcirdeba suraTisTvis aageT Tokis daWimvis manZilze Tu igive darCeba? SSM WWW kinetikuri •10. a) ra iqneba wignis siCqare xelSi daWerisas grafiki. v) daWimvis ra manZilze eqneba kldeze iqneba misi siCqare Wiqis risi tolia daWimvis energiis udidesi damokidebulebis me-2 amocanaSi? b) Tu orjer meti masis meore mcocavs maqsimaluri siCqare? wigns CamovagdebT, ra iqneba misi siCqare •16. 700 g masis bloki moZraobas iwyebs k = 400 n/m sixistis da umniSvnelo masis vertikaluri zambaridan aTvlili h0 simaR lidan. bloki zambaras ecema, 19 sm-iT kumSavs xelSi daWerisas? g) Tu wigns raime sawyisi siCqariT dabla visvriT, a)-s pasuxi gaizrdeba, Semcirdeba Tu igive darCeba? •11. ra siCqare eqneba manqanas a) A wertilSi, b) B wertilSi da g) C wertilSi me-5 amocanaSi? da wamierad Cerdeba. ra muSaobas asrulebs a) bloki zambaraze da b) zambara blokze? d) ra simaRleze ava manqana ukanasknel gorak g) risi tolia h0? d) Tu bloki 2h0 simaRlidan ze? e) Tu orjer meti masis meore manqanas Camovardeba, gavuSvebT, ra pasuxebs miviRebT a)-d) SekiT SeikumSeba zambara? xvebze? amocanaSi burTs, raTa mas umaRles wertilze •17. 5 g masis burTula zambaris saSualebiT vertikalurad maRla aisroles. zambaram 8 smiT unda SeikumSos, raTa burTma 20 m manZilze nulovani siCqare hqondes? ra iqneba am ganlagebul samiznes miaRwios. a) ramdeniT dros misi siCqare b) udables wertilze da icvleba burTula-dedamiwis sistemis mizi g) sawyisi wertilis mopirdapire wertilze? dulobis •12. a) ra sawyisi siCqare unda mivaniWoT me-4 ra maqsimaluri potenciuri energia manZiliT burTulis d) burTis masis gaormagebis SemTxvevaSi 20 m-iT maRla asvlis dros? b) ramdeniT rogor Seicvleba a)-g) pasuxebi? icvleba zambaris elastiuri potenciuri •13. 8.35 suraTze gamosaxul satvirTo manqanas muxruWebma umtyuna da misi siCqare 130 km/sT siCqares aRwevs, vidre manqana θ = 15° kuTxiT energia burTulis gasrolis dros? g) risi daxril, xaxunisgan Tavisufal avariul gzaze gava. satvirTo manqanis masaa 1,2×104 kg. a) ra umciresi L sigrZe unda hqondes avariul gzas, raTa manqana wamierad gaCerdes? b) satvirTo manqanis masis Semcirebis da g) misi siCqaris Semcirebis SemTxvevaSi minimaluri L sigrZe gaizrdeba, Semcirdeba Tu igive darCeba? SSM tolia zambaris sixiste? SSM •18. m = 2 kg masis bloki h = 40 sm simaRlidan Ca movarda da k = 1960 n/m sixistis zambaras daeca (sur. 8.36). gamoTvaleT zambaris SekumSvis maqsimaluri manZili. ••19. x RerZis gaswvriv moZrav nivTier wer sur. 8.36. amocana 18. tilze erTaderTi Semnaxveli  F = (6x – 12) n Zala moqmedebs, sadac x mocemulia metrebSi. sur. 8.35. amocana 13 •14. a) me-6 amocanaSi, energiebis saSualebiT gamoTvaleT burTis siCqare, rodesac burTi kldidan miwaze daecema. ra iqneba es siCqare, x=0 koordinatze, am ZalasTan dakavSirebuli U potenciuri energiaa 27 j. a) dawereT U-is x-ze damokidebulebis funqciis gamosaxuleba, sadac U joulebSia mocemuli, xolo x _ metrebSi. b) risi tolia maqsimaluri dadebiTi 259 potenciuri energia? x-is ra g) uaryofiTi magrebuli) manZilia 75 sm. rodesac Toki da d) dadebiTi mniSvnelobis dros iqneba horizontalur mdgomareobaSia, burTi moZ potenciuri energia nulis toli? raobas iwyebs da daStrixuli rkalis gaswvriv ••20. a) me-8 amocanaSi risi tolia burTis gadaadgildeba. ra iqneba burTis siCqare siCqare udables wertilze? b) masis gazrdis a) udables wertilze da b) umaRles wertilze, SemTxvevaSi siCqare izrdeba, mcirdeba Tu mas Semdeg rac wkiri Toks daiWers? ILW igive rCeba? ••21. me-7 amocanaSi risi tolia Q wertilSi ••24. t = 0 dros, 1 kg masis burTi maRali  koSkidan ν = (18 m/wm) + (24 m/wm) siCqariT is mdebare blokze moqmedi jamuri Zalis a) roles. risi tolia burTi-dedamiwis sistemis horizontaluri komponenti da b) vertikaluri komponenti? g) ra h simaRlidan ΔU, t = 0 drosa da t = 6 wm dros Soris? unda daiwyos moZraoba blokma, raTa rgolis ••25. 60 kg masis moTxilamure moZraobas gadasaxtomi adgilidan H = 20 m simaRleze wverze yofnis dros zedapirTan kontaqtis iwyebs (sur. 8.39). rodesac moTxilamure xteba, (kontaqtis misi siCqare horizontalisadmi θ = 28° kuTxes dakargvis zRvari niSnavs, rom zedapiridan adgens. haeris winaRoba ugulebelyaviT da blokze moqmedi drekadobis Zala nulis dauSviT, rom saTxilamuro trasaze xaxuni toli xdeba). d) aageT rgolis wverze yofnis ar arsebobs. a) ra maqsimalur h simaRleze dros blokze moqmedi drekadobis Zalis h axteba moTxilamure? b) Tu moTxilamure sawyis simaRleze damokidebulebis funqciis zurgCanTas moikidebs da, maSasadame, wonas grafiki (0-dan 6R-mde arealSi) intervalSi. moimatebs, h simaRle gaizrdeba, Semcirdeba ••22. Tu igive darCeba? dakargvis zRvarze 688 n tarzani 18 m mivides? wonis kldidan sigrZis lianas eWideba da qanaobs (sur. 8.37). qanaobis dros tarzani wveridan kldis 3,2 m-iT sur. 8.39. amocana 25. dabla eSveba. liana ••26. 8.34 suraTze naCvenebia L = 1,25 m sigrZis gawydeba, Tu masze qanqara. moqmedi Zala 950 n-s gadaaWarbebs. gawydeba Tu a) sur. 8.37. amocana 22. ara liana? b) Tu ar gawydeba, ra maqsimaluri Zala rodesac Toki vertikalisadmi θ0 = 40° kuTxes adgens, balansiris siCqarea ν0. ra siCqare aqvs balansirs ukiduresad dabal wertilze, Tu ν0 = 8 m/wm? ra umciresi ν0 mniSvnelobaa saWiro, raTa qanqara dabla moqmedebs wnelze qanaobis dros? Tu gawydeba, Camovides vertikalisadmi ra kuTxis dros moxdeba es? mdebareobas daubrundes da g) vertikalur ••23. 8.38 suraTze gamosaxuli Tokis sigrZea L = 120 sm. To- mdebareobas kis erTi bolo gazrdis SemTxvevaSi b) da g) SekiTxvebze fiqs irebulia, pasuxebi xolo meore bo Seicvleba? loze ••27. 2 kg masis qva umniSvnelo masis 4m sigrZis burTia m i m a g r e b u l i . f i q s i r e b u l i P bolodan P wer tilamde (sa dac wkiria da da Semdeg b) daubrundes, horizontalur rodesac Toki daWimulia? d) θ0 kuTxis ramdenime gradusiT Tokze gaizrdeba, qanaobs. Semcirdeba udables Tu wertilSi ar qvis siCqarea 8 m/wm. a) ra siCqare aqvs qvas, rodesac Toki vertikalisadmi 60° kuTxes adgens? sur. 8.38. amocanebi 23 da 64 b) vertikalisadmi ra maqsimaluri kuTxe 260 potenciuuri energia da energiis Senaxva Tavi 8 eqneba Toks qvis moZraobis ganmavlobaSi? mareobaSi dabrundeba (anu yuTsa da zambaras g) Tu qanqara-dedamiwis sistemis potenciuri Soris kontaqti gawydeba)? b) ra siCqare eqneba energia udables wertilze nulis tolia, ra yuTs daxrili sibrtyis boloSi? iqneba sistemis mTliani meqanikuri energia? ••31. m = 2 kg masis bloki θ = 30° ••28. 8.40 suraTze naCvenebia zambaraze dadebuli 8kg masis kuTxiT qva. xaxunisgan zambaras qva daxril, Tavi 10 sm-iT kumSavs. a) sufal sibrtyeze risi tolia zambaris damagrebul zamba sixiste? b) qva kidev rasTan dades (sur. 30 sm-iT dabla Cawies sur. 8.40. amocana 28. 8.43). k = 19,6 n/m sixistis zambara 20 sm-iT da Semdeg xeli gauSves. SekumSes da Semdeg xeli gauSves. a) ra iqneba ra iqneba SekumSuli zambaris elastiuri SekumSuli zambaris elastiuri potenciuri potenciuri energia? b) ramdeniT Seicvleba bloki-deda energia xelis gaSvebamde? sur. 8.43. amocana 31. g) ramdeniT Seicvleba qva-dedamiwis sistemis miwis mizidulobis potenciuri energia, rodesac energia qva gaSvebis adgilidan udides simaRles manZilze blokis moZraobisas? g) ra manZilia miaRwevs? d) risi tolia gaSvebis adgilidan blokis gaSvebis wertilsa da gadaadgilebis gazomili es udidesi simaRle? umaRles wertils Soris? ILW ••29. 8.41 suraTze gamosaxuli m = 12 kg masis ••32. 2 kg masis yuTi θ = 40° kuTxiT daxril, bloki xaxunisgan Tavisufal sibrtyeze Zevs da xaxunisgan sistemis mizidulobis gaSvebis potenciuri wertilidan maqsimalur Tavisufal, θ = 30° blokze kuTxiT k = 120 n/m sixistis zambarasTan aris da sibrtyeze obas daxril moZra iwyebs. rtyeze kavSirebuli (sur. Tokis 8.44). saSualebiT yuTi moZraobas sib iwyebs, rodesac zambara daWimuli ar aris. zambaraa davuSvaT blokis masa umniSvneloa da xaxuni damagrebuli, ro melsac 270 n Zala 2 sm-iT gadakiduli sur. 8.41. amocanebi 29 da 63. kumSavs. ar arsebobs. a) ra siCqare eqneba yuTs 10 smis gavlis Semdeg? b) ra manZils gaivlis yuTi daxril sibrtyeze, vidre wamierad gaCerdeba? bloki zambaras 5,5 sm manZiliT SekumSavs ra g) sidide da d) mimarTuleba eqneba yuTis da wamierad Cerdeba. a) ra manZili gaiara aCqarebas wamieri gaCerebis dros? blokma gaCerebamde? b) ra siCqare aqvs bloks zambarasTan Sejaxebis dros? SSM WWW ••30. 8.42 suraTze gamosaxuli k = 170 n/m sixistis zambara θ = 37° kuTxiT daxril, xaxunisgan Tavisufal sibrtyezea dama grebuli. zambaris bolodan sibrtyis bo lomde manZilia D = 1m (zambara moSvebul mdgomareobaSia). 2 kg masis yuTiT zambaras 0,2 m manZilze kum Saven da sur. 8.44. amocana 32. Semdeg ••33. 8.45a suraTze korpis sasroli Tofis yuTs xels uSveben. a) ra eqneba (sur. 8.45b) zambaris monacemebia asaxuli. siCqare masze naCvenebia drekadobis Zalis zambaris yuTs, SekumSvaze an daWimvaze damokidebulebis rodesac zambara grafiki. zambara 5,5 sm-iT ikumSeba da 3,8 g p i r v a n d e l (moSvebul) mdgo sur. 8.42. amocana 30. masis korps ubiZgebs. a) ra iqneba korpis 261 paragrafi 8.6. potenciuri energiis grafikis wakiTxva siCqare, rodesac zambara pirvandel m d g o m a r e o b a S i ••37. 8.48 suraTze naCvenebia U po d a b r u n d e b a ? b) davuSvaT kor pi zambaras tenciuri ewe giis beba da 1,5 sm-iT calkeveba deba. reobaze mox fiki. ra iqneba korpis •••34. gra sur. 8.48. amocana 37 nivTieri wertili x RerZis gaswvriv moZraobs da misi gaSvebi sas? damoki debulebis amjerad siCqare nivTieri wertilis x mdeba a) Wimavs, vidre gan ener b) masaa 0,9 kg (araSemnaxveli Zalebi ar moqmedebs). sur. 8.45. amocana 33. nivTieri wertili x = 4,5 m koordinatze bavSvebi 7 m/wm sawyisi siCqariT iwyebs moZraobs da x RerZis uaryofiTi mimarTulebiT gada adgildeba. a) Tu nivTieri wertili x = 1 m burTis srolisTvis magidaze damagrebul koordinats miaRwevs, ra iqneba misi siCqare zambaras iyeneben. magidis kididan samiznemde da, Tu am koordinats ver miaRwevs, ra iqneba horizontaluri manZilia D = 2,2 m (sur. 8.46). misi Semobrunebis wertili? ra b) sidide da ori bavSvi iatakze dadebuli patara yuTisTvis burTulis gartymas cdilobs. erT-erTi bavSvi zambaras 1,1 sm-iT kumSavs da g) mimarTuleba eqneba nivTieri wertilis burTula yuTis centrs 27 sm-iT cdeba. rogor siCqares, rodesac is x = 4 m koordinatidan unda SekumSos zambara meore bavSvma, raTa marcxniv ganagrZobs moZraobas? davuSvaT, burTula yuTis centrs moxvdes? dauSviT, nivTieri wertili x = 4,5 m koordinatidan rom zambarasa da burTze xaxuni ar moqmedebs. 7 m/wm siCqariT iwyebs moZraobas da x RerZis dadebiTi sur. 8.46. amocana 34. mimarTulebiT gadaadgildeba. d) Tu nivTieri wertili x = 7 m koordinats miaRwevs, ra iqneba misi siCqare da, Tu ver miaRwevs, ra iqneba misi Semobrunebis wertili? ra e) sidide da v) mimarTuleba eqneba nivTier wertilze moqmed Zalas, rodesac nivTieri wertili •••35. 25 sm sigrZis da 15 g masis Toki x = 5 m koordinatidan marjvniv ganagrZobs moZraobas? Werzea gakruli. mogvianebiT Toki Werze ••38. diatomuri molekulis (H2-is an O2-is vertikalurad magvari oratomiani molekulis) potenciuri Tokis kidia. mizidulobis rogor potenciuri icvleba energia energiaa: •••36. biWuna R = 13,8 m radiusis naxevarsfe A B – , r12 r 6 sadac r aris manZili molekulis or atoms ros formis yinulis gorakze zis da Semdeg Soris, xolo A da B dadebiTi mudmivebia. umniSvnelo sawyisi siCqariT dabla srialdeba potenciuri (sur. 8.47.). yinu SemakavSirebul Zalas ukavSirdeba. a) gamo mimarTulebis Secvlis dros? U= energia ori atomis erTad lze xaxunis Zala TvaleT wonasworuli dacileba anu atomebs ar moqmedebs. ra Soris manZili, rodesac TiToeul atomze simaRleze gaw moqmedi Zala nulis tolia. Tu atomebs Soris biWunasa manZili wonasworul dacilebaze b) naklebia da yinuls Soris da g) metia, es Zala ganizidavs (atomebs ydeba kontaqti? sur. 8.47. amocana 36. ganacalkevebs) Tu miizidavs (atomebs erT maneTisken ubiZgebs)? 262 Tavi 8 potenciuuri energia da energiis Senaxva •••39. x RerZis gaswvriv moZrav, 1kg masis nivTier wertilze erTaderTi F(x) Semnaxveli Zala moqmedebs. F(x)-Tan dakavSirebuli U(x) potenciuri energiaa: siTburi energia? g) ramdeniT izrdeba blokis kinetikuri energia? ••43. 3,57 kg masis bloks horizontalur zedapirze mudmivi siCqariT 4,06 m manZilze U(x) = –4xe–x/4 j , Tokis saSualebiT gadaaadgileben. Tokidan sadac x metrebSia mocemulia. x = 5 m koor blokze 7,68 n sididis da horizontalidan kinetikuri zemoT 15°-iT gadaxrili Zala moqmedebs. energiaa 2 j. a) risi tolia sistemis meqanikuri a) ra muSaobas asrulebs Tokidan moqmedi Zala, energia? b) aageT U(x)-is x-ze damokidebulebis b) ramdeniT izrdeba bloki-zedapiris sistemis funqciis grafiki, rodesac 0 ≤ x ≤ 10 m da siTburi energia da g) risi tolia bloksa da igive grafikze sistemis meqanikuri energiis zedapirs Soris arsebuli kinetikuri xaxunis dinatze nivTieri wertilis amsaxveli wrfe gaavleT. amocanis b) SekiTxvis koeficienti? SSM gamoyenebiT gamoTvaleT g) x-is umciresi da paragrafi 8.8. energiis Senaxva d) udidesi mniSvneloba, romelsac nivTieri •44. 60 kg masis moTxilamure tramplinidan 24 m/wm siCqariT da horizontalidan maRla 25° kuTxiT xteba. haeris wevis Sedegad miwaze daxtomisas moTxilamuris siCqarea 22 m/wm da tramplinidan vertikaluri manZilia 14 m. wertili miaRwevs. amocanis b) SekiTxvis gamoyenebiT gansazRvreT e) nivTieri wertilis maqsimaluri kinetikuri energia da v) x-is Sesabamisi mniSvneloba. z) miiReT F(x)-is xze damokidebulebis funqciis gamosaxuleba. T) x-is ra sasruli mniSvnelobisTvis iqneba F(x)=0? ramdeniT mcirdeba moTxilamure-dedamiwis sistemis meqanikuri energia gadaxtomidan miwaze daxtomamde? paragrafi 8.7. gare Zalis mier sistemaze Sesrulebuli muSaoba •45. 25 kg masis daTvi fiWvis xeze 12 m •40. ZaRli dasaZinebel yuTze 8 n hori misi siCqarea 5,6 m/wm. a) rogor icvleba daTvi- zontaluri ZaliT moqmedebs da iatakze dedamiwis sistemis mizidulobis potenciuri acurebs. yuTze moqmedi kinetikuri xaxunis energia Camosrialebis dros? b) risi tolia Zalaa 5 n. 0,7 m-iT gadaadgilebis Semdeg risi daTvis kinetikuri energia miwaze dacemamde? tolia a) Zalis mier yuTze Sesrulebuli g) ra saSualo xaxunis Zala moqmedebs daTvze? muSaoba da b) yuTisa da iatakis siTburi SSM ILW energiis nazrdi? gadaadgilebs. Tu bloksa da zedapirs Soris •46. 75 g masis burTula dedamiwis zedapiridan 1,1 m simaRlidan 12 m/wm siCqariT aisroles. 2,1 m simaRleze asvlis dros burTulis siCqare 10,5 m/wm-ia. ramdeniT Semcirdeba burTula-dedamiwis sistemis Emeq energia hae kinetikuri xaxunis koeficientia 0,2, risi ris wevis Sedegad? tolia a) muSis mier blokze Sesrulebuli •47. 81,8 •41. muSa 27 kg masis blokze horizontalidan qvemoT 32°-iT daxrili ZaliT moqmedebs da bloks 9,2 m manZilze mudmivi siCqariT muSaoba da b) bloki-zedapiris sistemis simaRlidan srialdeba da miwaze dacemamde autfilderi mili/sT beisbolis siCqariT isvris. burTs rodesac siTburi energiis nazrdi? infilderi ••42. 35 n sididis horizontaluri Zala 4 kg masis bloks zedapirze gadaaadgilebs, sadac kinetikuri xaxunis koeficientia 0,6. 100 futi/wm-ia. a) ra muSaobas asrulebs bloki-zedapiris bolis burTis wonaa 9 uncia) sistemaze modebuli Zala, rodesac bloki 3 m •48. mewyeris dros 520 kg masis klde 500 m sigrZis da 300 m simaRlis ferdobidan cur manZilze gadaadgildeba? b) gadaadgilebis dros blokis izrdeba. siTburi ramdeniT energia izrdeba 40 j-iT zedapiris Semcirda burTs iWers, ramdeni burTi-dedamiwis milis siCqare futi/funtiT sistemis meqa nikuri energia haeris wevis Sedegad? (beis deba. kldesa da ferdobis zedapirs Soris kinetikuri xaxunis koeficientia 0,25. a) Tu 263 ferdobis wverze klde-dedamiwis sistemis moZraobs, romlis kinetikuri xaxunis koe U mizidulobis potenciuri energia nulis ficientia μk = 0,25. D = 7,8 m manZilis gavlis tolia, risi tolia U kldis Camocurebamde? Semdeg bloks xaxunis Zala aCerebs. risi tolia b) ra energia gadadis siTbur energiaSi kldis a) bloki-zedapiris sistemis siTburi energiis Camocurebis dros? g) ra kinetikuri energia nazrdi, b) blokis maqsimaluri kinetikuri aqvs kldes, rodesac ferdobis fskerze energia da g) zambaris SekumSvis manZili? SSM Cavardeba? d) ra siCqare aqvs am dros kldes? WWW •49. 8.49 suraTze ••52. gamosaxuli blo zambaras awvebian da zambara 15sm-iT ikumSeba. ki ze Semdeg bloks xels uSveben da is magidaze dapirze curdeba. curdeba. 75sm-is gavlis Semdeg bloki Cerdeba. daxril A wertilidan B wertilamde 5m sur. 8.49. amocanebi 49 2kg masis blokiT horizontalur zambaris sixistea 200n/m. risi tolia bloksa da magidas Soris arsebuli kinetikuri xaxunis da 67  blokze 2 n sididis qveviT mimarTuli F Zala koeficienti? moqmedebs. blokze moqmedi xaxunis Zalis zontaluri zambaris moqmedebiT horizon manZilis gavlisas ••53. funTuSa k = 400 n/m sixistis hori sididea 10 n. Tu A da B wertilebs Soris talur zedapirze curdeba. zambaris meore blokis kinetikuri energia 35 j-iT izrdeba, bolo damagrebulia. zambaris wonasworuli ra muSaobas asrulebs mizidulobis Zala mdebareobis blokze? kinetikuri energiaa 20 j. srialis dros ••50. 8.50 suraTze Sualeduri simaRlidan zedapiri ufro xaxunisgan did dros funTuSis bloki funTuSaze 10 n xaxunis Zala moqmedebs. a) gavliT erTi ra manZils gaivlis funTuSa wonasworuli simaRleze adis. mdebareobidan, vidre wamierad gaCerdeba? vidre b) ra iqneba funTuSas kinetikuri energia gamosaxuli CaRrmavebis gavlis Tavisufalia, bloki ufro maRal simaRleze ava. d manZilis wonasworul mdebareobaze dabrunebisas? gavlis Semdeg bloks xaxunis Zala aCerebs. blokis sawyisi siCqarea 6 m/wm, simaRleebs ••54. 30°-iT daxril sibrtyeze zeviT moZravi 4 kg masis yuTis kinetikuri energiaa 128 j. ra Soris sxvaobaa h = 1,1 m da μk = 0,6. gamoTva manZils gaivlis yuTi, Tu yuTsa da sibrtyes leT d. Soris kinetikuri xaxunis koeficientia 0,3? ••55. 267 n wonis bavSvi 6,1 m simaRlis da horizontalisadmi 20° kuTxiT daxril sa srialoze srialdeba. bavSvsa da sasrialos Soris kinetikuri xaxunis koeficientia 0,1. a) ra energia gadadis siTbur energiaSi? b) Tu bavSvis sawyisi siCqarea 0,457 m/wm, ra iqneba sur. 8.50. amocana 50. misi siCqare Camosrialebis Semdeg? ••51. 8.51 suraTze gamosaxuli 3,5 kg masis bloki 640 n/m sixistis SekumSuli zambaris zemoqmedebiT Cqardeba. zambaris gasworebis ••56. qila 40°-iT daxril sibrtyeze maRla adis. 55 sm manZilis (sibrtyis gaswvriv) gavlis Semdeg qilis siCqarea 1,4 m/wm. qilasa Semdeg da bloki horizontalur zedapirze sibrtyes koeficientia Soris kinetikuri xaxunis 0,15. a) kidev ra manZils gaivlis qila daxril sibrtyeze? b) ra siCqare eqneba qilas ukan Camosrialebis Semdeg? g) kinetikuri xaxunis koeficientis Semcirebis SemTxvevaSi a) da b) pasuxebi gaizrdeba, sur. 8.51. amocana 51 Semcirdeba Tu ar Seicvleba? 264 potenciuuri energia da energiis Senaxva Tavi 8 ••57. 8.52 suraTze gamosaxuli m = 2,5 kg masis bloki k = 320 n/m sixis sur. 8.52. amocanebi 57 da 82. tis zambarisken monakveTze moxvdeba, sadac kinetikuri xaxunis koeficientia kvlav 0,5-ia. Semdeg bloki xaxunisgan Tavisufal aRmarTze adis da wamierad Cerdeba. sad gaCerdeba bloki? Tu bloki swor monakveTze Cerdeba, moZraobs. bBlokis gaCerebis momentSi zambara gansazRvreT romel maTganze da gamoTvaleT 7,5 sm-iT aris SekumSuli. bloksa da zedapirs Soris kinetikuri xaxunis koeficientia 0,25. swori monakveTis marcxena kididan gaCerebis zambarasTan Sexebidan blokis gaCerebamde adis, maSin swori monakveTidan gaCerebis a) ra muSaobas asrulebs drekadobis Zala adgilamde H simaRle gamoTvaleT. da b) risi tolia bloki-zedapiris sistemis •••61. adgilamde L manZili. Tu bloki aRmarTze 8.55 suraTze gamosaxuli bloki siTburi energiis nazrdi? g) risi tolia xaxunisgan Tavisufal zedapirze moZraobs da blokis siCqare zambarasTan Sejaxebisas? ILW ••58. 5,29 n wonis qva 20 m/wm siCqariT dedamiwis L = 0,75 m sigrZis monakveTs gadis, romelic θ = 30° kuTxiT daxrili zedapiris h = 2 m si donidan maRla aisroles. frenis dros qvaze maRleze iwyeba. am monakveTze kinetikuri 0,265 n sididis haeris weva moqmedebs. a) ra xaxunis koeficientia 0,4. A wertils bloki maqsimalur simaRleze ava qva da b) ra iqneba 8 m/wm siCqariT gaivlis. Tu bloki B wertils misi siCqare miwaze dacemis momentSi? miaRwevs, ra iqneba misi siCqare am dros da Tu •••59. ver miaRwevs, A-dan ra maqsimalur simaRleze nivTieri ava bloki? wertili 8.53 su raTze gamosaxuli formis zedapirze gadaadgildeba. ze dapiris sur. 8.53. amocana 59. brtyeli monakveTis sigrZea L = 40 sm. zedapiris mrud monakveTebze xaxuni ar moqmedebs, xolo brtyel monakveTze koeficientia μk = 0,2. kinetikuri nivTieri moZraobas A wertilidan iwyebs, romelic h = L/2 simaRleze mdebareobs. sur. 8.55. amocana 61. xaxunis wertili zedapiris •••62. 8.56 suraTze gamosaxuli 1800 kg masis liftis kabinis kabeli maSin wydeba, rodesac kabina pirvel sarTulze uZravad aris brtyeli monakveTis marcxena kididan ra gaCerebuli. am dros kabinasa da k = 0,15 mn/m manZilze gaCerdeba nivTieri wertili? sixistis zambaras Soris manZilia d = 3,7 m. •••60. 8.54 suraTze gamosaxuli bloki usafrTxoebis mowyobilobis momWerebi moZraobas d = 40 sm simaRlidan iwyebs da kabinaze moqmedeben da 4,4 kn sididis mudmivi igive d sigrZis swor monakveTze Cadis, sadac xaxunis Zala kabinis moZraobas ewinaaRmdegeba. kinetikuri xaxunis koeficientia 0,5. Tu a) gamoTvaleT kabinis siCqare, rodesac is bloki moZraobas ganagrZobs, is d/2 simaRlis zambaras meore daRmarTze Cava da d/2 sigrZis swor b) gamoTvaleT zam Seejaxeba. baris SekumSvis maq simaluri x manZili (SekumSvis dros xaxunis debas Zala moqme ganagrZobs). g) gamoTvaleT man Zili, romliTac ka bina sur. 8.54. amocana 60. zambarasTan Sejaxebis Semdeg sur. 8.56. amocana 62. 265 maRla avardeba. d) energiis Senaxvis kanonis da eskalatorTan gamoyenebiT erTad gamoTvaleT kabinis mier gadaad gavlili mTeli manZili gaCerebamde (dauSviT, gildeba. Eskala rom gaCerebul kabinaze moqmedi xaxunis Zala torze yuTis umniSvneloa). gaCerebamde •••63. 8.41 suraTze gamosaxuli m = 3,2 kg masis bloki θ = 30° kuTxiT daxril zedapirze d manZilze srialdeba da 431 n/m sixistis nakveTSi zambaras ejaxeba. rodesac bloki wamierad yuTs, b) ra sididis kinetikuri xaxunis Zala Cerdeba, zambara 21 sm-iT ikumSeba. risi tolia moqmedebs yuTze da g) ra energia miewodeba a) d manZili da b) blokisa da zambaris Sexebis eskalatoris pirvel wertilsa da im adgils Soris manZili, gansxvavdeba erTmaneTisgan a) da g) pasuxebi. sadac blokis siCqare udidesia? 67. 8.49 suraTze gamosaxuli bloki xaxunisgan 64. 8.38 suraTze gamosaxuli L = 120 sm sigrZis Tokis erT boloze burTia mimagrebuli, P Tokis meore bolo fiqsirebulia. wertilze wkiria damagrebuli. burTi ra sur. 8.58. amocana 66 kinetikuri ener gia miewodeba Tavisufal damatebiTi amocanebi xolo a) mo Zravas? daxril d) axseniT sibrtyeze ratom moZraobs. A da B wertilebze misi siCqarea 2 m/wm da 2,6 m/wm. meored Camosrialebis dros A wer tilze blokis siCqarea 4 m/wm. ra iqneba am jerad misi siCqare B wertilze? da Semdeg burTi wkiris garSemo ganagrZobs 68. 30 g masis tyvia 500 m/wm siCqariT moZraobs da 12 sm-iT daSorebul kedels ejaxeba. moZraobas. wkirs a) ramdeniT icvleba tyviis meqanikuri ener- mTlianad Semouaros, ra mniSvnelobas unda gia? b) ra saSualo Zala moqmedebs kedlidan aWarbebdes d? tyviaze? 65. sasrialos 12 m radiusis wriuli rkalis 69. 0,092 kg masis burTi 0,62 m sigrZis da forma umniSvnelo moZraobas iwyebs, vidre wkiri Toks daiWers imisTvis, aqvs. misi rom burTma maqsimaluri simaRlea masis Zelakis erT bolozea h = 4 m da dedamiwa wres exeba (sur. 8.57). 25kg damagrebuli, xolo Reros meore bolo sa masis bavSvi sasrialoze jdeba da miwaze yrden wertilzea fiqsirebuli. amis Semdeg Camosvlisas misi siCqarea 6,2 m/wm. a) ra Rero vertikalurad gaaswores da sayrdeni manZilze srialdeba bavSvi? b) ra saSualo wertilis garSemo abruneben. rodesac bur xaxunis Zala moqmedebs bavSvze Camosrialebis Ti udables wertilzea, ra iqneba a) misi dros? Tu dedamiwis nacvlad wres sasrialos siCqare da b) Reros daWimuloba? Rero wverze gavlebuli vertikaluri xazi exeba, amjerad risi tolia g) sasrialos sigrZe da d) bavSvze kvlav sayrdeni wertilis garSemo abruneben. moqmedi saSualo xaxunis Zala? g) vertikalisadmi ra kuTxis dros iqneba horizontalurad gaaswores da Reros daWimuloba burTis wonis toli? d) Tu burTis masas gavzrdiT, g)-s pasuxi gaizrdeba, Semcirdeba Tu ar Seicvleba? 70. 8.59 suraTze gamosaxuli jaWvi xaxunisgan Tavisufal magidaze Zevs da misi erTi meoTxedi nawili magidis kididan aris gadakidebuli. sur. 8.57. amocana 65. 66. qarxanaSi 300 kg masis yuTebi 1,2 m/wm siCqariT moZrav eskalatorze vertikalurad iyreba (sur. 8.58). eskalatorsa da TiToeul yuTs Soris kinetikuri xaxunis koeficientia 0,4. mcire drois Semdeg yuTi aRar srialebs Tu jaWvis sigrZea L = 28 sm da masaa m = 0,012 kg, ra mu Saobaa saWiro ga dakidebuli nawi lis magidaze asa tanad? sur. 8.59. amocana 70. 266 Tavi 8 potenciuuri energia da energiis Senaxva 71. x RerZis gaswvriv moZrav 2 kg masis nivTieri wertilze F(x) Semnaxveli Zala moqmedebs. F(x)-Tan dakavSirebuli U(x) potenciuri kunTebs daZabavs da zemoT xteba. am dros energiis grafiki 8.60 suraTzea naCvenebia. gadadis. axtomis dros tkacanis xma ismis x=2m koordinatze nivTieri wertilis siCqarea –1,5 m/wm. ra a) sidide da b) mimar Tuleba aqvs F(x) Zalas am koordinatze? romel koordinatebs Soris moZraobs 74. amotrialebuli tkacuna xoWo zurgis kunTebSi Senaxuli energia meqanikur energiaSi da swored aqedan modis xoWos dasaxelebac. m = 4×10–6kg masis xoWo 0,77 mm manZilze aiwia da h = 0,3 m videoCanawerma aCvena, rom nivTieri simaRleze axta. maRla awevis dros a) ra wertili g) marcxnisken da d) marjvnisken? saSualo sididis gare Zala moqmedebs xoWos e) ra siCqare aqvs nivTier wertils x = 7 m koordinatze? zurgidan zedapirze da b) risi tolia xoWos aCqareba g-s erTeulebSi? 75. avtomobilis moZraobas siCqarisgan damoukidebeli sagzao xaxuni da siCqaris kvadratis proporciuli haeris weva ewina aRmdegeba. 12000 n wonis avtomobilisTvis jamuri F winaRobis Zalaa F = 300 + 1,8ν2, sadac F mocemulia niutonebSi, xolo ν - m/wmebSi. gamoTvaleT 80 km/sT siCqariT moZravi manqanis 0,92 m/wm2 aCqarebiT asamoZraveblad saWiro simZlavre (cxenis ZalebSi). sur. 8.60. amocana 71. 72. aRmoCnda, rom garkveuli saxeobis zambara hukis kanons ar emorCileba. x manZiliT daWimvis dros zambara 52,8x + 38,4x2 ZaliT moqmedebs, romelic daWimvis mimarTulebis sawinaaRmdegoa. a) gamoTvaleT x = 0,5 m koor dinatidan x = 1 m koordinatamde zambaris daWimvisTvis saWiro muSaoba. b) zambaris erTi bolo fiqsirebulia, xolo meore boloze 2,17 kg masis nivTieri wertilia mimagrebuli. zambara x = 1 m manZiliTaa daWimuli. Tu nivTieri wertili moZraobas iwyebs, ra siCqare eqneba mas, rodesac zambara x = 0,5 m manZili Taa daWimuli? g) drekadobis Zala Semnaxvelia Tu araSemnaxveli? pasuxi daasabuTeT. 73. qarxnis muSam 3,7 m sigrZis da hori zontalisadmi 39°-iT daxrili sibrtyidan 180 kg masis yuTs SemTxveviT xeli gauSva. yuTsa da daxril sibrtyes Soris, aseve yuTsa da qarxnis horizontalur iataks Soris 76. 300 g masis burTi 1,4 m sigrZis da umniS vnelo masis Tokze Camokides, xolo Tokis meore bolo daafiqsires. Semdeg burTi erT mxares gawies, vidre Toki vertikalisadmi 30° kuTxes Seadgenda da xeli gauSves. gamo TvaleT a) burTis siCqare, rodesac Toki vertikalisadmi 200 kuTxes adgens da b) burTis maqsimaluri siCqare. g) ra kuTxe iqneba Toksa da vertikals Soris, rodesac burTis siCqare maqsimaluris erTi mesamedi gaxdeba? 77. 8.61 suraTze gamosaxuli bloki A wertilis gavliT 7 m/wm siCqariT moZraobs. blokis traeqtoriaze xaxuni ar moqmedebs, vidre is L = 12 m sigrZis monakveTs miaRwevs, sadac kinetikuri xaxunis koeficientia 0,7. naCvenebi simaRleebia h1 = 6 m da h2 = 2 m. ra siCqare eqneba bloks a) B wertilSi da b) C wertilSi? g) miaRwevs Tu ara bloki D wertilamde? Tu miaRwevs, ra iqneba misi siCqare am wertilSi? kinetikuri xaxunis koeficientia 0,28. a) ra siCqariT moZraobs yuTi daxrili sibrtyidan Camosvlisas? b) ra manZilze gacurdeba yuTi horizontalur iatakze? (dauSviT, rom da xrili sibrtyidan iatakze gadasvlisas yuTis kinetikuri energia ar icvleba). g) yuTis masis ganaxevrebis SemTxvevaSi a) da b) pasuxebi gaizrdeba, Semcirdeba Tu ar Seicvleba? sur. 8.61. amocana 77. 267 Tu ver miaRwevs, L sigrZis monakveTis ra gadaadgildeba, ra iqneba misi a) K da b) U manZils gaivlis bloki? x = 2 m koordinatze da g) K da d) U 78. nivTieri wertili x RerZis gaswvriv Tavdapirvelad x = 1 m koordinatidan x = 4 m koordinatze? Tu nivTieri wertili x RerZis koordinatisken gadaadgildeba da Semdeg x=1m koordinatze brundeba. am dros masze gare Zala moqmedebs. es Zala x RerZis gaswvrivaa mimarTuli da misi x komponentis x=0 dadebiTi mimarTulebiT gadaaadgildeba, ra iqneba misi e) K da v) U x = 11 m koordinatze, z) K da T) U x = 12 m koordinatze da i) K da k) U x = 13 m koordinatze? l) aageT U(x)-is xze damokidebulebis grafikis funqcia x-is 0- mniSvnelobebi x = 4 m koordinatisken gadaad dan 13 m-mde arealisTvis. gilebisas dabrunebisas nivTieri wertili moZraobas x = 0 koor gansxvavebulebia. cxrilSi mocemulia oTxi dinatidan iwyebs. ra iqneba misi m) kinetikuri SesaZlo varianti, sadac x metrebSia mocemuli; energia x = 5 m koordinatze da n) maqsimaluri xolo Zalis X komponentis mniSvneloba – dadebiTi xmax koordinati? o) rogor moiqceva da Semdeg ukan nivTieri wertili xmax-ze misvlis Semdeg? niutonebSi: a) b) g) d) x=4m koordinatisken gadaadgilebisas ukan dabrunebisas +3 +5 +2x +3x2 –3 +5 –2x +3x2 gamoTvaleT gare Zalis mier nivTier wertilze Sesrulebuli muSaoba mTeli traeqtoriis ganmavlobaSi (oTxive SemTxvevisTvis). areali 0-dan 2 m-mde 2 m-dan 3 m-mde 3 m-dan 8 m-mde 8 m-dan 11 m-mde 11 m-dan 12 m-mde 12 m-dan 15 m-mde Zala  F1 = (+3 n) F2 = (+5 n) F=0  F3 = –(4 n) F4 = –(1 n) F=0 81. me-80 amocanaSi mocemuli ZalTa ganlagebis dros, 2kg masis nivTieri wertili moZraobas 79. 2,5 kg masis bidoni 4 m simaRlidan 3 m/wm x=5m koordinatidan 3,45m/wm sawyisi siCqariT iwyebs da x RerZis uaryofiTi mimarTulebiT gadaadgildeba. a) Tu nivTieri wertili x=0 sawyisi siCqariT dabla isroles. bidonze koordinatamde miva, ra iqneba misi siCqare moqmedi haeris wevis Zala umniSvneloa. ra da, Tu ver miva, ra iqneba misi Semobrunebis kinetikuri energia eqneba bidons a) miwaze wertili? dacemisas da b) naxevari manZilis gavlisas? moZraobas x = 5 m koordinatidan 3,45 m/wm ra iqneba g) bidonis kinetikuri energia da sawyisi siCqariT iwyebs da x RerZis dadebiTi d) bidoni-dedamiwis sistemis mizidulobis mimarTulebiT gadaadgildeba. b) Tu nivTieri potenciuri energia bidonis miwaze dacemamde wertili x = 13 m koordinats miaRwevs, ra e) romel SemTxvevaSia gare Zala Semnaxveli? 2 wamiT adre? d) SekiTxvaze pasuxis gacemisas y = 0 aTvlis wertili dedamiwas daamTxvieT. davuSvaT, nivTieri wertili iqneba misi siCqare da, Tu ver miaRwevs, ra iqneba misi Semobrunebis wertili? 80. nivTieri wertili x RerZis gaswvriv 82. 8.52 suraTze gamosaxuli 30 j kinetikuri gadaadgildeba energiis mqone bloki moSvebul zambaras da moZraobisas masze sxvadasxva sididis Semnaxveli Zala moqmedebs (sur. 8.62 da TandarTuli cxrili). nivTieri wertili moZraobas x = 5 m koordinatidan iwyebs, sadac misi kinetikuri energiaa K = 14 j, xolo potenciuri energiaa U = 0. Tu nivTieri wertili x RerZis uaryofiTi mimarTulebiT sur. 8.62. amocanebi 80 da 81 sur. 8.63. amocana 82. 268 Tavi 8 potenciuuri energia da energiis Senaxva ejaxeba. zambaris SekumSvis Semdeg bloksa da maRla asvlis dros? g) ramdeniT icvleba zedapirs Soris xaxunis Zala moqmedebs. 8.63 buSti-dedamiwis suraTze naCvenebia blokis K(x) kinetikuri potenciuri energia maRla asvlis dros? d) energiis da zambaris U(x) potenciuri energiis Tu buStis gaSvebis adgilze mizidulobis x mdebareobaze damokidebulebis potenciuri energia nulis tolia, ra iqneba blokis funqciebis blokisa da grafikebi. zedapiris sistemis mizidulobis ramdeniT izrdeba igive energia buStis maqsimalur simaRleze siTburi energia, asvlis dros? e) Tu maqsimalur simaRleze rodesac a) bloki x = 0,1m koordinats miaRwevs mizidulobis da b) zambara maqsimalurad SeikumSeba? tolia, ra iqneba is buStis gaSvebis adgilze? 83. 1500 kg masis manqana 5°-iT daxril gzaze 30 km/sT siCqariT iwyebs moZraobas. Zrava v) risi tolia maqsimaluri simaRle? gamorTulia da manqanaze mxolod jamuri xaxunis Zala da mizidulobis Zala moqmedebs. potenciuri energia nulis 88. ferdobis kididan 0,55 kg masis sagani 1550 j sawyisi kinetikuri energiiT isroles. sagnis maqsimaluri gadaadgilebaa +140 m. 50 m-is gavlis Semdeg manqanis siCqarea 40 km/sT. a) ramdeniT Semcirda manqanis meqa risi tolia sagnis siCqaris a) horizontaluri nikuri energia jamuri xaxunis Zalis Sedegad? siCqaris vertikaluri komponentia 65m/wm, b) ra sididisaa jamuri xaxunis Zala? risi toli iqneba vertikaluri gadaadgileba 84. 1500 kg masis manqana horizontalur gzaze moZraobas da 30 wamis ganmavlobaSi 72 km/sT gasrolis adgilidan? 89. 8.64 suraTze siCqares anviTarebs. a) ra iqneba manqanis gamosaxuli blokis kinetikuri energia 30 wamis Semdeg? b) ra masa, saSualo simZlavre sWirdeba manqanas 30 wamis Tokia gadakiduli, ganmavlobaSi? g) risi tolia myisi simZlavre umniSvneloa. 30 wamis Semdeg, Tu davuSvebT, rom aCqareba blokze da daxril mudmivia? sibrtyeze xaxunis 85. 1,5 mgvt simZlavris orTqlmavali mata rebels 6 wuTis ganmavlobaSi 10 m/wm-dan 25 m/wm-mde aCqarebs. a) gamoTvaleT mata Zala ar moqmedebs. A yuTis masaa 1 kg, B yuTis masaa 2 kg, xolo θ kuTxea 30°. yuTebi moZraobas reblis masa. daadgineT b) matareblis si Toki daWimulia. ra iqneba yuTebis jamuri da b) vertikaluri komponenti? g) rodesac romelzec maSin iwyeben, sur. 8.64. amocana 89. rodesac damakavSirebeli Cqaris da b) matareblis amaCqarebeli Zalis kinetikuri energia, rodesac B yuTi 25sm-iT droze damokidebulebis funqciebi 6 wuTian dabla daiwevs? SualedSi. d) gamoTvaleT matareblis mier 90. 1,5 kg masis burTi horizontalisadmi 34°iT kuTxiT da 20 m/wm sawyisi siCqariT maRla ganvlili manZili am droSi. 86. 5 kg masis bloki horizontalisadmi 30°-iT daxril sibrtyeze 5 m/wm siCqariT maRla adis. ra manZils gaivlis sibrtyeze bloki, Tu a) xaxuni ar arsebobs da b) bloksa da sibrtyes Soris kinetikuri xaxunis koeficientia 0,4? g) am ukanasknel SemTxvevaSi ramdeniT gaiz rdeba blokisa da sibrtyis siTburi energia? d) Tu bloki ukan srialdeba, ra iqneba misi siCqare im adgilze, saidanac moZraoba daiwyo? 87. 1,5 kg masis wylis buSti 3 m/wm sawyisi siCqariT maRla aisroles. a) ra kinetikuri energia aqvs buSts gaSvebis momentSi? b) ra muSaobas asrulebs mizidulobis Zala buStze aisroles. a) ra sawyisi kinetikuri energia aqvs burTs? b) ramdeniT Seicvleba burTidedamiwis sistemis mizidulobis potenciuri energia, vidre burTi maqsimalur simaRles miaRwevs? g) ra maqsimalur simaRleze ava burTi? 91. sacirko warmodgenis dros 60 kg masis klouni zarbaznidan 16 m/wm sawyisi siCqariT da horizontalisadmi isroles. mcire drois ucnobi kuTxiT Semdeg klouni gasrolis adgilidan 3,9 m simaRleze gabmul badeSi aRmoCnda. haeris weva ugulvebelyaviT da badeSi moxvedris momentSi kinetikuri energia gamoTvaleT. klounis  92. nivTier wertilze mxolod F Semnaxveli Zala moqmedebs. rodesac nivTieri wertili A wertilze imyofeba, F -isa da nivTieri 269 96. vulkanidan amofrqveuli ferflis nakadi horizontalur zedapirze miiwevs da 10°-iT daxril aRmarTs ejaxeba. nakadis wina nawili dakavSirebuli aRmarTze 920 m manZils gaivlis da Cerdeba. potenciuri energiaa 40 j. rodesac nivTieri  davuSvaT nakadSi moqceuli airebi nakads wertilis sistemasTan wertili A-dan B-sken gadaadgildeba, masze F Zala +25 j muSaobas asrulebs. ra potenciuri energia aqvs sistemas, rodesac nivTieri wertili B wertilze imyofeba? 93. mudmivi, horizontaluri Zala 50 kg masis satvirTo manqanas 30°-iT daxril sibrtyeze mudmivi siCqariT 6 m manZilze gadaaadgilebs. manqanasa da sibrtyes Soris kinetikuri xaxunis koeficientia 0,2. a) ra muSaobas asrulebs modebuli Zala da b) ramdeniT izrdeba manqanisa da sibrtyis siTburi energia? 94. horizontalur zedapirze 20 kg masis bloki k = 4 kn/m sixistis horizontalur zambarazea mimagrebuli. bloks marjvnisken sweven da zambara 10 sm-iT iWimeba, ris Semdegac bloks xels uSveben. bloksa da maRla weven da miwidan moqmedi xaxunis Zala umniSvneloa. aseve davuSvaT, rom nakadis wina nawilis meqanikuri energia inaxeba. ra sawyisi siCqariT daiwyo moZraoba nakadis wina nawilma? 97. 0,5 kg masis banani 4 m/wm siCqariT maRla aisroles da 0,8 m maqsimalur simaRles miaRwia. ramdeniT Secvlis banani-dedamiwis sistemis meqanikur energias haeris wevis Zala bananis maRla asvlis dros? 98. 70 kg masis beisbolis moTamaSe 10 m/wm sawyisi siCqariT curdeba da miwaze ecema. a) ramdeniT mcirdeba moTamaSis kinetikuri energia da b) ramdeniT izrdeba miwisa da moTamaSis sxeulis siTburi energia? k = 200 n/m zedapirs Soris 80 n sididis xaxunis Zala 99. moqmedebs. a) ra iqneba blokis kinetikuri sixistis energia 2 sm-iT gadaadgilebis Semdeg? b) ra θ = 40° kuTxiT da iqneba blokis kinetikuri energia, rodesac xrili sibrtyis bloki sawyis mdebareobas daubrundeba? g) ra wverzea damagre maqsimalur kinetikur energias SeiZens bloki buli mTeli gadaadgilebis ganmavlobaSi? sibrtyeze 95. M = 2 kg da 2M ar moqmedebs. 1 kg zambara (sur. 8.66). xaxuni sur. 8.66. amocana 99. yuTi masis bloki zambaridan d = 0,6 m manZilze 16 j k = 200 n/m sixistis kinetikuri energiiT sibrtyeze maRla asvlas zambarasTan iwyebs. a) ra kinetikuri energia eqneba bloks, masis ori aris dakavSirebuli. zam rodesac zambaras 0,2 m-iT SekumSavs? b) ra baris bolo kinetikuri energiiT unda daiwyos moZraoba fiqsirebulia (sur. blokma, raTa is maSin gaCerdes, rodesac 8.65). talur erTi horizon sur. 8.65. amocana 95. zedapirze zambaras 0,4 m-iT SekumSavs? 100. manqana 40 kg masis tvirTs 40°-iT da blokze xaxuni ar moqmedebs da misi masa daxril gzaze mudmivi siCqariT 2 m manZilze umniSvneloa. yuTi moZraobas maSin iwyebs, gadaaadgilebs da manqanidan tvirTze moqmedi rodesac zambara moSvebul mdgomareobaSia. Zala gzis paraleluria. tvirTsa da gzas a) ra jamuri kinetikuri energia eqneba orive Soris kinetikuri xaxunis koeficientia 0,4. yuTs, rodesc Camokidebuli yuTi 0,09 m-iT a) ra muSaobas asrulebs manqana tvirTze da dabla daiwevs? b) ra kinetikuri energia eqneba b) ramdeniT izrdeba tvirTisa da gzis siTburi Camokidebul yuTs, rodesac is igive manZiliT energia? dabla daiwevs? g) ra maqsimalur manZilze 101. 3,2 kg masis zarmaca miwidan 3 m simaRleze daiwevs Camokidebuli yuTi, vidre wamierad gaCerdeba? kidia. a) ra mizidulobis potenciuri energia aqvs zarmaca-dedamiwis sistemas, Tu y = 0 270 Tavi 8 potenciuuri energia da energiis Senaxva aTvlis wertili dedamiwis donezea? zarmaca moZraobas maRali gorakis mwvervalze iwyebs. miwaze Camovarda da masze moqmedi haeris wevis ra Zala umniSvneloa. ra a) kinetikuri energia da gorakis mwvervalze, Tu is saTxilamuro g) siCqare eqneba zarmacas miwaze dacemamde? joxebs ar iyenebs? b) Tovlsa da Txilamurebs 102. 3200 n/m sixistis zambara daWimulia da misi elastiuri potenciuri energia 1,44 j-ia (moSvebul mdgomareobaSi U = 0). risi tolia ΔU, Tu zambara a) kidev 2 sm-iT daWimes, b) 2 smiT SekumSes da g) 4 sm-iT SekumSes? Soris daaxloebiT ra kinetikuri xaxunis 107. mocurave wyalSi 0,22 m/wm siCqariT micuravs. saSualo wevis Zalaa 110 n. ra 103. komfortuli laineri Queen Elizabeth 2 saSualo simZlavre esaWiroeba mocuraves? TboeleqtrosadguriTaa aRWurvili, romelic 32,5 sazRvao miliT moZraobis dros 92 mgvt 108. 16400 n wonis mgzavrebiT savse manqana 113 km/sT siCqariT moZraobs, rodesac mZRoli maqsimalur simZlavres gamoimuSavebs. ra Zala amuxruWebs siCqare eqneba moTxilamures dabali koeficienti gaaCerebs moTxilamures zustad dabali gorakis mwvervalze? da manqanas aCerebs. gzidan moqmedebs lainerze am siCqariT moZraobisas? borblebze moqmedi xaxunis Zalaa 8320 n. ra (1 sazRvao mili = 1,852 km/sT). manZilis gavlis Semdeg gaCerdeba manqana? 104. sabavSvo Tofis zambaris sixistea 700 n/m. burTis gasrolis win zambara ikumSeba 109. 60 kg masis cirkis msaxiobi 4 m simaRlis da burTi masze Tavsdeba. Semdeg Tofis energia Caxmaxi zambaras asworebs da isic burTs win Camosvlisas, Tu joxidan masze moqmedi xaxunis ubiZgebs. burTsa da zambaras Soris kontaqti Zala a) umniSvneloa da b) 500 n sididisaa? maSin wydeba, rodesac burTi Tofis lulidan 110. 1981 wels, daniel gudvini CikagoSi mdebare 443 m simaRlis Senobaze acocda. gamodis. horizontalisadmi 30°-iT daxrili joxidan dabla srialdeba. ra kinetikuri eqneba msaxiobs cirkis arenaze Tofi 57 g masis burTs 1,83 m maqsimalur a) simaRleze isvris. davuSvaT burTze moqmedi gamoTvaleT ra raodenobis energia gadaitana haeris wevis Zala umniSvneloa. a) ra siCqariT man biomeqanikuri (Siga) energiidan dedamiwa- isvris zambara burTs? b) Tu burTze moqmedi sakuTari xaxunis Zala TofSi umniSvneloa, ra manZiliT potenciur energiaSi, raTa am simaRleze ikumSeba zambara? amZvraliyo. b) ra raodenobis energiis gada 105. 15 n sididis horizontaluri Zala plast masis kubs mudmivi siCqariT 3 m manZilze tana mouxdeboda gudvins, Tu imave simaRleze gadaaadgilebs. am dros kubis temperaturas 111. 0,63 kg masis burTi 14 m/wm siCqariT aisroles da man 8,1 m maqsimalur simaRles akvirdebian. dakvirvebis Sedegad dgindeba, rom kubis siTburi energia 20 j-iT gaizarda. ramdeniT gaizrdeba zedapiris siTburi gudvinis masa Tavis miaxloebiT sistemis aiReT da mizidulobis asasvlelad Senobis kibeebiT isargeblebda? miaRwia. ramdeniT Seicvala burTi-dedamiwis sistemis meqanikuri energia burTis maRla energia? asvlis dros? 106. Tovliani gorakis mwvervalebi maT Soris mdebare velidan H = 850 m da h = 750 m 112. everestis mwvervali zRvis donidan 8850 m simaRlezea. a) ra energia unda daxarjos 90 kg masis mTamsvlelma mizidulobis Zalis simaRleebzea. gorakebs Soris saTxilamuro trasaa, romlis sigrZea 3,2 km da saSualo daxrilobaa θ = 30° (sur. 8.67.). a) moTxilamure sawinaaRmdegod, raTa zRvis donidan am simaRleze avides? b) ra raodenobis kanfeti uzrunvelyofs am energias, Tu TiToeuli kanfeti 1,25 j energias Seicavs? Tqvenma pasuxma unda cxadyos, rom mizidulobis Zalis winaaRmdeg Sesrulebuli muSaoba mTaze asvlisTvis saWiro energiis mxolod mcire sur. 8.67. amocana 106. nawilia. 271 113. 670 n wonis sprinteri 1,6 wamSi 7m-s SekumSvas potenciuri energia ukavSirdeba, garbis da am dros Tanabrad Cqardeba. ra CawereT a) siCqare da b) kinetikuri energia eqneba manZilze damokidebulebis funqcia. potenciuri energiis SekumSvis sprinters 1,6 wamis Semdeg? g) ra saSualo 1990-ian simZlavres gamoimuSaves sprinteri 1,6 wamis triceratops menjis Zvalze kbilebis kvali ganmavlobaSi? aRmoaCnda. kvalis forma mianiSnebda, rom 114. 8.68 suraTze gamosaxul 1400 kg is masis da wlebSi tiranozavrs ekuTvnoda. Sesamowmeblad granitis aRmoCenil dinozavr am varaudis mkvlevarebma brinjaosgan aluminisa tiranozavris kbilis bloks daxril sib modeli daamzades da Semdeg hidravlikuri 1,34 m/wm wnexis saSualebiT es kbili Zroxis ZvalSi siCqariT triceratopis Zvlis siRrmeze Casves. saWiro rtyeze Tanabari Tokis saSualebiT sur. 8.68. amocana 114. maRla sweven. naCve Zalis SeRwevis siRrmeze damokidebulebis grafiki 8.69 suraTzea gamosaxuli. siRrmis nebi manZilebia d1 = 40 m da d2 = 30 m. bloksa zrdasTan da daxril sibrtyes Soris kinetikuri xaxunis vinaidan erTad saWiro Zala izrdeba, konusis formis kbilis ZvalSi koeficientia 0,4. ra simZlavres warmoqmnis Casobisas, kbilsa da Zvals Soris kontaqti Tokidan blokze modebuli Zala? ufro mWidro xdeba. e) r muSaoba Seasrula 115. 0,42 kg masis Saflbordis disko uZravad hidravlikurma presma da, maSasadame, devs da moTamaSe kiis saSualebiT mis siCqares tiranozavrma (miaxloebiT)? v) ukavSirdeba 4,2 m/wm-mde zrdis (aCqareba mudmivia). aCqareba daaxloebiT 2 m-is manZilze grZeldeba, Tu ara potenciuri energia kbilis Casobas? Semdeg kiisa da diskos Soris kontaqti wydeba. gaCerebamde davuSvaT disko kidev Saflbordis 12 m-s gaivlis. moedani brtyelia da diskoze moqmedi xaxunis Zala ucvlelia. ramdeniT gaizrdeba disko-moedanis sistemis siTburi energia a) damatebiTi 12 m-is gavlis dros da b) mTliani 14 m manZilis gavlis dros? g) ra muSaobas asrulebs kii diskoze? 116. 70 kg masis adamiani fanjridan gadaxta da fanjridan 11 m-iT qvemoT gaSlil samaSvelo badeSi Cavarda. bade 1,5 m-iT daiWima da am dros adamiani wamierad gaCerda. davuSvaT sur. 8.69. amocana 117. meqanikuri energia inaxeba da bade idealuri 118. 70 kg masis mexanZre 4,3 m simaRlis zambaris saxiT moqmedebs. gamoTvaleT badis vertikalur boZze srialdeba. a) Tu mexanZre 1,5 m-iT boZs kargad ar eWideba da masze moqmedi elastiuri potenciuri energia daWimvisas. xaxunis Zala umniSvneloa, ra siCqare eqneba 117. ukana kbiliT saganze maqsimum 750 n ZaliT mexanZres miwaze Casvlamde? b) Tu mexanZre SegiZliaT imoqmedoT. warmoidgineT, rom boZs ufro magrad eWideba da masze 500 n Zirtkbilas bards ReWavT. am dros Zirtkbila sididis k = 2,5×105 n/m sixistis zambarasaviT moqmedebs moqmedebs, ra siCqare eqneba mexanZres miwaze da kbils ewinaaRmdegeba. gamoTvaleT a) Zir Casvlamde? tkbilas SekumSvis manZili da b) kbilis mier 119. Zirtkbilaze Sesrulebuli muSaoba. g) CawereT xaxunisgan Tavisufal zedapirze 2 m/wm2-iT kbilidan moqmedi Zalis SekumSvis manZilze Cqardeba da misi siCqare 10 m/wm-dan 30 m/wm- damokidebulebis funqcia. d) Tu Zirtkbilas mde izrdeba. a) ramdeniT icvleba blokis maRla 15 kg mimarTuli masis bloki xaxunis Zala horizontalur, 272 potenciuuri energia da energiis Senaxva Tavi 8 meqanikuri energia da b) ra saSualo tempiT gadaecema bloks energia? risi tolia energiis gadacemis myisi tempi, rodesac blokis 124. 600 n wonis moTxilamure R = 20 m radiusis, xa siCqarea g) 10 m/wm da d) 30 m/wm? xunisgan 120. 119-e amocana Tavidan amoxseniT, oRond fal amjerad miiCnieT, rom xaxunisgan Tavisufali rakze gadadis (sur. zedapiri horizontalisadmi 5°-iT aris Tavisu wriul 8.70). go davuSvaT, daxrili. moTxilamureze 121. aSS-s kontinentis zedapiris farTobi daaxloebiT 8×106km2-ia, xolo zRvis donidan moqmedi aTvlili saSualo simaRle daaxloebiT 500 m-s aRwevs. naleqebis raodenoba weli wadSi daaxloebiT 75 sm-ia. atmosferoSi aor Tqlebis Sedegad wvimis wylis daaxloebiT 2/3 brundeba, xolo danarCeni nawili okeaneSi Caedineba. Tu wyali-dedamiwis sistemis mizi dulobis potenciuri energiis wvimis wylis dinebasTan dakavSirebul Semcirebas eleq troenergiad gardavqmniT, ra iqneba misi saSualo simZlavre? (1 m3 wylis masa 1000kg-ia). 122. 20 kg masis saganze F = –3x – 5x2 Semna xveli Zala moqmedebs, sadac F mocemulia niutonebSi, xolo x – metrebSi. rodesac sagani x = 0 koordinatze imyofeba, ZalasTan dakavSirebuli potenciuri energia nulis tolia. a) ra iqneba ZalasTan dakavSirebuli sur. 8.70. amocana 124. haeris winaRobis Zala umniSvneloa. B wertilSi, θ = 20° kuTxeze moTxilamuris siCqarea 8 m/wm. a) ra siCqare eqneba moTxilamures gorakis mwvervalze (A wertilSi), Tu is saTxilamuro joxebs ar iyenebs? b) ra minimaluri siCqare SeiZleba hqondes moTxilamures B wertilSi, raTa man moZraoba ganagrZos? g) Tu moT xilamuris wonaa 700 n, zemoT miRebuli ori pasuxi gaizrdeba, Semcirdeba Tu ar Seicvleba? 125. 100 m simaRlis CanCqeridan TiToeuli wamis ganmavlobaSi 1200 m3 wyali Camodis. vardnis dros wylis mier SeZenili kinetikuri energiis 3/4-s hidroeleqtruli generatori eleqtroenergiad warmoqmnis gardaqmnis. generatori ra tempiT eleqtroenergias? (1 m3 wylis masaa 1000 kg). sistemis potenciuri energia, rodesac sagani 126. 68 kg masis paraSutiT mxtomeli 59 m/wm x = 2 m koordinatze imyofeba? b) Tu sagani x saboloo siCqariT vardeba. a) ra tempiT mci RerZis uaryofiTi mimarTulebiT moZraobs rdeba dedamiwa-mxtomelis sistemis mizi da x = 5 m koordinatze misi siCqarea 4m/wm, dulobis potenciuri energia? b) ra tempiT ra iqneba misi siCqare aTvlis saTaveze? mcirdeba sistemis meqanikuri energia? g) ra pasuxebs miviRebT a) da b) SekiTxvebze, Tu 127. 50 g masis burTi fanjridan 8 m/wm sawyisi siCqariT da horizontalisadmi 30° kuTxiT x = 0 koordinatze sagnis mdebareobis dros sistemis potenciuri energia –8 j-ia? 123. k = 200 n/m sixistis zambara vertikalu maRla isroles. gamoTvaleT a) burTis kine tikuri energia frenis umaRles wertilSi da rad kidia, misi zeda bolo Werze damagrebulia, b) burTis siCqare fanjridan 3 m-iT dabla. y = 0. aris Tu ara b)-s pasuxi g) burTis masaze an xolo qveda bolos koordinatia zambaris qveda boloze 20 n wonis bloki d) srolis kuTxeze damokidebuli? daamagres da xeli gauSves. risi tolia iqneba 128. 9,4 kg masis sagani vertikalurad maRla zambara-blokis sistemis a) K kinetikuri aisroles. maRla asvlis dros sagani-de energia, b) mizidulobis potenciuri energiis damiwis sistemis meqanikuri energia haeris ΔUg cvlileba da g) elastiuri potenciuri wevis Sedegad 68kj-iT mcirdeba. ra manZiliT cvlileba, rodesac blokis ufro maRla avidoda sagani haeris wevis energiis ΔUe koordinatia y = –5 sm? risi tolia d) K, e) ΔUg da v) ΔUe, rodesac y = –10 sm? risi tolia z) K, T) ΔUg da i) ΔUe, rodesac y = –15sm? risi tolia k) K, l) ΔUg da m) ΔUe, rodesac y = –20 sm? ararsebobis SemTxvevaSi? 129. L sigrZis da umniSvnelo masis Zelakis erT boloze burTia damagrebuli (sur. 8.71). 273 Zelakis meore bo siCqarea 3,2 m/wm, xolo Cqerobebis gavlis lo sayrden wer- Semdeg wylis siCqarea 13 m/wm. dedamiwa-wylis tilzea damagre- sistemis mizidulobis potenciuri energiis buli da burTi ver- ra procenti gadadis kinetikur energiaSi tikalur wreze Cqerobebze? moZraobs. sayrden ( m i n i S n e b a : wertilze d a v u S v a T xaxuni ar moqmedebs. bur Ti sur. 8.71. amocana 129. moZraobas A wertilidan ν0 sa 10kg Cqerobebze wyali Cadis). L-iT, m-iT da g-Ti gamosaxuli ν0-is formula. 133. 8.72a suraTze r-iT dacilebuli m da M masis mqone b) risi tolia Zelakis daWimuloba, rodesac ori atomisgan Se wyisi siCqariT iw yebs da D wertilSi Cerdeba. a) gamoiyvaneT burTi B wertils gaivlis? g) patara gogona sayrden wertilze Camojda da xaxuni gaCnda. amis Semdeg A wertilidan igive siCqariT daZruli burTi ramdeniT C wertilSi mcirdeba burTis Cerdeba. meqanikuri naCvenebia mdgari molekula (m << M). 8.72b su raTze gamosaxulia sur. 8.72. amocana 133. U(r) molekulis potenciuri energiis r-ze damokidebulebis energia moZraobis ganmavlobaSi? d) ramdeniT funqciis mcirdeba burTis meqanikuri energia, vidre moZraoba a) Tu ori atomisgan Semdgari sis burTi ramdenime rxevis Semdeg B wertilSi sabolood gaCerdeba? 130. liTonis xelsawyos saxexi danadgaris borblis fersoze xexaven. grafiki. aRwereT atomebis temis E mTliani meqanikuri energia nulze metia (aseve E1-c) da b) Tu E nulze naklebia E2-c). (aseve E1=1×10–19j r = 0,3 nm, da danadgaridan gamoTvaleT g) sistemis potenciuri energia, xelsawyoze 180 n Zala moqmedebs. fersosa d) atomebis mTliani kinetikuri energia da da xelsawyos Soris moqmedi xaxunis Zalebi e) TiToeul atomze moqmedi Zalis sidide xelsawyos patara nawilebs aclis. borblis da mimarTuleba. r-is ra mniSvnelobebisTvis radiusia 20 sm da 2,5 br/wm siCqariT moZraobs. xdeba atomebs Soris v) ganzidva, z) mizidva borbalsa da xelsawyos Soris kinetikuri T) r-is ra mniSvnelobisaTvis aris es Zala xaxunis koeficientia gadascems borblis 0,32. ra mamoZravebeli tempiT Zrava siTbur energias borbals da xelsawyodan moclil patara nawilebs - kinetikur energias? 131. niagaras CanCqeris 50 m simaRlidan nulis toli? 134. x RerZis gaswvriv rav moZ nivTier wertilze F(x) Semnaxveli Za TiToeuli wamis ganmavlobaSi 5,5×106 kg wyali la Camodis. a) ramdeniT mcirdeba TiToeuli 8.73 wamis ganmavlobaSi wyali-dedamiwis sistemis n a C v e n e b i a mizidulobis F(x)-Tan Tu mTeli potenciuri es energia energia? b) eleqtroenergiad moqmedebs. suraTze sur. 8.73. amocana 134. da kavSirebuli U(x) potenciuri energiis gardaiqmneba, ra tempiT iqneba SesaZlebeli cvlileba nivTieri wertilis mdebareobaze eleqtroenergiis miwodeba? (1m3 wylis masaa damokidebulebis grafiki. a) aageT F(x) 1000kg). g) Tu eleqtroenergiis fasia 1 centi/ funqciis kvt•sT, ra iqneba wliuri Semosavali? b) sistemis E meqanikuri energiaa 4j. 8.73 132. mdinaris done Cqerobebis gavlis Semdeg 15 m-iT dabldeba. Cqerobebis win wylis kinetikuri energiis funqciis grafiki. suraTze grafiki aageT 0<x<6m nivTieri arealisTvis. wertili K(x) 274 Tavi 8 potenciuuri energia da energiis Senaxva 135. m1 da m2 masis nivTier wertilebs 136. vertikaluri zambaris erTi bolo Werze Soris Semdegi sididis mizidulobis Zala daamagreT, xolo meore boloze kombosto moqmedebs: CamokideT. Semdeg kombosto qvemoT Camo F(x) = G m 1m 2 , x2 sadac G mudmivaa da x aris nivTier werti wieT, vidre zambaridan masze moqmedi, maRla lebs Soris manZili. a) miiReT Sesabamisi U(x) wonasworebs. daamtkiceT, rom kombosto-de potenciuri davuSvaT damiwis sistemis mizidulobis potenciuri U(x)→0, rodesac x→∞ da x dadebiTia. energiis danakargi zambaris mier SeZenili b) ra muSaobaa saWiro, raTa nivTier wertilebs potenciuri energiis orze namravlis tolia. Soris dacileba x = x1-dan x = x1+d-mde gai ratom ar aris es ori sidide erTmaneTis zardos? energiis funqcia. mimarTuli Zala mizidulobis Zalas gaa toli? 9 masis centri da impulsi avaria yvelaze saxifaToa, rodesac manqanebi erTmaneTs wina nawilebiT Seaskdebian. amgvari Sejaxeba orive mZRolisTvis sasikvdilo SeiZleba aRmoGndes. raoden gasakviric ar unda iyos, gamokvlevebis Sedegad dadginda, rom sasikvdilo riski naklebia, Tu manqanaSi mZRolTan erTad mgzavric imyofeba ratom xdeba ase? pasuxs am TavSi SeityobT 9.1. ra aris fizika? sagzao SemTxvevebis gaanalizebisTvis mowveuli yvela inJiner-meqanikosi fizikas iyenebs. baletis nebismieri maswavlebeli, romelic naxtomis gakeTebaSi avarjiSebs gogonebs, aseve fizikas eyrdnoba. zogadad, nebismieri saxis rTuli moZraobis analizi fizikis daxmarebiT martivdeba. me-9 TavSi Cven ganvixilavT, Tu rogor xdeba saganTa sistemis rTuli moZraobis gamartiveba sistemis specialuri wertilis _ masis centris gansazRvris saSualebiT. magaliTad, Tu burTs haerSi maRla aisvriT, misi moZraoba martivi, xolo traeqtoria parabolis formis iqneba (sur. 9.1). es SemTxveva me-4 paragrafSi ukve ganvixileT da am dros burTs nivTier wertilad miviCnevdiT. beisbolis tafelas haerSi moqnevis dros moZraoba ufro rTulia (sur. 9.1b). tafelas sxvadasxva nawili sxvadasxva formis traeqtorias Semowers da tafelas nivTier wertilad ver ganvixilavT. tafela ufro nivTieri wertilebis sistemaa, romelic haerSi sxvadasxva traeqtoriiT moZraobs. miuxedavad amisa, tafelas erTi gansakuTrebuli wertili sur. 9.1. a) haerSi asrolili burTi parabolis formis traeqtoriiT moZraobs. b) haerSi moqneuli beisbolis tafelas masis centri (Savi wertili) parabolis formis traeqtoriiT moZraobs, magram tafelas yvela sxva wertili ufro rTul traeqtorias Semowers. 275 276 Tavi 9 masis centri da impulsi - masis centri aqvs, romelic martivi, parabolis formis traeqtoriiT moZraobs. tafelas sxva nawilebi masis centris garSemo moZraoben (masis centris mdebareobis dasadgenad tafela gaSlil TiTze SeimagreT. masis centri Tqveni TiTis gaswvriv, tafelas centralur RerZze mdebareobs). beisbolis tafelas haerSi qneviT bevrs verafers miaRwevT, samagierod Zalian karg karieras gaikeTebT, Tu did manZilze mxtomelebs an mocekvaveebs haerSi sworad xtomas aswavliT. am saqmis Seswavla ki adamianis masis centriT unda daiwyoT. 9.2. masis centri nivTieri wertilebis sistemis (magaliTad, adamianis) masis centris gansazRvris mizani sistemis SesaZlo moZraobis winaswari amocanobaa. nivTieri wertilebis sistemis masis centri is wertilia, romelzec sistemis mTeli masaa koncentrirebuli da yvela Zala masze moqmedebs. am TavSi SeviswavliT nivTieri wertilebis sistemis masis centris dadgenis xerxs. Tavdapirvelad mxolod ramdenime nivTieri wertilisgan Semdgar sistemas ganvixilavT, xolo Semdeg ufro rTul sistemebsac gavarCevT. ufro mogvianebiT imasac davadgenT, Tu rogor gadaadgildeba sistemis masis centri, rodesac sistemaze gare Zalebi moqmedebs. nivTier wertilTa sistema 9.2a suraTze naCvenebia d manZiliT dacilebuli, m1 da m2 masebis nivTieri wertilebi. x RerZis aTvlis saTave m1 masis nivTier wertils emTxveva. am ori nivTieri wertilisgan Semdgari sistemis masis centris mdebareobaa: m2 xcom* = m +m d 1 (9.1) 2 davuSvaT, m2=0. am SemTxvevaSi mxolod m1 masis nivTieri wertili arsebobs da masis centri am wertilis mdebareobas unda emTxveodes. 9.1 formula marTlac gvaZlevs xcom=0. Tu m1=0, am SemTxvevaSi mxolod erTi nivTieri wertili rCeba da rogorc mosalodnelia xcom=d. rodesac m1=m2, masis centri ori nivTieri wertilis SuaSia da marTlac 9.1 formula xcom=1/2d pasuxs gvaZlevs. 9.1 formula aseve gvauwyebs, rom Tu arc m1 da arc m2 nulis toli ar aris, maSin xcom-is mniSvneloba nulsa da d-s Soris meryeobs anu masis centri or nivTier wertils Soris imyofeba. 9.2b suraTze gamosaxulia ufro zogadi SemTxveva, rodesac koordinatTa sistema marcxniskenaa daZruli. masis centris mdebareoba Semdegi formuliT ganisazRvreba: xcom= m1 x1 + m2 x2 m 1 + m2 . (9.2) xcom-is indeqsi „com“ aris inglisuri frazis „center of mass“ abreviatura da qarTuli `mc~ abreviaturiT ar SevcvaleT. * sur. 9.2. a) m1 da m2 masebis or nivTier wertils Soris manZilia d. „com“ saxeliT dasaTaurebuli wertili 9.1 formuliT gamoTvlili masis centria. b) amjerad aTvlis saTave nivTieri wertilebidan ufro moSorebiT aris ganlagebuli. masis centris mdebareoba 9.2 formuliT gamoiTvleba. orive SemTxvevaSi masis centris mdebareoba nivTieri wertilebisadmi ucvlelia. 9.2. masis centri 277 yuradReba miaqcieT, rom Tu am formulaSi x1=0 CavsvamT, x2=d da 9.2 formula 9.1 formulamde daiyvaneba. aseve yuradReba miaqcieT, rom miuxedavad koordinatTa sistemis daZvrisa, masis centri TiToeuli nivTieri wertilidan igive manZilzea ganlagebuli. 9.2 formula Semdegi saxiT SeiZleba Caiweros: m1 x1 + m2 x2 xcom= , M (9.3) sadac M aris sistemis mTliani masa (am SemTxvevaSi M=m1+m2). es formula SegviZlia ganvazogadoT da x RerZis gaswvriv ganlagebuli n raodenobis nivTieri wertilebisTvis davweroT. am SemTxvevaSi sistemis mTliani masaa M=m1+m2+...+mn da masis centris mdebareobaa: xcom= m1 x1 + m2 x2 + m3 x3 + . . . + mn xn = M . (9.4) i indeqsi cvladi ricxvia da is 1-dan n-mde mTel mniSvnelobebs moicavs. es indeqsi sxvadasxva nivTier wertilebs, maT masebs da x koordinatebs gansazRvravs. Tu nivTieri wertilebi sam ganzomilebaSi aris ganlagebuli, maTi masis centri sami koordinatis saSualebiT ganisazRvreba. 9.4 formulis ganvrcoba gvaZlevs: xcom = , ycom = , zcom = (9.5) masis centris gansazRvra veqtorebis saSualebiTac SeiZleba. Tavdapirvelad gavixsenoT, rom xi, yi da zi koordinatze ganlagebuli nivTieri wertilis mdebareoba mdebareobis veqtoris saSualebiT moicema: . am SemTxvevaSi indeqsi nivTier wertils gansazRvravs, xolo , da (9.6) erTeulovani veqtorebia, romlebic Sesabamisad x, y da z RerZebis dadebiTi mimarTulebisaa. nivTier wertilTa sistemis masis centris mdebareoba aseve mdebareobis veqtoris saSualebiT moicema: (9.7) 9.5 formulaSi Semavali sami skalaruli gantoleba erTi veqtoruli gantolebiT SegviZlia SevcvaloT: , (9.8) sadac M aris sistemis mTliani masa. am formulis SemowmebisTvis masSi 9.6 da 9.7 formulebi CasviT da Semdeg x, y da z komponentebi ganacalkeveT. Sedegad kvlav 9.5 formula miiReba. myari sxeulebi Cveulebrivi sagani, magaliTad beisbolis tafela, uricxvi raodenobis nivTieri wertilisgan (atomebisgan) Sedgeba da umjobesia is materiis uwyvet ganawilebad miviCnioT. am SemTxvevaSi, `nivTieri wertilebi~ dm masis mqone elementebia, 9.5 formulaSi Semavali jamebi integralebi xdeba da masis centris koordinatebi Semdegi formulebiT ganisazRvreba: , , , (9.9) sadac M aris sagnis masa. araerTgvarovani sxeulebisaTvis am integralebis gamoTvla sakmaod rTuli iqneba, amitom amjerad mxolod erTgvarovan sxeulebs ganvixilavT. erTgvarovani sxeulebis simkvrive yovel 278 Tavi 9 masis centri da impulsi wertilSi erTi da igivea anu sxeulis nebismieri mocemuli elementis simkvrive mTeli sxeulis ρ simkvrivis tolia: dm M = , (9.10) dV V sadac dV aris dm masis elementis moculoba, xolo V - sagnis mTliani moculoba. 9.10 formulidan dm=(M/V)dV 9.9 formulaSi CavsvaT da miviRebT: ρ= , , . (9.11) erT an met integrals gverds avuvliT, Tu sagans simetriis wertili, wrfe an sibrtye gaaCnia. am SemTxvevaSi, sagnis masis centri simetriis wertilze, wrfeze an sibrtyezea ganlagebuli. magaliTad, erTgvarovani sferos (romelsac simetriis wertili gaaCnia) masis centri sferos centrzea ganlagebuli. erTgvarovani konusis (romlis RerZi simetriis wrfea) masis centri konusis RerZzea ganlagebuli. bananis (romelsac simetriis sibrtye aqvs) simetriis centri am sibrtyezea ganlagebuli. ar aris aucilebeli sagnis masis centri sagnis SigniT iyos ganlagebuli. y sakontrolo wertili 1. suraTze naCvenebia kvadratuli formis firfita, romelsac kuTxeebidan oTxi identuri kvadrati CamoaWres. 1 2 a) sad aris sawyisi firfitis masis centri? sad aris firfitis masis x centri b) pirveli kvadratis, g) pirveli da meore kvadratis, d) pirveli da mesame kvadratis, e) pirveli, meore da mesame kvadratis da 4 3 v) oTxive kvadratis moWris Semdeg? amocanis amoxsnis nimuSi 9.1 m1=1,2kg, m2=2,5kg da m3=3,4kg masis nivTieri wertilebi tolgverda samkuTxeds qmnis gverdis sigrZe a=140sm. sad iqneba am sistemis masis centri? amoxsna: Cven nivTier wertilebTan gvaqvs Sexeba da ara ganvrcobil myar sxeulTan, amitom maTi masis centris dasadgenad 9.5 formula SegviZlia gamoviyenoT. wertilebi tolgverda samkuTxedis ganlagebulni, ori amitom formula sibrtyeSi mxolod gvWirdeba. arian pirveli gamoTvlebi ufro martivi gaxdeba Tu x da y RerZebs ise gavavlebT, rom erT-erTi wertili iqneba aTvlis saTave da x RerZi samkuTxedis erT-erT gverds daemTxveva (sur. 9.3.). am SemTxvevaSi, nivTieri wertilebis koordinatebia: sur. 9.3. sami nivTieri wertili a sigrZis gverdis tolgverda samkuTxeds qmnis. masis centris  mdebareobas rcom mdebareobis veqtori gansazRvravs. sistemis mTliani M masaa 7,1kg. 9.5 formulidan gamomdinare, masis centris koordinatebia: nivTieri wertili masa (kg) x (sm) y (sm) 1 1 ,2 0 0 2 2,5 140 0 3 3,4 70 120 = = m1 x1 + m2 x2 + m3 x3 M (1,2 kg)(0) + (2,5 kg)(140 sm) + (3,4 kg)(70 sm) 7,1 kg = 83 sm (pasuxi) 9.2. masis centri = 58 sm m1 y1 + m2 y2 + m3 y3 M = da suraTze gamosaxulia P firfita, 2R mdebareobis veqtori gansazRvravs, romlis komponentebia xcom da ycom. 9.2 amocanis amoxsnis nimuSi erTgvarovani (pasuxi)  9.3 suraTze masis centris mdebareobas rcom (1,2 kg)(0) + (2,5 kg)(0) + (3,4 kg)(120 sm) = 7,1 kg 9.4a 279 radiusis romelsac R radiusis disko CamoaWres. xy koordinatTa sistemis daxmarebiT daadgineT firfitis masis centris mdebareoba comp. amoxsna: Tavdapirvelad simetriis gamoye nebiT firfitis P centris mdebareoba davadginoT. Semdeg gaviTvaliswinoT, rom firfita x RerZisadmi simetriulia. aqedan gamomdinare, comp wertili x RerZze unda iyos ganlagebuli. firfita (diskos CamoWris (a) Semdeg) y RerZisadmi simetriuli ar aris. miuxedavad amisa, y RerZis marjvniv ufro meti masaa Tavmoyrili da, amitom, comp wertili am RerZidan marjvniv unda iyos ganlagebuli. yovelive zemoTqmulidan gamomdinare, comp wertili daaxloebiT iq mdebareobs, sadac 9.4a suraTzea naCvenebi. P firfitis masis centris (comp) namdvili koordinatebis dasadgenad 9.11 formulebi SegviZlia gamoviyenoT. es meTodi sakmaod gaarTulebs amocanas, amitom umjobesia davuSvaT, rom erTgvarovani sagnis masa sagnis masis centrze mdebare nivTier wertilSia (b) koncentrirebuli. Tavdapirvelad amoWrili disko (davarqvaT mas S) ukan davabrunoTY (sur. 9.4b) da firfitis pirvandeli saxe aRvadginoT (am firfitas C davarqvaT). S diskos comS masis centri wriuli simetriulobis Sedegad S-is centrze, x = –R koordinatze mdebareobs. aseve C firfitis comC masis centri C-is centrze anu aTvlis saTavezea. modiT, yovelive zemoTqmuls erTianad movuyaroT Tavi: masis masis centris centri mdebareoba P comP xP=? mP S comS xS = –R mS C comC xC=0 mC=mS+mP firfita masa (g) sur. 9.4. a) P firfitis radiusia 2R, xolo wriuli xvrelis radiusia R. P firfitis masis centri comp wertilzea ganlagebuli. b) S disko ukan daabrunes da C firfitam pirvandeli saxe miiRo. naCvenebia S diskos comS masis centri da C firfitis comC masis centri. g) S-is da P-is gaerTianebis comS+P masis centri C firfitis comC-s anu aTvlis saTaves emTxveva. axla davuSvaT, rom S diskos mS masa xS = –R koordinatze mdebare nivTier wertilSia koncentrirebuli, xolo mP masa xP koordinatze mdebare nivTier wertilSia koncentrirebuli (sur. 9.4g). amis Semdeg es 280 masis centri da impulsi Tavi 9 ori nivTieri wertili sistemad miviCnioT farTobebs da maTi xS+P masis centris dasadgenad 9.2 gaviTvaliswinebT, rom: formula gamoviyenoT: m x + mP xP xS+P = S S . mS + mP (9.12) Semdeg gavixsenoT, rom S diskos da P firfitis gaerTianeba C firfitas gvaZlevs. maSasadame, comS+P masis centris xS+P mdebareoba comC-s xC mdebareobas (aTvlis saTaves) unda daemTxves, amitom xS+P=xC=0. miRebuli 9.12 Sedegi formulaSi CavsvaT da xP-sTvis amovxsnaT: xP = – xS mS . mP (9.13) amjerad kidev erTi problema gaCnda, vinaidan 9.13 formulaSi Semavali masebi ar viciT. samagierod masebi S-is da P-is zedapiris SegviZlia davukavSiroT Tu masa = simkvrive × moculoba = simkvrive × sisqe × farTobi am SemTxvevaSi: mS = mP simkvrive S × simkvrive P sisqe S sisqe P × farTobi S farTobi P vinaidan firfita erTgvarovania, simkvrive ρS = ρP da S diskosa da P firfitis sisqe erTmaneTis tolia: mS = mP = farTobi S farTobi P = farTobi S farTobi C πR 1 = π(2R)2 – πR2 3 2 – farTobi S 9.13 formulaSi es Sedegi da xS = –R CavsvaT da miviRebT: 1 xP = R 3 (pasuxi) amocanis amoxsnis taqtika taqtika 1: masis centrTan dakavSi rebuli amocanebi 9.1 da 9.2 amocanis amoxsnis nimuSebSi mocemulia sami strategia, Tu rogor unda gamartivdes masis centrTan dakavSirebuli amocanebi: 1) gamoiyeneT sagnis simetria, iqneba es wertili, wrfe Tu sibrtye; 2) Tu SesaZlebelia sagnis ramdenime nawilad dayofa, TiToeuli es nawili nivTier wertilad miiCnieT; 3) koordinatTa RerZebi Wkvianurad gaavleT. Tu sistema nivTieri wertilebisgan Sedgeba, erT-erTi nivTieri wertili aTvlis saTaves daamTxvieT. Tu Tqveni sistema simetriis wrfis mqone sxeulia, am wrfes x an y RerZi daamTxvieT. aTvlis saTavis arCeva mTlianad Tqvens survilzea da masis centris mdebareoba masze damokidebuli ar aris. 9.3. niutonis meore kanoni nivTier wertilTa sistemisTvis Cven ukve viciT nivTier wertilTa sistemis masis centris mdebareobis dadgena da amjerad ganvixilavT, Tu rogor gadaaadgilebs masis centrs gare Zala. modiT, biliardis ori burTisgan Semdgari martivi sistema gavarCioT. Tu kiis saSualebiT biliardis erT burTs meores moartyamT, ori burTisgan Semdgari sistema moZraobas dartymis mimarTulebiT ganagrZobs. albaT Zalian gagikvirdebaT, rom orive burTi ukan dagibrundeT an, magaliTad, marcxnisken amoZravdes. ori burTisgan Semdgari sistemis masis centri aris is wertili, romelic dartymis mimarTulebiT moZraobas ganagrZobs da Sejaxeba masze gavlenas ar axdens. es wertili burTebs Soris Sua wertilzea (vinaidan burTebis masebi Tanabaria) da Tu kargad daakvirdebiT, darwmundebiT, rom Cveni mosazreba sworia. ar aqvs mniSvneloba Sejaxeba iribia, pirdapiria Tu raRac Sualeduria, masis centri mainc dartymis mimarTulebiT moZraobas ganagrZobs. modiT masis centris moZraoba ufro detalurad ganvixiloT. Tavdapirvelad biliardis burTebis wyvili sxvadasxva masebis n raodenobis nivTieri wertilebiT SevcvaloT. Cven am nivTieri wertilebis individualuri moZraoba ki ar gvainteresebs, aramed maTi erTobliobis masis centris moZraoba. masis centri sistemis mTliani masis toli masis mqone 9.3. niutonis meore kanoni nivTier wertilTa sistemisTvis 281 nivTieri wertiliviT moZraobs. am nivTier wertils mdebareoba, siCqare an aCqareba SegviZlia mivaniWoT. nivTier wertilTa sistemis masis centris moZraobas Semdegi veqtoruli formula asaxavs: t =M com (9.14) (nivTier wertilTa sistema) es formula nivTier wertilTa sistemis masis centris moZraobisTvis niutonis meore kanonia. yuradReba miaqcieT, rom mas erTi nivTieri wertilis moZraobisTvis gankuTvnili formulis ( t = m ) saxes aqvs. miuxedavad amisa, 9.14 formulaSi Sesuli sami sidide frTxilad unda gamoiTvalos: 1. t aris sistemaze moqmedi yvela gare Zalis veqtoruli jami (Zalebis tolqmedi). sistemis erTi nawilidan meoreze moqmedi Zalebi (Sida Zalebi) 9.14 formulaSi ar Sedis. 2. M aris sistemis jamuri masa. Cven vuSvebT, rom moZraobis sistemaSi gare nawilaki ar Sedis da sistemis arcerTi nawilaki gareT ar gamodis. sistema Caketilia da M masa ucvlelia. 3. com sistemis masis centris aCqarebaa. 9.14 formula sistemis raime sxva wertilis aCqarebaze informacias ar Seicavs. 9.14 formula -is da t -is koordinatTa RerZebis gaswvrivi komponentebis Semcveli sami com formulis identuria. es formulebia: Ft,x=Macom,x , Ft,y=Macom,y , Ft,z=Macom,z . (9.15) amjerad SegviZlia ukan davbrundeT da biliardis burTebis moZraoba ganvixiloT. rodesac burTi moZraobas iwyebs, ori burTisgan Semdgar sistemaze gare Zala ar moqmedebs. maSasadame  t = 0 da 9.14 formulis mixedviT acom = 0. aCqareba siCqaris cvlilebis tempia, amitom vaskvniT, rom ori burTisgan Semdgari sistemis masis centris siCqare ar icvleba. burTebs Soris Sejaxebis dros erTi burTidan meoreze Sida Zalebi iwyebs moqmedebas. es Zalebi t jamur Zalaze zegavlenas ar axdens da jamuri Zala nulis toli rCeba. maSasadame, sistemis masis centri, romelic Sejaxebamde moZraobda, Sejaxebis Semdegac igive siCqariT da mimarTulebiT imoZravebs. 9.14 formula, nivTier wertilTa sistemasTan erTad 9.1b suraTze gamosaxuli beisbolis tafelas msgavs myar sagnebsac miesadageba. am SemTxvevaSi, 9.14 formulaSi Semavali M tafelas masaa, xolo t tafelaze moqmedi mizidulobis Zalaa. igive 9.14 formulidan gamomdinare   acom = g anu tafelas masis centri ise moZraobs, TiTqos is  M masis nivTieri wertilia, romelzec Fg Zala moqmedebs. 9.5 suraTze naCvenebia kidev erTi saintereso SemTxveva. warmoidgineT, rom feierverkis dros raketa parabolis formis traeqtoriiT moZraobs. garkveul adgilze raketa feTqdeba da nawilebad iSleba. Tu raketa ar afeTqda, maSin is naCvenebi traeqtoriiT moZraobas ganagrZobs. afeTqebis Zalebi sistemis Sida Zalebia anu sistemis erTi nawilidan sur. 9.5. feierverkis raketa fre nis dros feTqdeba. haeris wevis ararsebobis SemTxvevaSi raketis nawilebis masis centri sawyisi pa raboluri traeqtoriiT ganagrZobs moZraobas, vidre nawilebi miwaze dai wyeben dacemas. meoreze moqmedeben. haeris wevis Zalis ugulebelyofis SemTxvevaSi, sistemaze moqmedi jamuri t gare Zala mizidulobis Zalaa da mniSvneloba ar aqvs raketa afeTqdeba Tu ara. 9.14 formulidan gamomdinare, raketis nawilebis masis centris   acom aCqareba g -s tolia. es niSnavs, rom raketis nawilebis masis centri igive paraboluri traeqtoriiT moZraobs, romliTac raketa imoZravebda. 282 Tavi 9 masis centri da impulsi sur. 9.6. didi jete. (kenet lousi, cekvis fizika, Schirmer Books, 1984). scenaze naxtomis Sesrulebis dros balerina mxrebs maRla swevs da xelebs horizontalurad Slis (sur. 9.6). amis Semdeg balerinas masis centri maRla gadaadgildeba. masis centri paraboluri traeqtoriiT scenis gaswvriv moZraobs da sxeulisadmi misi moZraoba balerinas Tavisa da torsis mier aRebul simaRles amcirebs. Sedegad balerinas Tavi da torsi horizontaluri traeqtoriiT moZraobs da farfatis iluzia iqmneba. 9.14 formulis damtkiceba n raodenobis nivTier wertilTa sistemisTvis 9.8 formula gvaZlevs: (9.16)  sadac M aris sistemis jamuri masa, xolo rcom - sistemis masis centris mdebareobis ganmsazRvreli veqtori. 9.16 formulis droiT diferencireba mogvcems: aq (9.17) (=d /dt) aris i rigis nivTieri wertilis siCqare, xolo i (=d com /dt) - masis centris com siCqare. 9.17 formulis droiT diferencireba mogvcems: (9.18)   aq ai (=d i /dt) aris i rigis nivTieri wertilis aCqareba, xolo acom (=d com /dt) - masis centris aCqareba. miuxedavad imisa, rom masis centri geometriuli wertilia, mas mdebareoba, siCqare da aCqareba nivTieri wertilis msgavsad gaaCnia.   niutonis meore kanonidan gamomdinare, mi ai aris i rigis nivTier wertilze moqmedi Fi Zalis toli. maSasadame, 9.18 formula ase SeiZleba Caiweros: (9.19) 9.19 formulis marjvena mxareze mocemuli Zalebi sistemis nivTier wertilebs Soris moqmedi 9.3. niutonis meore kanoni nivTier wertilTa sistemisTvis 283 Zalebi (Sida Zalebi) da sistemis garedan nivTier wertilze moqmedi Zalebia (gare Zalebia). niutonis mesame kanonidan gamomdinare, Sida Zalebi mesame kanonis ZalTa wyvilebs qmnis da erTmaneTs abaTileben. 9.19 formulis marjvena mxareze mxolod sistemaze moqmedi gare Zalebis veqtoruli jami rCeba. am SemTxvevaSi, 9.19 formula 9.14 formulaze daiyvaneba. 9.14 formula damtkicebulia. sakontrolo wertili 2. or mociguraves xaxunisgan Tavisufal yinulze umniSvnelo masis joxi sawinaaRmdego boloebiT uWiravs. RerZi joxis gaswvrivaa gavlebuli da misi aTvlis saTave mociguraveebis sistemis masis centrs emTxveva. erT-erTi mociguravis - fredis wona, meore mociguravis - eTelis wonaze orjer metia. sad Sexvdebian mociguraveebi Tu a) fredi joxs gamowevs da eTelisken gadaadgildeba, b) eTeli joxs gamowevs da fredisken gadaadgileba da g) orive mocigurave erTdroulad gamowevs joxs? amocanis amoxsnis nimuSi 9.7a suraTze gamosaxuli sami 9.3 nivTieri wertili ar moZraobs. sistemis gareT mdebare sxeulebidan TiToeul nivTier wertilze gare Zala moqmedebs. Zalebis mimarTulebebi naCvenebia, xolo maTi sidideebia F1 = 6 n, F2 = 12 n da F3=14 n. ra mimarTulebiT moZraobs sistemis masis centri da ra aCqareba aqvs mas? amoxsna: masis centris mdebareoba 9.1 amocanis amoxsnis nimuSis meTodiTaa gansazRvruli da wertiliT aRiniSneba. masis centri realur nivTier wertilad SegviZlia miviCnioT, romlis masa sistemis mTliani masis (M = 16 kg) tolia. aseve SegviZlia davuSvaT, rom samive gare Zala masis centrze moqmedebs (sur. 9.7b). masis centrisTvis niutonis meore kanoni Semdegi saxiT Caiwereba: = M com t     F1 + F2 + F3 = Macom com an aqedan gamomdinare:  acom (9.20) sur. 9.7. a) naCveneb adgilebze ganlagebul uZrav     F1 + F2 + F3 = Macom = (9.21) M 9.20 formula gvauwyebs, rom masis centris  acom aCqarebis mimarTuleba sistemaze moqmedi jamuri t nivTier wertilebze gare Zalebi moqmedebs. sistemis masis centri aRniSnulia. b) sistemis masis centri sistemis mTliani M masis toli nivTieri wertilis msgavsad moqmedebs da Zalebic masze moqmedebs.  naCvenebia jamuri t gare Zala da masis centris acom aCqareba. gare Zalis mimarTulebas emTxveva 9.7b). vinaidan nivTieri wertilebi SesaZlebelia misi komponenturi formiT Ca Tavdapirvelad ar moZraoben, masis centric wera, Semdeg acom -is komponentebis da bolos, (sur. ar moZraobs. mas Semdeg rac masis centri  aCqarebas iwyebs, is acom -is da, e.i. -is t mimarTulebiT gadaadgildeba. 9.21 formulis marjvena mxare veqtorul kalkulatorze SegviZlia gamovTvaloT. aseve   Tavad acom -is gamoTvla. x RerZis gaswvriv gveqneba: F1x + F2x + F3x M –6 n + (12 n) cos 45° + 14 n = = 1,03 m/wm2 16 kg acom,x = 284 masis centri da impulsi Tavi 9 y RerZis gaswvriv gveqneba: F + F + F3y acom,y = 1y M2y 0 + (12 n) sin 45° + 0 = = 0,53 m/wm2 16 kg mniSvnelobas da kuTxes gamovTvliT: = 1,16 m/wm2 ≈ 1,2 m/wm2 Semdeg komponentebis saSualebiT  acom -is θ = tg–1 acom,y = 27° acom,x (pasuxi) (pasuxi) 9.4. impulsi mocemul TavSi nivTier wertilTa sistemis nacvlad mxolod erT nivTier wertils ganvixilavT, rac ori mniSvnelovani sididis gansazRvris saSualebas mogvcems. pirveli gansazRvreba kargad nacnob sityvas - impulss exeba, romelsac yoveldRiur enaSi  ramdenime mniSvneloba aqvs. nivTieri wertilis impulsi p aris veqtoruli sidide:   p=m mν (nivTieri wertilis impulsi), sadac m aris nivTieri wertilis masa, xolo (9.22) - misi siCqare (xSirad impulsTan Asityva wrfivsac xmaroben, raTa es sidide kuTxuri impulsisgan ganasxvavon). vinaidan m yovelTvis dadebiTi  skalaruli sididea, 9.22 formulidan gamomdinare p -s da -s erTi da igive mimarTuleba aqvT. aseve 9.22 formula gvamcnobs, rom SI sistemaSi impulsis erTeulia kg·m/wm. niutonma moZraobis meore kanoni impulsis saSualebiT Camoayaliba: nivTieri wertilis impulsis cvlilebis tempi nivTier wertilze moqmedi jamuri Zalis tolia da am Zalis mimarTulebisaa. formulis saSualebiT es wesi ase Caiwereba: t nivTier wertilze moqmedi jamuri (9.23)  t gare Zala nivTieri wertilis p impulss cvlis da, Sesabamisad, impulsi mxolod jamuri gare Zalis moqmedebis Sedegad icvleba. Tu jamuri gare  Zala ar arsebobs, p ar icvleba. 9.7 paragrafSi gamoCndeba, rom es faqti amocanebis amoxsnaSi Zalian gvexmareba.  9.22 formulidan 9.23 formulaSi p CavsvaT da mudmivi m masis pirobebSi gveqneba: = t maSasadame, t  dp d  d = (m ) = m = m acom . dt dt dt    = d p /dt da Fnett = ma eqvivalenturi formulebia da isini nivTieri wertilis moZraobisTvis niutonis meore kanons gamoxataven. sakontrolo wertili 3. suraTze naCvenebia p impulsis t droze damokidebulebis grafiki RerZis gaswvriv moZravi nivTieri wertilisTvis. nivTier wertilze RerZis gaswvriv mimarTuli Zala moqmedebs. a) daalageT danomrili monakveTebi Zalis sididis mixedviT dididan mcireze. b) romel monakveTze neldeba nivTieri wertili? 9.5. nivTier wertilTa sistemis impulsi 285 9.5. nivTier wertilTa sistemis impulsi erTi nivTieri wertilis impulsi ukve ganvsazRvreT, amjerad ki Cveni gansazRvreba nivTier wertilTa sistemaze ganvavrcoT. ganvixiloT n raodenobis nivTieri wertilisgan Semdgari sistema, romelSic yvela nivTier wertils sakuTari masa, siCqare da impulsi aqvs. nivTieri wertilebi erTmaneTze moqmedeben da maTze gare Zalebic aris modebuli. sistemis mTliani  impulsia P , rac individualuri nivTieri wertilebis impulsebis veqtoruli jamia: (9.24)   P = mν com com (impulsi, nivTier wertilTa sistema) (9.25) Tu am formulas 9.17 formulas SevadarebT, gamoCndeba, rom: nivTier wertilTa sistemis impulsi asec SeiZleba ganisazRvros: nivTier wertilTa sistemis impulsi sistemis jamuri M masis da masis centris siCqaris namravlis tolia. Tu 9.25 formulas droiT gavawarmoebT, miviRebT: (9.26) 9.14 da 9.26 formulebis Sedareba saSualebas iZleva niutonis meore kanoni nivTier wertilTa sistemisTvis Caiweros: sadac t t (nivTier wertilTa sistema), (9.27) aris sistemaze moqmedi jamuri gare Zala. 9.27 formula erTi nivTieri wertilisTvis miRebuli  = d p /dt formulis ganzogadebaa mravali nivTieri t wertilisgan Semdgari sistemisTvis. am formulis Tanaxmad, nivTieri wertilebis sistemaze moqmedi jamuri gare Zala t  P iwevs sistemis impulsis cvlilebebs anu impulsi mxolod jamuri gare Zalis moqmedebiT icvleba. Tu gare jamuri Zala  tafelasa da burTs Soris Sejaxeba burTis nawils amtvrevs ar moqmedebs, P impulsi ar icvleba. 9.6. Sejaxeba da impulsi  nivTieri wertilis magvari nebismieri sagnis p impulsi ar icvleba, Tu jamuri gare Zala ar moqmedebs. magaliTad, Cven SegviZlia sxeuls vubiZgoT da misi impulsi SevcvaloT. ufro dramatulad Tu mivudgebiT sakiTxs, maSin SegviZlia sxeuls beisbolis tafela CavartyaT. amgvari Sejaxebis dros sxeulze moqmedi gare Zalis sidide didia, xanmokle moqmedebisaa da sxeulis impulss swrafad cvlis. Sejaxeba Cvens samyaroSi xSirad xdeba da sakmaod Zviradac ujdeba kacobriobas, rodesac saqme manqanebs exeba. manqanebis Sejaxebas mogvianebiT ganvixilavT, amjerad ki Sejaxebis ufro martivi saxeoba unda SeviswavloT, rodesac nivTieri wertilis magvari moZravi sxeuli raime sxva sxeuls (samiznes) ejaxeba. 286 masis centri da impulsi Tavi 9 erTi Sejaxeba davuSvaT, moZravi sxeuli burTia, xolo samizne - tafela. Sejaxeba xanmoklea da burTze moqmed Zalas misi Seneleba, SeCereba da moZraobis mimarTulebis Secvla SeuZlia. 9.8 suraTze naCvenebia amgvari Sejaxebis erTi momenti. burTze  F (t ) Zala moqmedebs, romelic Sejaxebis ganmavlobaSi icvleba  sur. 9.8. burTisa da da burTis p wrfiv impulss cvlis. impulsis cvlileba Zalas  Sejaxebis dros burTze  niutonis meore kanonis ( F (t=)d p /dt) saSualebiT ukavSirdeba. moqmedebs. maSasadame, dt drois SualedSi burTis impulsis cvlilebaa:   d p = F (t ) dt tafelas F (t ) Zala (9.28) Sejaxebis Sedegad gamowveul impulsis jamur cvlilebas gamovTvliT, Tu movaxdenT 9.28 formulis orive mxaris integrirebas Sejaxebamde ti droidan Sejaxebis Semdeg tf dromde: (9.29)     mxares Sejaxebis J Zalis impulsi ewodeba:  J (Zalis impulsis gansazRvreba). am formulis marcxena mxareze impulsis p f - pi = Dp cvlilebaa mocemuli. formulis marjvena (9.30) maSasadame, sagnis impulsis cvlileba Zalis impulsis tolia:   Δ p = J (impulsi-Zalis impulsis Teorema) (9.31) es gamosaxuleba veqtoruli saxiTac SeiZleba Caiweros:    p f - pi = D J p (9.32) veqtoruli gamosaxulebis komponenturi formaa: Δpx = Jx pf x – pi x da  Fx dt (9.33) (9.34)  (a) Tu F (t ) funqcia viciT, maSin am funqciis integrebiT J -s  gamovTvliT. Tu F -is t-ze damokidebulebis funqciis grafiki gvaqvs mocemuli, maSin grafiksa da t RerZs Soris farTobis  (sur. 9.9a) gamoTvla aseve J -s mogvcems. xSirad ar viciT, Tu rogor icvleba Zala droSi, magram viciT Zalis FsaS saSualo sidide da Sejaxebis Δt(=tf – ti) xangrZlivoba. am SemTxvevaSi Zalis impulsis sididea: J=FsaS Δt (9.35) 9.9b suraTze naCvenebia saSualo Zalis droze damokidebule bis funqciis grafiki. grafikis qveda arealis farTobi 9.9a suraTze gamosaxuli F(t) realuri Zalis grafikis qveda arealis farTobis tolia, vinaidan orive farTobi Zalis impulsis J sididis tolia. burTis nacvlad aqcenti tafelaze SegviZlia gadavitanoT (sur. 9.8). niutonis mesame kanoni gvauwyebs, rom nebismieri momentSi tafelaze moqmedi Zala burTze moqmedi Zalis toli da mimarTulebiT sapirispiroa. 9.30 formulidan gamomdinare (b) sur. 9.9. a) grafiki 9.8 suraTze gamosaxuli Sejaxebis dros burTze moqmed droSi cvalebad F(t) Zalas asaxavs. grafikis qveda arealis  farTobi burTze moqmedi J Zalis impulsis tolia. b) marTkuTxedis Δt drois SualedSi simaRle burTze moqmedi FsaS saSualo Zalaa. marTkuTxedis farTobi grafikis qveda arealis farTobis tolia da,  maSasadame, isic J Zalis impulss gvaZlevs. 287 9.6. Sejaxeba da impulsi es niSnavs, rom tafelaze moqmedi Zalis impulsis sidide burTze moqmedi Zalis impulsis toli da mimarTulebiT sapirispiroa. sakontrolo wertili 4. sahaero medesantes para sur. 9.10. Tanabari wrfivi impulsebis Suti ar gaexsna da TovlSi CaeSva. medesanti msu mqone Wurvebi uZrav samiznes ejaxeba. FsaS saSualo samizneze moqmedi Zala marjvniskenaa mimarTuli da misi sidide sagnebis Sejaxebis anu masebis samiznesTan Sejaxebis tempzea damokidebuli. buqad daSavda. SiSvel miwaze dacemis SemTxvevaSi medesantis gaCerebis dro 10-jer naklebi iqneboda, xolo Sejaxeba sasikvdilo aRmoCndeboda. Tovlis arseboba a) medesantis impulsis cvlilebas, b) mede santis SemaCerebel Zalis impulss da g) medesantis SemaCerebel Zalas zrdis, amcirebs Tu ar cvlis? SejaxebaTa seria amjerad sxeulze moqmedi Zala ganvixiloT, rodesac sxeuli identur SejaxebaTa serias ganicdis. magaliTad, SegviZlia ganvixiloT CogburTis burTebis kedelTan Sejaxeba. TiToeuli amgvari Sejaxebis Sedegad kedelze Zala moqmedebs. Cven SejaxebaTa mravalricxovani seriis dros Zalis saSualo mniSvneloba gvainteresebs. 9.10 suraTze gamosaxulia m masis da m wrfivi impulsis mqone Wurvebis nakadi, romelic x RerZis gaswvriv gadaadgildeba da uZrav samizne sxeuls ejaxeba. Δt drois SualedSi samiznesTan Sejaxebuli Wurvebis raodenobaa n. vinaidan moZraoba mxolod x RerZis gaswvriv xdeba, impulsis RerZis gaswvrivi komponentebi SegviZlia gamoviyenoT. maSasadame, TiToeuli Wurvis sawyisi impulsia mν da Sejaxebis Sedegad misi wrfivi impulsiT Δp-iT icvleba. Δt drois SualedSi n raodenobis Wurvebis wrfivi impulsebis jamuri cvlilebaa nΔp. igive drois SualedSi moqmedi  J Zalis impulsi x RerZis gaswvrivaa mimarTuli, misi sididea nΔp, xolo mimarTuleba wrfivi impulsebis sawinaaRmdegoa. Zalis impulsis gamosaxuleba komponentur formaSi ase Caiwereba: J = –nΔp , (9.36) minus niSani asaxavs, rom J-s da Δp-s sapirispiro mimarTulebebi aqvT. 9.35 formulis gardaqmna da masSi 9.36 formulis Casma Sejaxebebis dros samizneze moqmed FsaS saSualo Zalas mogvcems: FsaS = J Δt =– n Δt Δp = – n Δt m Δv , (9.37) FsaS-s 9.37 formula n/Δt-s saSualebiT gvaZlevs, rac Wurvebis samiznesTan Sejaxebis tempia. Δν Wurvebis siCqaris cvlilebaa. Tu Wurvebi Sejaxebis dros Cerdeba, maSin 9.37 formulaSi Δν-s nacvlad SegviZlia CavsvaT: Δν = νf – νi = 0 – ν = –ν , (9.38) sadac νi (= ν) da νf (= 0) Wurvis siCqareebia Sejaxebamde da Sejaxebis Semdeg. Tu Wurvi samiznidan siCqaris sidids cvlilebis gareSe ukuigdeba, maSin νf = –ν da gveqneba: Δν = νf – νi = – ν – ν = –2ν (9.39) Δt drois SualedSi samiznes Δm = nm masa ejaxeba. maSasadame, 9.37 formula ase SeiZleba Caiweros: FsaS = – Δm Δv Δt (9.40) 288 masis centri da impulsi Tavi 9 9.40 formula FsaS Zalas Δm/Δt-s saSualebiT gvaZlevs, rac samiznesTan masis Sejaxebis tempia. amjeradac SegviZlia Δν 9.38 an 9.39 formulis saSualebiT SevcvaloT, rac Wurvebis moqmedebazea damokidebuli. sakontrolo wertili 5. suraTze naCvenebia vertikaluri kedlidan siCqaris sididis cvlilebis gareSe asxletili  burTi. ganixileT burTis wrfivi impulsis Δ p cvlileba. a) Δpx dadebiTia, uaryofiTia Tu nulis tolia? b) Δpy dadebiTia, uaryofiTia Tu nulis tolia? g) ra mimarTulebas  aqvs Δ p -s? amocanis amoxsnis nimuSi 9.4 9.11a suraTze naCvenebia Sejibris dros am kedelTan Sejaxebuli manqanis traeqtoria. vTvaloT, da νi=70m/wm siCqariT wrfivad moZraobs. Semdeg manqana kedlisadmi formulis kalkulatorze mZRoli Sejaxebamde kedlisadmi 30° kuTxiT Sejaxebis    JJ = = p f - pi (9.41) marjvena pirdapir mxare veqtorul SegviZlia gamo m=80kg, f =50m/wm-s i =70m/wm-s 30° kuTxiT. am vinaidan (–10°) kuTxiT da SemTxvevaSi 9.41 formulas komponenturi formiT gamovTvliT. x RerZis gaswvriv 10° kuTxiT da νf = 50m/wm siCqariT ganagrZobs moZraobas. mZRolis masaa 80kg.  gveqneba: a) ra J Zalis impulsi moqmedebs mZRolze Jx = m(νfx – νix ) = (80kg)[(50m/wm)cos(–10°) – (70m/wm)cos30°] = –910 kg·m/wm Sejaxebis Sedegad? amoxsna: Cven SegviZlia mZRoli nivTieri wertilis magvar sagnad miviCnioT, Tumca 9.30 y RerZis gaswvriv gveqneba: Jy = m(νfy – νiy ) =(80kg)[(50m/wm)sin(–10°) – (70m/wm)sin30°] = –3495 kg·m/wm ≈ –3500 kg·m/wm  formuliT J -s ver gamovTvliT, vinaidan  Sejaxebis dros mZRolze moqmedi F (t) Zala  ar viciT. aqedan gamomdinare, F (t) funqcia an  funqciis grafiki mocemuli ar gvaqvs da J -s am SemTxvevaSi Zalis impulsi iqneba:  3500ˆj ) kg·m/wm J = (–910 (-910iˆ -– 3500 gamoTvla am gziT SeuZlebelia.   samagierod, J -s gamoTvla mZRolis p im  pulsis cvlilebis da 9.32 formulis ( J =   J = p f - pi ) saSualebiT SegviZlia gamovTvaloT.  9.11b suraTze naCvenebia mZRolis pi impulsi  Sejaxebamde da p f impulsi Sejaxebis Semdeg. 9.32 da 9.22 formulebidan gamomdinareobs: (pasuxi) Zalis impulsis sididea: J= = 3616 kg·m/wm ≈ 3600 kg·m/wm  J Zalis impulsis kuTxea: J θ = tg–1 y (pasuxi) Jx kalkulatori gvaZlevs 75,4°-s, magram gaixse neT, rom Sebrunebuli tangesis fizikurad (a) (b) (g)  sur. 9.11. a) mZRolisa da manqanis mier ganvlili traeqtoria kedelze Sejaxebis dros. b) mZRolis pi sawyisi   impulsi da p f saboloo impulsi. g) mZRolze moqmedi J Zalis impulsi Sejaxebis dros. 289 9.7. impulsis mudmivoba swori pasuxis misaRebad am Sedegs 180° unda = 2,583 × 105 n ≈ 2,6 × 105 n (pasuxi) gamoviyenoT F = ma formula (sadac m = 80kg)  daematos. Tu J -s komponentebs gavavlebT (sur. 9.11g), aRmoCndeba, rom swori pasuxia 75,4° + 180° = 255,4° anu: θ = –105° (pasuxi) da davadgenT, rom Sejaxebis ganmavlobaSi mZRolis aCqarebaa daaxloebiT 3,22 × 10 m/wm = 329g. savaraudod, mZRoli 2 sTvis Sejaxeba sasikvdilo iqneboda. b) Sejaxeba 14 miliwamis ganmavlobaSi grZel inJiner-meqanikosebi safrTxis Semcirebis deba. ra saSualo Zala moqmedebs mZRolze Tvis trasis kedlebs ise ageben, rom Sejaxeba Sejaxebis ganmavlobaSi? ufro amoxsna: 9.35 formulidan gamomdinare, FsaS didxans gagrZeldes. magaliTad, Tu Cvens amocanaSi Sejaxeba 10-jer ufro saSualo Zala J Zalis impulsis da Sejaxebis Δt didxans gagrZeldeba da sxva monacemebi ar xangrZlivobis Sefardebis tolia: FsaS = saSualo 3 Seicvleba, saSualo Zala da saSualo aCqareba J 3616 kg·m/wm = 0,014 wm Δt 10-jer naklebi iqneba, rac mZRols gadarCenis saSualebas miscems. 9.7. impulsis mudmivoba davuSvaT, nivTier wertilTa sistemaze moqmedi jamuri  t gare Zala (da maSasadame jamuri J Zalis impulsi) nulia (sistema izolirebulia) da garedan sistemaSi nivTieri wertilebi ar Sedis da sistemidan ar gamodis. 9.27 formulaSi  t = 0 Casma dP /dt=0 pasuxs mogvcems anu:  P = mudmiva (Caketili, izolirebuli sistema) (9.42)  Tu nivTier wertilTa sistemaze jamuri gare Zala ar moqmedebs, sistemis mTliani P impulsi ar icvleba. miRebul Sedegs impulsis Senaxvis kanoni ewodeba. es kanoni Semdegi formuliTac SeiZleba Caiweros:   P i = P f (Caketili, izolirebuli sistema) (9.43) maSasadame, Caketili, izolirebuli sistemisTvis: ( mTliani impulsi raime sawyis ti droSi )( = mTliani impulsi raime momdevno tf droSi ) gafrTxileba: impulsi da energia erTmaneTSi ar aurioT. mocemuli Tavis amocanis amoxsnis nimuSebSi impulsi inaxeba, xolo energia ara. 9.42 da 9.43 formulebi veqtoruli gantolebebia da TiToeuli maTganis eqvivalenturi sami formula arsebobs, romelic Tavis mxriv xyz koordinatTa sistemis urTierTperpendikularuli sami mimarTulebiT impulsis Senaxvas Seesabameba. impulsi SeiZleba erTi an ori mimarTulebiT inaxebodes, xolo yvela mimarTulebiT ara. amis mizezi sistemaze moqmedi Zalebia. Tu Caketil sistemaze moqmedi jamuri gare Zalis komponenti RerZis gaswvriv nulis tolia, maSin am RerZis gaswvriv sistemis impulsis komponenti ar icvleba. magaliTad mogvyavs oTaxSi gasrolili greipfruti. frenis dros greipfrutze (romelsac  sistemad miviCnevT) mxolod vertikalurad qvemoT mimarTuli Fg mizidulobis Zala moqmedebs. maSasadame, greipfrutis impulsis vertikaluri komponenti icvleba da vinaidan greipfrutze horizontaluri gare Zala ar moqmedebs, impulsis horizontaluri komponenti ucvlelia. 290 Tavi 9 masis centri da impulsi yuradReba miaqcieT, rom Cven aqcents Caketil sistemaze moqmed gare Zalebze vakeTebT. Sida Zalebs sistemis nawilebis impulsis Secvla SeuZlia, magram erTiani sistemis mTlian impulss ver cvlis. mocemuli Tavis amocanis amoxsnis nimuSebSi ganxilulia erTganzomilebiani da organzomile biani afeTqebebi. erTganzomilebiani niSnavs, rom afeTqebamde da afeTqebis Semdeg moZraoba erTi RerZis gaswvriv xdeba, xolo organzomilebiani Sesabamisad miuTiTebs, rom moZraoba ori RerZis Semcvel sibrtyeze mimdinareobs. momdevno TavebSi erTganzomilebian da organzomilebian Sejaxebebs ganvixilavT. sakontrolo wertili 6. Tavdapirvelad uZravi sagani xaxunisgan Tavisufal zedapirze afeTqda da or nawilad daiSala. erTi nawili x RerZis dadebiTi mimarTulebiT gasrialda. a) risi tolia nawilebis mTliani impulsi afeTqebis Semdeg? b) SesaZlebelia Tu ara meore nawilma x RerZisadmi raime kuTxiT imoZraos? g) ra mimarTuleba aqvs meore nawilis impulss? amocanis amoxsnis nimuSi 9.5 erTganzomilebiani afeTqeba: m = 6 kg masis mudmivia da x RerZis gaswvriv 9.43 formula xmis misacemi yuTi xaxunisgan Tavisufal SegviZlia gamoviyenoT. zedapirze, x RerZis dadebiTi mimarTulebiT sistemis sawyisi impulsi yuTis impulsis ν = 4 m/wm siCqariT misrialebs. Semdeg yuTi feTqdeba da or nawilad iSleba. m1 = 2 kg masis erTi nawili x RerZis dadebiTi mimarTulebiT ν1 = 8 m/wm siCqariT moZraobs. ra siCqare aqvs m2 masis meore nawils? tolia: amoxsna: meore nawilis siCqares miviRebdiT, Tu gvecodineboda misi impulsi, vinaidan m2 = m – m1 = 4 kg. nawilebis impulsebi da yuTis impulsi erTmaneTs daukavSirdeba, Tu impulsi mudmivia. modiT SevamowmoT. Cveni aTvlis saTave zedapiria. yuTisa da Semdeg ori nawilisgan Semdgari sistema Caketilia, magram izolirebuli ar aris, vinaidan yuTsa da nawilebze drekadobis Zala da mizidulobis Zala moqmedebs. es Zalebi vertikalurad aris mimarTuli da sistemis impulsis horizontalur komponents ver   Pi = mν , yuTis nawilebis saboloo impulsebia:   Pf 1 = m1ν11 da   Pf 2 = m 2ν2 2  sistemis saboloo, mTliani Pf impulsi nawi lebis impulsebis veqtoruli jamia:    Pf = Pf 1 + Pf 2 =m 1 +m 2 vinaidan siCqareebi da impulsebi x RerZis gaswvriv gavlebuli veqtorebia, isini x komponentebis saSualebiTac SeiZleba gamo isaxos. gamoviyenoT es SesaZlebloba, 9.43 formula da miviRebT: Pi = Pf anu mν = m1ν1 + m2ν2 monacemebis Casmis Semdeg miviRebT: (6 kg)(4 m/wm) = (2 kg)(8 m/wm) + (4 kg)ν2 Secvlis. arc afeTqebis Sedegad miRebul ν2 = 2 m/wm Zalebs SeuZlia impulsis horizontaluri da komponentis Secvla, vinaidan isini sistemis vinaidan pasuxi dadebiTia, (pasuxi) yuTis meore Sida Zalebia. aqedan gamomdinare, sistemis nawilic x RerZis dadebiTi mimarTulebiT impulsis moZraobs. horizontaluri komponenti 291 9.7. impulsis mudmivoba amocanis amoxsnis nimuSi 9.6 erTganzomilebiani afeTqeba: 9.12a suraTze naCvenebia M jamuri masis kosmosuri gada MS 1 HS mzidi da tvirTi, romlebic x RerZis gaswvriv moZraoben. mzis mimarT maTi sawyisi siCqarea 2100 km/sT. mcire afeTqebis Sedegad gadamzidi 0,2M masis tvirTs gareT gamoisvris (sur. 9.12b). amis Semdeg gadamzidi tvirTze 500km/sT-iT ufro swrafad moZraobs anu gadamzidsa da tvirTs Soris fardobiTi νfar siCqarea 500 km/sT. ra HS siCqare aqvs am dros i= gadamzidi 0,20 M tvirTis modulu 0,80 M x x (a) (b) sur. 9.12. a) datvirTuli kosmosuri gadamzidi i sawyisi siCqariT moZraobs. b) gadamzidi tvirTs gamoisvris. tvirTis siCqare mzis mimarT MS-ia, xolo gadamzidi mzis mimarT HS siCqariT moZraobs gadamzids mzis mimarT? amoxsna: gadamzidi-tvirTis sistema Caketili es toloba simboloebiT gamovsaxoT: da izolirebulia, amitom misi jamuri impulsi νHS=νfar + νMS anu νMS = νHS – νfar mudmivia:   Pi = Pf , (9.44) (9.47) da gamosrolis Semdeg momentebs asaxavs. 9.46 formulaSi νMS-is nacvlad es formula CavsvaT, xolo Semdeg 9.45 da 9.46 formulebi 9.44 formulaSi gadavitanoT. Sedegad mi moZraoba erTi RerZis gaswvriv xdeba, amitom viRebT: sadac i da f indeqsebi tvirTis gamosrolamde Mνi = 0,2M(νHS – νfar) + 0,8MνHS impulsebi da siCqareebi x komponentebis saSualebiT SeiZleba Caiweros. tvirTis rac gvaZlevs: νHS = νi + 0,2νfar gamosrolamde gvaqvs: Pi = Mνi (9.45) anu νHS = 2100 km/sT + (0,2)(500 km/sT) = 2200 km/sT modiT, νMS iyos gamosrolili tvirTis siCqare mzis mimarT. tvirTis gamosrolis Semdeg sistemis jamuri impulsia: Pf=(0,2M)νMS + (0,8M)νHS sakontrolo (9.46) tvirTis impulsia, xolo meore wevri – gada mzidis impulsi. SegviZlia dakavSiroT: = ( gadamzidis siCqare tvirTis mimarT tvirTs Soris siCqareebi (km/sT) tvirTi gadamzidi 1500 2000 3000 b) + tvirTis siCqare mzis mimarT ) amocanis amoxsnis nimuSi organzomilebiani afeTqeba: M masis qoqosis fardobiTi siCqareebi. SeavseT gamotovebuli mniSvnelobebi. a) = )( cxrilSi gamosrolis Semdeg) da gadamzidsa da magram es siCqare sxva, cnobil siCqareebTan ) 7. tvirTis siCqareebi (mzis mimarT, tvirTis Cven ar viciT tvirTis νMS siCqare mzis mimarT, ( wertili mocemulia kosmosuri gadamzidisa da formulis marjvena mxaris pirveli wevri gadamzidis siCqare mzis mimarT (pasuxi) g) 1000 fardobiTi siCqare (km/sT) 400 600 9.7 9.13). samive nawili zedapiris gaswvriv kaklis SigniT moTavsebulma asafeTqebelma, gasrialda. 0,3M masis C nawilis saboloo xaxunisgan Tavisufal zedapirze uZravad siCqarea νfC = 5 m/wm. dadebuli kakali sam nawilad daSala (sur. a) ra iqneba 0,2M masis B nawilis siCqare? 292 masis centri da impulsi Tavi 9 amoxsna: modiT, SevamowmoT, inaxeba Tu ara impulsi. amisTvis gaviTvaliswinoT, rom 1) qoqosis kakali da misi nawilebi Caketil sistemas qmnian, 2) afeTqebis Zalebi sistemis SigniT moqmedeben da 3) sistemaze jamuri gare Zala ar moqmedebs. aqedan gamomdinare, sistemis impulsi mudmivia. Tavdapirvelad xy koordinatTa sistema 9.13b suraTis msgavsad gavavloT da x RerZis uaryofiTi mimarTuleba fA-s mimarTulebas davamTxvioT. am SemTxvevaSi, x RerZi 80°-s adgens, xolo fB-sadmi fC-sadmi kuTxe 50°-ia. sur. 9.13. afeTqebuli qoqosis kakalis sami nawili xaxunisgan Tavisufal zedapirze sami mimarTulebiT gadaadgildeba. a) movlenis zedxedi. b) igive movlena organzomilebian koordinatTa sistemaSi. impulsi TiToeuli RerZis gaswvriv calcalke inaxeba. modiT y RerZi gamoviyenoT da inaxeba, gveqneba: CavweroT: Pfy-is gamosaxulebis misaRebad, 9.22 formulis y komponentiani forma (py = mνy) gamoviyenoT da TiToeuli nawilis saboloo impulsis y Pix=Pfx , (9.49) sadac Pix=0, vinaidan qoqosis kakali Tavidan uZravad devs. Pfx-is misaRebad saboloo impulsis x komponentebi gamovTvaloT. amisTvis gavixsenoT, rom A nawilis masa 0,5M (= M – 0,2M – 0,3M) unda iyos: pfA,x = –0,5MνfA, pfB,x = 0,2MνfB,x = 0,2MνfB cos 50°, pfC,x = 0,3MνfC,x = 0,3MνfC cos 80°. amis Semdeg, 9.49 formula aseTi saxiT komponenti gamovTvaloT: Caiwereba: Piy=Pfy , (9.48) sadac i indeqsi impulsis sawyis mniSvnelobas  (afeTqebamde) asaxavs, xolo y indeqsi Pi -s an  Pf -s y komponentze mianiSnebs. sawyisi impulsis Piy komponenti nulis tolia, vinaidan qoqosis kakali Tavidan uZravad devs. Pix = Pfx = pfA,x + pfB,x + pfC,x νfC = 5 m/wm da νfB = 9,64 m/wm da gveqneba: 0 = –0,5MνfA + 0,2M(9,64 m/wm)cos 50° +0,3M(5 m/wm)cos 80° pfA,y = 0, pfB,y = – 0,2MνfB,y = –0,2MνfB sin50°, pfC,y = 0,3MνfC,y = 0,3MνfC sin80°. (PfA,y = 0 pasuxi RerZebis arCevidan gamomdi nareobs). amjerad 9.48 formula Semdegi saxiT saidanac gamovTvliT: SeiZleba Caiweros: Piy = Pfy = pfA,y + pfB,y + pfC,y sakontrolo νfC = 5 m/wm da gveqneba: 0 = 0 – 0,2MνfB sin 50° + (0,3M)(5 m/wm) sin 80°, saidanac miviRebT: νfB = 9,64 m/wm ≈ 9,6 m/wm νfA=3m/wm (pasuxi) wertili (pasuxi) 8. davuSvaT, afeTqebuli qoqosis kakali 9.13 suraTze y RerZis uaryofiTi gadaadgildeba. inaxeba mimarTulebiT Tu ara misi impulsi a) x RerZis gaswvriv (rogorc amas b) ra siCqare aqvs A nawils? 9.49 formula amtkicebs da b) y RerZis gaswvriv (rogorc amas 9.48 formula amoxsna: vinaidan impulsi x RerZis gaswvriv amtkicebs)? amocanis amoxsnis taqtika taqtika 2. impulsis mudmivoba izolirebulia. Caketil sistemaSi nivTiereba izolirebul sistemaze jamuri gare Zala ar moqmedebs. araizolirebul sistemaSi impulsis TiToeuli komponenti cal-calke inaxeba, Tu jamuri gare Zalis Sesabamisi komponenti nulis tolia. maSasadame, erTi komponenti SeiZleba Seinaxos, xolo meore (nivTieri wertilebi) ar Sedis da ar gamodis. ara. impulsis SenaxvasTan dakavSirebuli amo canebis amoxsnisas Tavdapirvelad darwmu ndiT, rom Tqveni sistema Caketili da 9.8. impulsi da kinetikuri energia Sejaxebebis dros amis Semdeg sistemis ori mdgomareoba (sawyisi da saboloo) airCieT da sistemis impulsis gamosaxulebebi orive mdgomareobisTvis CawereT. gamosaxulebis Cawerisas Tqvens mier arCeuli aTvlis inerciuli sistema ar dagaviwydeT da aseve darwmundiT, rom mTel 293 sistemas moicavT da mis arc erT nawils ar tovebT an iseT sagnebs ar ganixilavT, romlebic sistemas ar ekuTvnis.   sabolood Pi da Pf erTmaneTs gautoleT da gamoTvaleT, rasac gekiTxebian. 9.8. impulsi da kinetikuri energia Sejaxebebis dros 9.6 paragrafSi nivTieri wertilis magvari sxeulebis Sejaxeba ganvixileT, magram mTeli yuradReba mxolod erT sxeulze gvqonda gadatanili. Semdgom paragrafebSi mTlianad sistemas gavarCevT da davuSvebT, rom sistema Caketili da izolirebulia. 9.7 paragrafSi  amgvari sistemisTvis davadgineT, rom misi mTliani P ar icvleba, vinaidan jamuri gare Zala ar moqmedebs. es wesi Zalian gamosadegia da winaswar gvamcnobs Sejaxebis Sedegebs detalebis codnis gareSe. CvenTvis ori Sejaxebuli sxeulisgan Semdgari sistemis mTliani kinetikuri energiac sainteresoa. Tu Sejaxeba mTlian kinetikur energias ar cvlis, maSin sistemis kinetikuri energia inaxeba. amgvar Sejaxebas elastiuri (drekadi) ewodeba. yoveldRiur cxovrebaSi, magaliTad manqanebis an tafelasa da burTis Sejaxebis dros, kinetikuri energiis raRac nawili yovelTvis gadadis energiis sxva formaSi (magaliTad siTbur energiasa da bgeris energiaSi) da, maSasadame, sistemis kinetikuri energia ar inaxeba. amgvar Sejaxebas araelastiuri (aradrekadi) ewodeba. zogierT SemTxvevaSi, sxeulis Sejaxeba miaxloebiT SegviZlia elastiurad miviCnioT. warmoidgineT, rom burTi myar zedapirs daeca. Tu zedapirsa da burTs Soris Sejaxeba elastiuria, maSin burTi kinetikur energias ar dakargavs da sawyis simaRleze aisxliteba. realur cxovrebaSi asxletis simaRle naklebia da es cxadyofs, rom Sejaxebis dros kinetikuri energiis nawili daikarga da Sejaxebac araelastiuria. miuxedavad zemoTqmulisa, Cven mainc SegviZlia kinetikuri energiis mcire danakargi ugulebelvyoT da Sejaxeba elastiurad miviCnioT. sxeulebis araelastiur Sejaxebas sistemis kinetikuri energiis danakargi yovelTvis Tan axlavs. danakargi ufro metia, Tu sxeulebi erTmaneTs Seewebeba. am SemTxvevaSi, Sejaxebas sruliad araelastiuri ewodeba. beisbolis burTis da tafelas Sejaxeba araelastiuria, xolo Sejaxeba svel, sagozavian burTsa da tafelas Soris sruliad araelastiuria, vinaidan burTi tafelas ewebeba. swored amitom, beisbols sveli, sagozaviani burTebiT ar TamaSoben. 9.9. araelastiuri Sejaxebebi erT ganzomilebaSi Sejaxebamde araelastiuri Sejaxeba erT ganzomilebaSi 9.14 suraTze naCvenebia ori sxeuli erTganzomilebian Sejaxebamde da Sejaxebis Semdeg. aseve gamosaxulia siCqare Semdeg Sejaxebamde (i indeqsiT) da Sejaxebis Semdeg (f indeqsiT). es ori sxeuli Caketil da izolirebul sistemas qmnis. ori sxeulisgan Semdgari sistemisTvis impulsis Senaxvis kanoni SegviZlia CavweroT: (  mTliani Pi impulsi Sejaxebamde )( =  mTliani Pf impulsi Sejaxebis Semdeg ) sur. 9.14. 1 da 2 sxeuli x RerZis gaswvriv moZraobs. suraTze naCvenebia mdgomareoba araelastiur Sejaxebamde da Sejaxebis Semdeg. 294 Tavi 9 masis centri da impulsi impulsis Senaxvis kanoni asec SeiZleba gamoisaxos:     p1i + p 2i = p1 f + p 2 f (9.50) (impulsis mudmivoba) moZraoba erT ganzomilebaSi xdeba, amitom veqtorebis isrebi SegviZlia wavSaloT da mxolod RerZis gaswvrivi komponentebi gamoviyenoT. vinaidan p=mν, 9.50 formula aseT saxes miiRebs: m1ν1i+m2ν2i=m1ν1f+m2ν2f (9.51) Tu magaliTad masebi, sawyisi siCqareebi da erT-erTi saboloo siCqare viciT, meore saboloo siCqaresac iolad gamovTvliT. sruliad araelastiuri Sejaxeba erT ganzomilebaSi 9.15 suraTze gamosaxulia ori sxeuli sruliad araelastiur Sejaxebamde da Sejaxebis Semdeg. m2 masis sxeuli Tavdapirvelad uZravia (ν2i=0). es sxeuli samizned SegviZlia miviCnioT, xolo meore sxeuls sasroli vuwodoT. Sejaxebis Sedegad Sewebebuli sxeuli V siCqariT ganagrZobs moZraobas. Sejaxebamde Sejaxebis Semdeg amgvar situaciaSi 9.51 formula Semdeg saxes iRebs: m1ν1i = (m1 + m2)V V= anu m1 ν m1 + m2 1i (9.52) sur. 9.15. ori sxeulis sruliad (9.53) Tu, magaliTad, masebi da sasrolis ν1i sawyisi siCqare viciT, 9.53 formulis daxmarebiT V siCqaresac gamovTvliT. yuradReba miaqcieT, rom V siCqare ν1i-ze naklebi unda iyos, radgan m1/(m1 + m2) masebis Sefardeba erTze naklebia. araelastiuri Sejaxeba. m2 masis sxeuli Sejaxebamde uZravia, xolo m1 masis meore sxeuli misken moZraobs. Sejaxebis Sedegad Sewebebuli ori  sxeuli, rogorc erTi, V siCqariT ganagrZobs moZraobas. masis centris siCqare Caketil, izolirebul sistemaSi, sistemis masis centris icvleba, vinaidan jamuri gare Zala, romlis moqmedebac Secvlida siCqares, ar arsebobs. misaRebad 9.14 suraTze com siCqaris gamosaxulebis gamosaxul, ori sxeulisgan Semdgar sistemas da erT ganzomilebaSi momxdar Sejaxebas  davubrundeT. 9.25 formulidan ( P = M com) gamomdinare,  P com siCqaresa da ori sxeulisgan Semdgari sistemis jamur impulss Soris Semdegi kavSiria:  P=M (9.54) com = (m1 + m2) com  jamuri P impulsi Sejaxebis dros inaxeba, ase rom misi gamoTvla 9.50 formulis orive mxaris saSualebiT SeiZleba. sur. 9.16. 9.15 suraTze naCvenebi sistemis kadrebi, rodesac ori sxeuli sruliad araelastiurad ejaxeba erTmaneTs. sistemis masis centri yvela kadrSia gamosaxuli. masis centris com siCqareze Sejaxeba gavlenas ar axdens. Sejaxebis Semdeg sxeulebi erTmaneTs  ewebeba da maTi saerTo V siCqare com-is tolia. com siCqare Sejaxebis Sedegad ar 295 9.9. araelastiuri Sejaxebebi erT ganzomilebaSi modiT am formulis marcxena mxaris gamoyenebiT CavweroT:    P = p1i + p 2i  9.54 formulaSi P -s nacvlad es gamosaxuleba CavsvaT da com-sTvis amovxsnaT:     P P = p1i + p 2i = com = m1 + m2 m1 + m2 miRebuli formulis marjvena mxare mudmivaa, e.i. com-is (9.55) (9.56) mniSvnelobac Sejaxebamde da Sejaxebis Semdeg ar icvleba. 9.16 suraTze gamosaxul kadrebze sruliad araelastiuri Sejaxebis dros masis centris moZraobaa naCvenebi. meore sxeuli samiznea da 9.56 formulaSi misi sawyisi impulsia     p2i == m m22ν 2i2i = 0. pirveli sxeuli sasrolia da 9.56 formulaSi misi sawyisi impulsia p1i = m1ν 11ii . yuradReba miaqcieT, rom Sejaxebis Semdeg masis centri mudmivi siCqariT moZraobas ganagrZobs. Sejaxebis Semdeg sxeulebis V saerTo siCqare com-is tolia, vinaidan masis centri Sewebebul sxeulebTan erTad moZraobs. sakontrolo wertili 9. pirveli da meore sxeuli erTmaneTs sruliad araelastiurad ejaxeba (Sejaxeba erT ganzomilebaSi xdeba). ra iqneba maTi saboloo impulsi, Tu sxeulebis sawyisi impulsebia a) 10kg·m/wm da 0, b) 10kg·m/wm da 4kg·m/wm, g) 10kg·m/wm da –4kg·m/wm? amocanis amoxsnis nimuSi 9.8 balistikuri qanqara tyviebis siCqaris gasa zomad gamoiyeneboda, vidre drois sazom eleqtronul xelsawyoebs gamoigonebdnen. 9.17 suraTze gamosaxuli qanqara or grZel Tokze Camokidebuli M = 5,4 kg masis xis didi blokisgan Sedgeba. bloks m = 9,5 g masis tyvias esvrian. tyviis moxvedris Semdeg bloki+tyviis sistema amoZravdeba da maTi masebis centri qanqaris gaCerebamde h = 6,3 sm vertikalur manZils gadis. ra siCqariT moZraobda tyvia blokTan Sejaxebamde? amoxsna: tyviis ν siCqare h simaRles gansaz Rvravs, magram am ori sididis erTmaneTTan sur. 9.17. balistikuri qanqara tyviis siCqaris gasa zomad gamoiyeneba dakavSirebisTvis meqanikuri energiis Senaxvis princips ver gamoviyenebT, vinaidan tyviis awonasworebs. aqedan gamomdinare, Seja moxvedris dros meqanikuri energia sxva saxis xebis dros tyvia-blokis sistemaze moqmedi energiebSi gadadis. gamosavali is aris, rom jamuri gare Zalis impulsi nulis tolia. 1) tyviisa da blokis Sejaxeba da 2) tyviisa maSasadame, sistema izolirebulia da da blokis amoZraveba (rodesac meqanikuri jamuri impulsi mudmivia; 2) Sejaxeba erT energia inaxeba) cal-calke unda ganvixiloT. ganzomilebaSi xdeba da Sejaxebis Semdeg pirveli safexuri. tyvia-blokis sistemaSi tyviisa da blokis mimarTuleba tyviis Sejaxeba Zalze xanmoklea, amitom Sesa sawyisi moZraobis mimarTulebas emTxveva. Zlebelia ori ramis daSveba: 1) Sejaxebis Sejaxeba erT ganzomilebaSi xdeba, bloki dros blokze moqmedi mizidulobis Zala da Tavdapirvelad uZravia da tyvia bloks Tokebidan blokze moqmedi Zala erTmaneTs ewebeba, amitom impulsis Senaxvis principi 296 masis centri da impulsi Tavi 9 9.53 formuliT gamovsaxoT. Sesabamisi simboloebis Casmis Semdeg, 9.53 formula kinetikuri energia moZraobis umaRles aseT saxes iRebs: potenciuri energiis tolia. tyviisa da V= m1 v m1 + m2 (9.57) wertilSi myofi sistemis mizidulobis blokis siCqare Sejaxebis Semdeg aris V da gveqneba: 1 (m 2 meore safexuri. rodesac tyvia da bloki erTad iwyebs dedamiwis moZraobas, sistemis tyvia-bloki- meqanikuri energia inaxeba (Tokebidan blokze moqmedi Zala meqanikur energias ar cvlis, vinaidan es V-s nacvlad 9.57 formula CavsvaT da miviRebT: m+M m 0,0095 kg + 5,4 kg = 0,0095 kg v= Zala yovelTvis blokis moZraobisadmi perpendikularulad aris mimarTuli). = 630 m/wm blokis mdebareobis sawyisi done aTvlis doned miviCnioT, sadac mizidulobis + M)V 2 = (m + M)gh. (pasuxi) balistikuri qanqara ~gardamqmnelia~, romel am sac msubuqi sagnis (tyviis) maRali siCqare SemTxvevaSi, meqanikuri energiis mudmivoba masiuri sagnis (blokis) siCqareze gadayavs da niSnavs, rom moZraobis dasawyisSi sistemis gazomvas amartivebs. potenciuri energia nulis tolia. amocanis amoxsnis nimuSi 9.18 suraTze gamosaxuli ori 9.9 manqana erTmaneTs sruliad araelastiurad, x RerZis gaswvriv dros erTganzomilebiani ejaxeba. Sejaxebis dros moZraobis manqanebi Caketil sistemas qmnian. modiT davuSvaT, rom sur. 9.18. ori manqana Sejaxebamde Sejaxebis dros manqanebs Soris moqmedi Zalis impulsi Zalze didia da gzidan saburavebze moqmedi xaxunis Zalebis impulsebi SegviZlia formulidan gamomdinare: ugulebelvyoT. amgvari daSvebis Semdeg ori manqanisgan Semdgar sistemaze jamuri gare Zala ar moqmedebs. pirveli manqanis sawyisi siCqaris x RerZis gaswvrivi komponentia ν1i=+25m/wm, xolo meore manqanisa – ν2 i= –25m/wm. Sejaxebis dros TiToeul manqanaze moqmedi Zala manqanis siCqaris Δν cvlilebas gardacvalebis iwvevs. albaToba swored Δν-s da Δν2 cvlilebebi unda (9.58) Sejaxeba sruliad araelastiuria, Sejaxebis Semdeg orive manqana moZraobs, rogorc erTi sxeuli masiT m1 + m2 da siCqariT V. 9.58 formulaSi ν1f-s da ν2f-s nacvlad V CavsvaT da misTvis amovxsnaT: V= mZRolis mniSvnelobazea damokidebuli. Cven manqanebis siCqareebis Δν1 m1ν1i + m2ν2i = m1ν1f + m2ν2f m1ν1i + m2ν2i m1 + m2 (9.59) monacemebis Casmis Sedegad miviRebT: (1400kg)(+25m/wm) + (1400 g)(–25m/wm) V= m 1 + m2 = 0. gamovTvaloT. maSasadame, pirveli manqanis siCqaris cvli a) davuSvaT orive manqanaSi mxolod mZRoli lebaa: zis da TiToeuli manqanis mTliani masaa Δν1 = ν1f – ν1i = V– ν1i = 0 – (+25m/wm) = –25m/wm 1400kg. risi tolia manqanebis siCqareebis Δν1 da Δν2 cvlilebebi? meore manqanis siCqaris cvlilebaa: amoxsna: sistema Caketili da izolirebulia, Δν2 = ν2f – ν2i = V – ν2i (mZRolis CaTvliT) amitom misi jamuri impulsi mudmivia. 9.51 = 0 – (–25m/wm) = +25m/wm (pasuxi) (pasuxi) 297 9.10 elastiuri Sejaxebebi erT ganzomilebaSi b) modiT, Sejaxeba Tavidan ganvixiloT, oRond amoxsna: Sejaxebis dros impulsi inaxeba da am amjerad pirvel manqanaSi 80kg masis mgzavri faqtTan dakavSirebuli 9.58 formula Semdegi zis. risi tolia Δν1 da Δν2? saxiT SeiZleba Caiweros: amoxsna: am SemTxvevaSic zustad igive meTods viyenebT, rac amocanis a) SekiTxvis dros gamoviyeneT, oRond amjerad m1=1480kg ricxvs vsvamT formulebSi: V = 0,694 m/wm Δν2 = +25,7 m/wm Δν1-s mniSvneloba CavsvaT da gardaqmnebis Semdeg miviRebT: m2 Δν1 =– (9.61) m1 Δν2 cvlilebazea damokidebuli. 9.61 formulaSi Δν1 = –24,3 m/wm g) da Δν2 = ν2f – ν2i mZRolis gardacvalebis riski siCqaris Δν es Sedegi gvaZlevs: da m1(ν1f – ν1i) = –m2(ν2f  – ν2i) am formulaSi Δν1 = ν1f – ν1i (pasuxi) naklebia, rodesac Cans, Δν rom mniSvnelobebis Sefardeba masebis Sefardebis Sebrunebulia da 9.60 formulaSi albaTobis (r) masebis Sefardebaze manqanaSi mgzavri zis. mZRolis gardacvalebis damokidebuleba amiTaa ganpirobebuli. albaToba Δν1-zea damokidebuli da amitom amocanis a) SekiTxvidan da 9.60 formulidan pirvel mZRols naklebi safrTxe emuqreba. gamomdinare, samanqano pirveli mZRoli imyofeba, misi gardacvalebis avariebTan dakavSirebul mona cemebSi Δν aris Sedis, magram manqanebis masa riskia: r1 = c da ubeduri SemTxvevis aRwera namdvilad aris. mkvlevarebma miRebuli monacemebi Sekribes da gamoTvales, rom pirveli mZRolis gardacvalebis r1 albaToba Semdegi formuliT moicema: r1 = c ( ) m2 m1 1,79 , (9.60) sadac c mudmivaa. axseniT ratom Sedis am formulaSi m2/m1 Sefardeba da Semdeg igive formulis daxmarebiT pirveli mZRolis gardacvalebis albaTobebi SeadareT, ro desac manqanaSi mxolod mZRoli zis da mZRolTan erTad mgzavric imyofeba. amocanis b) rodesac ( 1400 kg 1400 kg ) 1,79 manqanaSi mxolod =c. SekiTxvidan da (9.62) igive 9.60 formulidan gamomdinare, rodesac manqanSi pirvel mZRolTan erTad mgzavric imyofeba, mZRolis gardacvalebis riskia: 1,79 1400 kg r1 = c 9.62 ( ) = 0,9053 c . 1400 kg + 80 kg formula c-s nacvlad CavsvaT da gveqneba: r1´ = 0,9053r1 ≈ 0,91r1 . (pasuxi) amrogad, pirveli mZRolis gardacvalebis riski daaxloebiT 9%-iT naklebia, rodesac manqanaSi mgzavric imyofeba. 9.10 elastiuri Sejaxebebi erT ganzomilebaSi 9.8 paragrafSi ukve vTqviT, rom realur cxovrebaSi Sejaxebebi araelastiuria, magram zogierTi maTgani miaxloebiT SegviZlia elastiurad miviCnioT anu davuSvaT, rom Sejaxebuli sxeulebis jamuri kinetikuri energia ar icvleba da energiis sxva saxeobebSi ar gadadis: ( )( jamuri kinetikuri energia Sejaxebamde = jamuri kinetikuri energia Sejaxebis Semdeg es ar niSnavs, rom calkeuli sxeulis kinetikuri energia ar icvleba. ) elastiuri Sejaxebis dros TiToeuli sxeulis kinetikuri energia SesaZloa Seicvalos, magram sistemis jamuri kinetikuri energia ar icvleba. magaliTad, biliardis TamaSis dros erTi burTis meoresTan Sejaxeba miaxloebiT SegviZlia elastiurad miviCnioT. Tu Sejaxeba pirdapir xdeba, Sejaxebuli burTis kinetikuri energia TiTqmis mTlianad samizne burTs gadaecema (miuxedavad Cveni daSvebisa, Sejaxebis dros gamocemuli xma niSnavs, rom kinetikuri energiis raRac nawili mainc gadavida bgeris energiaSi). 298 Tavi 9 masis centri da impulsi uZravi samizne 9.19 suraTze gamosaxulia ori sxeuli erT ganzomilebaSi Sejaxebamde da Sejaxebis Semdeg. Sejaxeba biliardis burTebis Sejaxebis magvaria. m1 masis da ν1i sawyisi siCqaris sasroli sxeuli m2 masis da uZravi (ν2i=0) samizne sxeulisken moZraobs. davuSvaT, rom ori sxeulisgan Semdgari sistema Caketili da izolirebulia. am SemTxvevaSi, sistemis jamuri impulsi inaxeba da 9.51 formulidan gamomdinare: m1ν1i = m1ν1f + m2ν2f (9.63) (impulsi) sur. 9.19. pirveli sxeuli x RerZis gaswvriv moZraobs da uZrav meore sxeuls elastiurad ejaxeba. Sejaxebis Semdeg orive sxeuli RerZis gaswvriv moZraobas ganagrZobs. Tu Sejaxeba elastiuria, maSin jamuri kinetikuri energiac inaxeba da gveqneba: 1 m ν 2 = 1 m1ν1f2 + 1 m2ν2f2 (kinetikuri energia) 2 1 1i 2 2 (9.64) TiToeul am formulaSi i indeqsi sxeulebis sawyis siCqareebs asaxavs, xolo f indeqsi - saboloo siCqareebs. Tu sxeulebis masebi da pirveli sxeulis ν1i sawyisi siCqare viciT, ucnobi sidideebi mxolod sxeulebis ν1f da ν2f saboloo siCqareebia. Cvens gankargulebaSi ori formulaa da maSasadame ori ucnobis gamoTvla SesaZlebelia. Tavdapirvelad 9.63 formula Semdegi saxiT CavweroT: m1(ν1i – ν1f) = m2ν2f (9.65) 9.64 formulas ki aseTi saxe mivceT: m1(ν1i – ν1f)(ν1i + ν1f) = m2ν2f2 (9.66) Semdeg 9.66 formula 9.65 formulaze gavyoT da gardaqmnebi SevasruloT: v1f = m1 – m 2 v m1 + m2 1i (9.67) da v2f = 2m1 υ . m1 + m2 1i (9.68) 9.68 formulaSi Cans, rom ν2f siCqare yovelTvis dadebiTia (Tavdapirvelad uZravi, m2 masis samizne sxeuli yovelTvis moZraobas iwyebs). 9.67 formulidan gamomdinare ν1f siCqares orive niSani SeiZleba hqondes (m1 masis sasroli sxeuli moZraobas ganagrZobs, Tu m1>m2, xolo Tu m1<m2, sasroli aisxliteba). modiT ramdenime gansakuTrebuli SemTxveva ganvixiloT. 1. Tanabari masebi. Tu m1=m2, 9.67 da 9.68 formulebi martivdeba: ν1f = 0 da ν2f = ν1i amgvar Sedegs mxolod biliardis kargi moTamaSeebi iReben. am formulebis Tanaxmad, Tanabari masebis sxeulebis Sejaxebis Semdeg pirveli sxeuli (Tavdapirvelad moZravi) Cerdeba, xolo meore sxeuli (Tavdapirvelad uZravi) pirveli sxeulis siCqariT iwyebs moZraobs. sxvagvarad rom vTqvaT, Tanabari masebis sxeulebi ubralod siCqareebs cvlian. igive Sedegs miviRebT, Tu meore sxeuli Tavdapirvelad uZravi ar aris. 2. masiuri samizne. 9.19 suraTze masiuri samizne niSnavs, rom m2 >> m1. magaliTad, Cven SegviZlia golfis burTi zarbaznis uZrav birTvs movaxvedroT. am SemTxvevaSi, 9.67 da 9.68 formulebi mogvcems: ν1f ≈ –ν1i da ν2f ≈ ( ) 2m1 ν1i m2 (9.69) 299 9.10 elastiuri Sejaxebebi erT ganzomilebaSi maSasadame, pirveli sxeuli (golfis burTi) aisxliteba da sawyisi traeqtoriiT sawinaaRmdego mimarTulebiT moZraobs, xolo misi siCqaris sidide ar icvleba. uZravi meore sxeuli (zarbaznis birTvi) dabali siCqariT win gadaadgildeba, vinaidan 9.69 formulis frCxilebSi Casmuli sidide erTze bevrad naklebia. 3. masiuri sasroli. am SemTxvevaSi m1 >> m2 da ukve zarbaznis birTvs vesvriT golfis burTs. 9.67 da 9.68 formulebi gvaZlevs: ν1f ≈ ν1i da ν2f ≈ 2ν1i . (9.70) 9.70 formulis Tanaxmad, pirveli sxeuli (zarbaznis birTvi) moZraobas ganagrZobs da misi siCqare TiTqmis ucvlelia. meore sxeuli (golfis burTi) zarbaznis birTvis win orjer ufro swrafad gagordeba. SesaZloa gagiCndeT SekiTxva: ratom maincdamainc orjer swrafad? pasuxis gasacemad gavixsenoT, rom 9.69 formuliT aRwerili Sejaxebis dros msubuqi sxeulis (golfis burTis) siCqare +ν-dan –ν-mde anu 2ν-iT icvleboda. igive xdeba am SemTxvevaSic, magram amjerad siCqare 0-dan 2ν-mde icvleba. moZravi samizne Cven ukve ganvixileT sasroli sxeulis da uZravi samiznis elastiuri Sejaxeba. modiT amjerad davakvirdeT SemTxvevas, rodesac elastiur Sejaxebamde orive sxeuli moZraobs. 9.20 suraTze gamosaxuli mdgomareobis SemTxvevaSi, impulsis Senaxvis kanoni ase Caiwereba: m1ν1i + m2ν2i = m1ν1f + m2ν2f sur. 9.20. ori sxeuli elastiur Se jaxebas erT ganzomilebaSi ganicdis (9.71) kinetikuri energiis Senaxvis kanoni ki ase saxes miiRebs: 1 m ν 2 + 1 m2ν2i2 = 1 m1ν1f2 + 1 m2ν2f2 2 1 1i 2 2 2 (9.72) ν1f-sa da ν2f-is gamoTvlisTvis 9.71 formula Semdegi saxiT CavweroT: m1(ν1i – ν1f) = –m2(ν2i – ν2f) (9.73) 9.72 formulas ki aseTi saxe mivceT: m1(ν1i – ν1f)(ν1i + ν1f) = –m2(ν2i – ν2f)(ν2i + ν2f) (9.74) Semdeg 9.74 formula 9.73 formulaze gavyoT da gardaqmnebis Sedegad miviRebT: v1f = m1 – m 2 2m2 v + v m1 + m2 1i m1 + m2 2i (9.75) v2f = 2m1 m2 – m1 v + v . m1 + m2 1i m1 + m2 2i (9.76) da yuradReba miaqcieT, rom sxeulebisTvis 1 da 2 indeqsis miniWeba pirobiTia. Tu 9.20 suraTze da 9.75-9.76 formulebSi indeqsebs gadavanacvlebT, miRebuli Sedegi ar Seicvleba. aseve gaiTvaliswineT, rom Tu ν2i = 0 anu meore sxeuli uZravi samizne xdeba, maSin 9.75-9.76 formulebi 9.67-9.68 formulebze daiyvaneba. sakontrolo wertili 10. ra iqneba 9.19 suraTze gamosaxuli samiznis saboloo impulsi, Tu sasroli sxeulis sawyisi impulsia 6kg·m/wm da saboloo impulsia a) 2kg·m/wm da b) –2kg·m/wm? g) ra iqneba samiznis saboloo kinetikuri energia, Tu sasroli sxeulis sawyisi da saboloo kinetikuri energiebia Sesabamisad 5j da 2j? 300 masis centri da impulsi Tavi 9 amocanis amoxsnis nimuSi 9.10 vertikalur Tokebze Camokidebuli liTonis ori sfero Tavdapirvelad erTmaneTs exeba (sur. 9.21). Semdeg m1=30g masis pirveli sfero h1 =8sm simaRleze marcxniv gaswies da xeli gauSves. pirveli sfero m2=75g masis meore sferos elastiurad ejaxeba. ra iqneba pirveli sferos ν1f siCqare Sejaxebis Semdeg? amoxsna: sferoebis moZraoba or etapad SegviZlia davyoT da Semdeg isini calcalke gavaanalizoT. es etapebia: 1) pirveli sferos moZraoba Sejaxebamde da 2) sferoebis Sejaxeba. pirveli etapi. pirveli sferos moZraobis dros sfero-dedamiwis sistemis meqanikuri energia inaxeba (meqanikur energias sur. 9.21. Tokebze Camokidebuli liTonis sferoebi Tavdapirvelad erTmaneTs exeba. Semdeg m1 masis pirveli sfero h1 simaRleze marcxniv gawies da xeli gauSves. meore etapi. Cven ukve davuSviT, rom Sejaxeba elastiuria, magram amjerad kidev ori Tokidan pirvel sferoze moqmedi Zala daSveba gvesaWiroeba. pirvel rigSi Cven ar cvlis, vinaidan es Zala yovelTvis sferos moZraobis SegviZlia davuSvaT, rom Sejaxeba erT mimarTulebisadmi perpendikularulia). pirveli ganzomilebaSi xdeba, vinaidan sferoebi sferos Sejaxebis mdebareobis udablesi done aTvlis doned pirveli nulis tolia. sferos am energiis tolia, da izolirebulad SegviZlia miviCnioT. es niSnavs, rom sistemis jamuri impulsi rodesac inaxeba. pirveli sfero h1 simaRleze imyofeba: 1 m ν2= 2 1 1i amis gamoyenebiT m1gh1 Semdeg Sejaxebis 9.67 formulis Semdeg pirveli sferos siCqare gamovTvaloT: es formula pirveli sferos Sejaxebamde ν1i siCqarisTvis amovxsnaT: ν1i = Semdeg sferoebisgan Semdgari sistema Caketilad energia udables doneze sistemis mizidulobis potenciuri Sejaxebis garda amisa, Sejaxeba Zalian xanmoklea da SemTxvevaSi, kinetikuri da miaxloebiT horizontalurad moZraoben. miviCnioT, sadac mizidulobis potenciuri energia win v1f = m 1 – m2 0,03 kg – 0,075 kg v = (1,252 m/wm) m1 + m2 1i 0,03 kg + 0,075 kg = –0,537 m/wm ≈ –0,54 m/wm = (pasuxi) minus niSani gvamcnobs, rom pirveli sfero = 1,252 m/wm Sejaxebis Semdeg marcxnisken moZraobs. 9.11. Sejaxebebi or ganzomilebaSi ori sxeulis Sejaxebis dros, maT Soris moqmedi Zalis impulsi sxeulebis Semdgomi moZraobis mimarTulebas gansazRvravs. rodesac Sejaxeba pirdapiri ar aris, sxeulebi sawyisi RerZis gaswvriv moZraobas ar ganagrZoben. Caketil, izolirebul sistemaSi momxdari organzomilebiani Sejaxebebis dros jamuri impulsi kvlav inaxeba:     P1i + P2i = P1 f + P2 f (9.77) sur. 9.22. or sxeuls Soris momxdari iribi, elastiuri Sejaxeba. m2 masis sxeu li (samizne) Tavdapirvelad uZravia. 301 9.12. cvladi masis mqone sistemebi: raketa Tu Sejaxeba elastiuria, maSin jamuri kinetikuri energiac inaxeba: K1i + K2i = K1f + K2f (9.78) 9.77 formula or ganzomilebaSi momxdari Sejaxebis gaanalizebisTvis ufro gamodgeba Tu mas xy koordinatTa sistemis komponentebis saSualebiT CavwerT. magaliTad, 9.22 suraTze gamosaxulia sasrol sxeulsa da Tavdapirvelad uZrav samizne sxeuls Soris momxdari iribi Sejaxeba. sxeulebs Soris moqmedi Zalis impulsebi sxeulebs x RerZisadmi, romlis gaswvriv sasroli sxeuli moZraobda, θ1 da θ2 kuTxeebiT gadaaadgileben. am SemTxvevaSi, x RerZis gaswvrivi komponentebisTvis 9.77 formula aseT saxes miiRebs: m1ν1i = m1ν1f cos θ1 + m2ν2f cos θ2 (9.79) xolo y RerZis gaswvriv gveqneba: 0 = –m1ν1f sin θ1 + m2ν2f sin θ2 9.78 formula siCqareebis meSveobiTac SeiZleba Caiweros: (9.80) 1 m ν 2 = 1 m1ν1f2 + 1 m2ν2f2 (kinetikuri energia) 2 1 1i 2 2 (9.81) 9.79-9.81 formulebi ramdenime cvlads Seicavs, esenia: ori masa (m1 da m2), sami siCqare (ν1i, ν1f da ν2f) da ori kuTxe (θ1 da θ2). Tu am cvladebidan oTxi mainc viciT, maSin danarCen samsac gamovTvliT. sakontrolo wertili 11. 9.22 suraTze sasroli sxeulis sawyisi impulsia 6kg·m/wm, impulsis saboloo x komponentia 4kg·m/wm da impulsis saboloo y komponentia –3kg·m/wm. ra iqneba samiznis impulsis a) saboloo x komponenti da b) saboloo y komponenti? 9.12. cvladi masis mqone sistemebi: raketa aqamde ganxiluli sistemebisTvis vuSvebdiT, rom maTi jamuri masa ucvlelia. zogjer, magaliTad raketis SemTxvevaSi es ase ar xdeba. gasrolis dros raketis masis umetes nawils saraketo sawvavi Seadgens, romlis umetesi nawili iwveba da raketis Zravis gamosaSvebi naxvretidan gamodis. raketis aCqarebis dros masis cvlilebis asaxvisTvis niutonis meore kanons ara mxolod raketisTvis vwerT, aramed raketisa da misi wvis produqtebisTvis. raketis aCqarebis dros am sistemis masa ar icvleba. aCqarebis gamoTvla davuSvaT, Cven aTvlis inerciuli sistemisadmi uZravad (a) vimyofebiT da raketis aCqarebas vakvirdebiT. raketa kosmosSi moZraobs da masze mizidulobis an atmosferuli wevis Zala ar moqmedebs. amgvari erTganzomilebiani moZraobis SemTxvevaSi raketis masaa M, xolo raime t droSi misi siCqarea ν (sur. 9.23a). 9.23b suraTze naCvenebia dt drois Semdeg mimdinare movlenebi. amjerad raketis siCqarea ν + dν, xolo misi masaa M + dM, sadac masis dM cvlileba uaryofiTi sididea. dt droSi raketidan gamosuli gamonabolqvis masaa –dM da Cveni aTvlis inerciuli sistemisadmi misi siCqarea U. (b) sur. 9.23. a) M masis aCqarebuli raketa t droSi aTvlis inerciuli sistemidan. b) igive mdgomareoba t + dt droSi. dt drois SualedSi gamosuli gamonabolqvi naCvenebia. 302 Tavi 9 masis centri da impulsi Cveni sistema raketisa da dt droSi gamosuli gamonabolqvisgan Sedgeba. sistema Caketili da izolirebulia, amitom dt droSi misi impulsi inaxeba: Pi=Pf (9.82) sadac i da f indeqsebi dt drois Sualedis dasawyissa da dasasruls asaxavs. 9.82 formula Semdegi saxiT SeiZleba Caiweros: Mν = –dM U + (M + dM)(ν + dν) (9.83) formulis marjvena mxares gamosaxuli pirveli wevri dt SualedSi gamosuli gamonabolqvis impulsia, xolo meore wevri raketis impulss dt Sualedis bolos asaxavs. 9.83 formula gamartivdeba, Tu raketasa da gamonabolqvs Soris arsebul fardobiT siCqares νfar gamoviyenebT. es siCqare aTvlis inerciuli sistemisadmi arsebul siCqareebs Semdegi saxiT ukavSirdeba: ( raketis siCqare aTvlis sistemisadmi )( = )( raketis siCqare gamonabolqvisadmi + gamonabolqvis siCqare aTvlis sistemisadmi ) modiT es yvelaferi simboloebis saSualebiT CavweroT: (ν + dν) = νfar + U, U = ν + dν – νfar anu (9.84) U-s gamosaxuleba 9.83 formulaSi CavsvaT da miviRebT: –dMνfar = M dν (9.85) 9.85 formulis orive mxare dt-ze gavyoT: dM dv νfar = M (9.86) dt dt dM/dt (raketis masis dakargvis tempi) –R-iT SevcvaloT, sadac R aris sawvavis moxmarebis masuri tempi, xolo dν/dt raketis aCqarebaa. amgvari gardaqmnebis Semdeg 9.86 formula gvaZlevs: – Rνfar = Ma (raketis pirveli formula) (9.87) 9.87 formula nebismieri momentisTvis gamosadegia. yuradReba miaqcieT, rom 9.87 formulis marcxena mxaris erTeuli Zalis erTeulia (kg/wm· m/wm=kg·m/wm2=n) da mxolod raketis Zravis maxasiaTeblebzea damokidebuli anu ra R tempiT moixmars sawvavs Zrava da raketisadmi ra νfar siCqariT gamodis masa raketidan. Rνfar wevrs raketis Zravis Zalvas vuwodebT da mas T simboloTi gamovsaxavT. Tu 9.87 formulas T = Ma saxiT CavwerT, sadac a aris raketis aCqareba im momentSi, rodesac misi masaa M, niutonis meore kanoni naTlad warmogvidgeba Tvalwin. siCqaris gamoTvla rogor Seicvleba raketis siCqare sawvavis moxmarebasTan erTad? 9.85 formulidan gvaqvs: dνfar = – νfar dM M integreba gvaZlevs: – νfar , sadac Mi aris raketis sawyisi masa, xolo Mf - saboloo masaa. integralebis gamoTvlis Semdeg miviRebT: νf – νi = – νfar ln Mi Mf (raketis meore formula) (9.88) 303 9.12. cvladi masis mqone sistemebi: raketa am formulis mixedviT izrdeba raketis siCqare, rodesac misi masa Mi-dan Mf-ze icvleba (9.88 formulaSi ln simbolo naturalur logariTms asaxavs). igive formula kargad asaxavs ramdenimesafexuriani raketis upiratesobas, romelSic Mf mimdebare safexurebis moSorebis Sedegad aris Semcirebuli. safexurebi raketas maSin Sordeba, rodesac maTi sawvavi amoiwureba. idealur raketas daniSnulebis adgilze mxolod sasargeblo tvirTi unda darCes. mimoxilva da Sejameba masis centri. n raodenobis nivTieri wer tilisgan Semdgari sistemis masis centri aris wertili, romlis koordinatebia: xcom = , ycom = , zcom = , (9.8) sadac M aris sistemis jamuri masa. sistemisTvis. nivTier wertilTa sistemis masis centris moZraobas nivTier wertilTa niutonis meore kanoni gansazRvravs: sadac t =M t impulsi da niutonis meore kanoni. erTi   mν nivTieri wertilis p =impulsia:   p=m mν (9.22) niutonis meore kanoni impulsis saSualebiT SeiZleba Caiweros: moqmedi saSualo Zalaa: FsaS = (9.23) nivTier wertilTa sistemisTvis gveqneba:   P = mν com com da t (9.25, 9.27) Sejaxeba da Zalis impulsi. nivTieri wertilis magvari sxeulis Sejaxebis dros J Δt sadac n/Δt =– n Δt Δp = – n Δt m Δv , (9.37) aris sxeulTa nakadis uZrav sxeulTan Sejaxebis tempi, xolo Δν - TiToeuli sxeulis siCqaris cvlileba. saSualo Zala asec SeiZleba gamoisaxos: FsaS = – sadac Δm/Δt Δm Δv , Δt (9.40) aris uZrav sxeulTan masis Sejaxebis tempi. Tu sxeulebi Sejaxebis dros Cerdeba, maSin 9.37 da 9.40 formulebSi Δν = –ν, xolo Tu sxeulebi siCqaris cvlilebis gareSe ukan aisxliteba, maSin Δν = –2ν. impulsis mudmivoba. Tu sistema izo lirebulia t (9.35) rodesac m masis da ν siCqaris mqone sxeulTa aris sistemaze moqmedi yvela gare - sistemis masis centris aCqareba. J = FsaS Δt . Sejaxebuli (9.14) com , Zalis jami, M - sistemis jamuri masa, xolo com nakadi uZrav sxeuls ejaxeba, uZrav sxeulze niutonis meore kanoni nivTier wertilTa sistemisTvis . (9.30)  Tu F (t ) Zalis saSualo sididea FsaS da Seja xeba Δt drois ganmavlobaSi grZeldeba, maSin erT ganzomilebaSi moZraobis dros: (9.5) an  J da masze jamuri gare Zala ar moqmedebs, maSin sistemis P impulsi ucvlelia:  P = mudmiva (Caketili, izolirebuli sistema) (9.42) impulsis mudmivoba asec SeiZleba Caiweros:   Pi = Pf (Caketili, izolirebuli sistema), (9.43)  niutonis meore kanonis gamoyeneba Zalis sadac i da f indeqsebi P -s mniSvnelobas raime impulsi-impulsis Teoremas gvaZlevs: sawyis da momdevno droSi asaxaven. 9.42 da     (9.31, 9.32) p f - pi = Dp = J ,    sadac p f - pi = Dp aris sxeulis impulsis  cvlileba, xolo J – Sejaxebis dros erTi  sxeulidan meoreze moqmedi F (t ) Zalis Sedegad miRebuli Zalis impulsi: 9.43 impulsis mudmivobis kanonis amsaxveli eqvivalenturi formulebia. araelastiuri Sejaxeba erT ganzomilebaSi. ori sxeulis araelastiuri Sejaxebis dros, ori sxeulisgan Semdgari sistemis kinetikuri 304 Tavi 9 masis centri da impulsi energia ar inaxeba. Tu sistema Caketili da izolirebulia, misi jamuri impulsi inaxeba v1f = m 1 – m2 v m1 + m2 1i (9.67) da v2f = 2m1 v . m1 + m2 1i (9.68) da es faqti veqtoruli formiT SegviZlia CavweroT:     p1i + p 2i = p1 f + p 2 f , (9.50) sadac i da f indeqsebi impulsis mniSvnelobebs asaxavs Sejaxebamde da Sejaxebis Semdeg. Tu sxeulebi erTi RerZis gaswvriv moZraoben, maSin Sejaxeba erT ganzomilebaSi xdeba da 9.50 formula siCqaris am RerZis gaswvrivi komponentebiT SeiZleba Caiweros: m1ν1i+m2ν2i=m1ν1f+m2ν2f . (9.51) Sejaxeba or ganzomilebaSi. Tu ori sxeuli erTmaneTs ejaxeba da erTi RerZis gaswvriv ar moZraobs (Sejaxeba iribia), maSin Sejaxeba or ganzomilebaSi xdeba. Tu ori sxeulisgan Semdgari sistema Caketili da izolirebulia, Sejaxebis dros impulsi inaxeba:     P1i + P2i = P1 f + P2 f (9.77) am formulis komponenturi forma Sejaxebis Tu Sejaxebisas sxeulebi erTmaneTs ekroba, aRmwer or formulas gvaZlevs (TiToeuli Sejaxeba sruliad araelastiuria da gaerTia ganzomilebisTvis erTi formulaa gankuT nebul sxeuls Tanabari V saboloo siCqare vnili). aqvs. rodesac Sejaxeba elastiuria, kinetikuri masis centris moZraoba. ori Sejaxebuli energiis Senaxvis principi mesame formulasac sxeulis Caketili, izolirebuli sistemis gvaZlevs: K1i+K2i=K1f+K2f masis centrze Sejaxeba gavlenas ar axdens. kerZod, masis centris com siCqare Sejaxebis Sedegad ar icvleba. (9.78) cvladi masis mqone sistemebi. gare Zalebis ararsebobis SemTxvevaSi raketa myisi elastiuri Sejaxebebi erT ganzomile baSi. elastiuri Sejaxebis dros Sejaxebuli tempiT Cqardeba: sxeulebis sistemis kinetikuri energia inaxeba. sadac M aris raketis myisi masa (sawvavis Tu sistema Caketili da izolirebulia, maSin misi impulsic inaxeba. erT ganzomilebaSi momxdari Sejaxebis dros, rodesac meore sxeuli samiznea da pirveli sxeuli _ sasroli, kinetikuri energiis da impulsis Senaxvis principi Sejaxebis Semdgomi siCqareebis gamosaxulebebs gvaZlevs: Rνfar = Ma (raketis pirveli formula), (9.87) CaTvliT), R – sawvavis moxmarebis tempi da νfar - sawvavis gamonabolqvis fardobiTi siCqare raketisadmi. Rνfar wevrs raketis Zravis Zalva ewodeba. rodesac R da νfar mudmivia, raketis siCqare νi-dan νf-ze icvleba da masa Mi-dan Mf xdeba: νf – νi = – νfar ln Mi Mf (raketis meore formula) (9.88) SekiTxvebi 1. 9.24 suraTze gamosaxuli, identuri masis zedapirze oTxi nivTieri wertili xaxunisgan Tavisufal siCqareebis mimarTulebebi naCvenebia, xolo mudmivi siCqariT moZraobs. maTi sidideebi erTi da igivea. ganixileT nivTier wertilTa wyvilebi. romeli wyvili qmnis iseT sistemas, romlis masis centri a) uZravia, b) uZravia da aTvlis saTaveze mdebareobs da g) aTvlis saTaveze gadis? 2. 9.25 suraTze gamosaxul sam nivTier wertilze gare Zalebi moqmedebs. or nivTier wertilze moqmedi Zalebis sidideebi da sur. 9.24. SekiTxva 1. mimarTulebebi naCvenebia. ra sididis da 305 mimarTulebis grafikebi. Zala moqmedi Zalis impulsis mixedviT dididan moqmedebs mesame nivTier wertilze, nivTieri tilebis daalageT grafikebi sxeulze mcireze. Tu wer siste sur. 9.25. SekiTxva 2. mis masis centri a) uZravia, b) mudmivi siCqariT marjvniv moZraobs da g) marjvniv Cqardeba? sur. 9.28. SekiTxva 5. 3. 9.26 suraTze naCvenebia sami an oTxi identuri nivTieri wertilisgan Seqmnili 6. 9.5 amocanis amoxsnis nimuSSi mocemuli yuTi x RerZis gaswvriv mudmivi, dadebiTi oTxi jgufi. wertilebi sididiT erTi da siCqariT moZraobs da afeTqebis Sedegad igive siCqariT x an y RerZis paralelurad or nawilad iSleba. m1 masis pirveli nawili moZraoben. daalageT jgufebi masis centris 1 dadebiTi siCqariT ganagrZobs moZraobas, xolo m2 masis meore nawilma SesaZlebelia a) siCqaris mixedviT dididan mcireze. dadebiTi siCqariT (sur. 9.29a), b) 2 2 uaryofiTi siCqariT (sur. 9.29b) an g) nulovani siCqariT (sur. 9.29g) imoZraos. sami SesaZlo Sedegi 1-is Sesabamisi sidideebis mixedviT dididan mcireze daalageT. sur. 9.29. SekiTxva 6. 7. horizontalur zedapirze ganlagebuli bloki an uZravia, an x RerZis dadebiTi mimarTulebiT moZraobs, an igive RerZis uaryofiTi mimarTulebiT moZraobs. Semdeg bloki or nawilad iSleba da nawilebi x RerZis sur. 9.26. SekiTxva 3. 4. 9.27 suraTze naCvenebia gaswvriv ganagrZobs moZraobas. davuSvaT, Tavisufali bloki da ori nawili Caketil, izolirebul sxeulis grafikebi, rodesac Sokoladis sam sistemas yuTze moqmedebs impulsebis droze damokidebulebis eqvsi da yuTebi xaxunisgan Tavisufal zedapirze SesaZlo grafiki 9.30 suraTzea naCvenebi. gadaadgildeba. TiToeuli yuTis SemTxvevaSi romeli grafiki asaxavs fizikurad SeuZlebel horizontaluri Zalebi impulsi x RerZis gaswvriv inaxeba Tu y RerZis qmnis. blokis da mdgomareobas da ratom? gaswvriv? sur. 9.27. SekiTxva 4. 5. 9.28 suraTze naCvenebia Sejaxebul sxeulze moqmedi Zalis droze damokidebulebis sur. 9.30. SekiTxva 7. nawilebis 306 Tavi 9 masis centri da impulsi suraTze naCvenebia pirveli blokis x mdebare- 8. ori sxeuli er TmaneTs x RerZis gaswvriv, erT ga obis t droze damokidebulebis grafiki, vidre xc adgilze da tc droSi Sejaxeba moxdeba. sura- nzomilebaSi Tis stiurad ela Cvenebia dasaTaurebul arealSi gagrZeldeba grafiki (Sejaxebis Semdeg) Tu Seejaxa. 9.31 suraTze na romel sur. 9.31. SekiTxva 8 a) m1<m2 da b) m1>m2? g) romeli daStrixuli wrfis gaswvriv gagrZeldeba grafiki, Tu sxeu lebis da masis centris mdebareobis droze m1=m2? damokidebulebis grafiki. a) sxeulebi Tav 11. dapirvelad naCvenebia xaxunisgan moZraobdnen Tu erT-erTi 9.34 suraTze sur. 9.34. SekiTxva 11. maTgani uZravi iyo? wrfis romeli monakveTi Tavisufal Seesabameba masis centris moZraobas b) Se pirze, x RerZis gaswvriv moZravi pirveli jaxebamde da g) Sejaxebis Semdeg? d) Seja bloki, xebamde ufro swrafad moZravi sxeulis masa elastiurad meore sxeulis masaze metia, naklebia Tu misi gamosaxulia ori blokisgan Semdgari sistemis tolia? masis centris sami SesaZlo mdebareoba (B 9. 9.32 suraTze na wertili Cvenebia ori sxeulis manZilzea ganlagebuli). Sejaxebis Semdeg da maTi masis centris pirveli bloki uZravia, win gadaadgildeba Tu mdebareobis vidre is uZrav Seejaxeba. blokebis meore bloks suraTze centrebidan aseve Tanabar ukan brundeba, rodesac masis centri a) A, b) B droze da g) C wertilzea ganlagebuli? damokidebulebis grafikebi. zeda sxeule 12. bloki xaxunisgan Tavisufal zedapirze bi Caketil, izoli gadaadgildeba da igive masis meore uZrav rebul sistemas qmni bloks an da erT ganzomi lebian, sur. 9.32. SekiTxva 9. sruliad ejaxeba. blokebis K 9.35 suraTze naCvenebia kinetikuri energiebis oTxi SesaZlo grafiki Sejaxebamde da Sejaxebis araelastiur Sejaxebas ganicdian. pirvel grafikze a) ori sxeuli da b) masis centri x Semdeg. a) romeli grafiki asaxavs fizikurad SeuZlebel mdgomareobas da ratom? romeli RerZis dadebiTi mimarTulebiT moZraobs Tu grafiki asaxavs saukeTesod a) elastiur uaryofiTi mimarTulebiT? g) romeli grafiki Sejaxebas da g) araelastiur Sejaxebas? asaxavs fizikurad SeuZlebel mdgomareobas da ratom? 10. m1 masis pirveli bloki xaxu nisgan Tavisufal zedapirze, x Rer Zis gaswvriv gada adgildeba da m2 masis uZrav meore bloks elastiurad ejaxeba. 9.33 sur. 9.33. SekiTxva 10. sur. 9.35. SekiTxva 12. 307 amocanebi SSM amoxsnebi mocemulia Student Solutions Manual-Si. www amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday ILW Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley. com/college/halliday. • – ••• wertilebis raodenoba amocanis sirTules gviCvenebs sur. 9.38. amocana 4. paragrafi 9.2. masis centri suraTze naCvenebi firfitis masis centri, Tu ·1. 9.36 suraTze L=5sm? naCvenebia ··5. amiakis molekulaSi (sur. 9.39) wyalbadis (H) m1=3kg, m2=4kg da m3=8kg ma sami atomi tolgverda samkuTxeds qmnis da samkuTxedis centri sebis nivTieri wertilebis sis tema. ra a) x da b) y sur. 9.36. amocana 1. koordinatze wyalbadis mdebareobs sistemis masis centri? g) Tu m3 masa TandaTanobiT izrdeba, sistemis masis centri am nivTieri wertilisken gadaadgildeba, Sordeba mas Tu adgilze rCeba? SSM ·2. 2kg masis nivTieri wertilis xy koordinatebia (–1,2m,05m), xolo 4kg masis meore nivTieri wertilis koordinatebia orive nivTieri wertili (0,6m;–0,75m). horizontalur sibrtyezea ganlagebuli. ra a) x da b) y koordinatze unda ganlagdes 3kg masis nivTieri wertili, raTa sami nivTieri wertilisgan Semd gari sistemis masis centris koordinatebi iyos (–0,5m;–0,7m)? ··3. 9.37 suraTze, L=22m sigrZis Zelaki atomur masasTan Sefardebaa 13,9, xolo azotsa da wyalbadis atomebs Soris manZilia L=10,14×10-11m. ra a) x da b) y koordinatze mdebareobs molekulis masis centri? ILW ··6. 9.40 suraTze naCvenebia firfita da misi zomebia d1=11sm, d2=2,8sm da d3=13sm. firfitis naxevari (simkvrive=2,7g/sm ), 3 aluminisgan xolo meore Sedgeba naxevari rkinaa (simkvrive=7,85g/sm ). ra a) x, b) y da 3 sami U-s qmnis. vertikaluri Zelakebis masaa 14g, xolo horizontaluri Zelakis masa 42g-ia. ra a) x da b) y koordinatze mdebareobs sistemis masis erTnairi TiToeuli atomidan d=9,4×10-11m sur. 9.39. amocana 5. manZiliTaa daSorebuli. azotis atomi naCvenebi piramidis wverzea gan lagebuli, xolo wyalbadis atomebi piramidis fuZes qmnian. azotis atomuri masis wyalbadis Sebrunebul g) z koordinatze mdebareobs firfitis masis centri? centri? sur. 9.37. amocana 3. ··4. ra a) x da b) y koordinatze mdebareobs 9.38 sur. 9.40. amocana 6. 308 Tavi 9 masis centri da impulsi ··7. 9.41 suraTze gamosaxuli ku bis formis yuTi umniSvnelo sisqis liTonis firfi tisgan aris damza debuli. yuTi ze modan gaxsnilia da misi gverdis sigrZea L=40sm. ra masis satvirTo manqana avtomobils daeweva da uswrebs. a) ra manZilzea avtomobilisatvirTo manqanis sistemis masis centri SuqniSnidan, rodesac t = 3 wm? b) ra siCqariT moZraobs am dros masis centri? ·12. m = 0,5 kg masis zeTisxili xy koordinatTa sistemis aTvlis saTaveze devs, xolo M = 1,5 kg masis braziliuri Txilis koordinatebia  2iˆ ++33ˆj )) n Zala (1,2 m, 2,0 m). rodesac t = 0, F0 = ((2 sur. 9.41. amocana 7. a) x, b) y da g) z koordinatze mdebareobs yuTis masis centri? zeTisxilze iwyebs moqmedebas, xolo Txilze ···8. werSi ra iqneba zeTisxili-Txilis sistemis 0,14kg, xolo simaRlea 12sm (sur. 9.42). WurWeli 1,31kg masis sodiT ··13. 9.43 suraTze, erTi bloki Camokidebul erTgvarovani li Tonis WurWlis masaa gaavses da Semdeg zemoT da qvemoT naxvretebi gaukeTes (liTonis dana masis centris gadaadgileba, rodesac t=4 wm? meore blokTan gorgolaWze gadakidebuli TokiT aris gadabmuli. m1=0,6kg masis pirveli blokis centris xy koordinatebia (-0,5m,0m), xolo m2=0,4kg masis Camokidebuli blokis sur. 9.42. amocana 8. kargi umniSvneloa). ra h simaRleze mdebareobs WurWlis masis centri a) Tavdapirvelad da b) WurWlidan sodis mTlianad gamosvlis Semdeg? g) ra mosdis h simaRles sodis daclis dros? d) Tu nebismier momentSi darCenili  Fn = ((–3 -3iˆ -–22ˆj )) n. erTeulovan-veqtorul Cana sodis simaRlea x, gamoTvaleT x, rodesac masis centri udables wertilze mdebareobs. paragrafi 9.3. niutonis meore kanoni nivTier wertilTa sistemisTvis ·9. qva vardnas iwyebs, rodesac t = 0. orjer meti masis qva igive wertilidan maSin vardeba, rodesac t = 100 mwm. a) vardnis dawyebis adgilidan ra manZilzea qvebis masis centri, centris xy koordinatebia (0,-0,1m). Tokis da gorgolaWis masebi umniSvneloa. pirveli bloki moZraobas iwyebs da orive bloki manamde moZraobs, vidre pirveli bloki gorgolaWs Seejaxeba. pirvel bloksa da zedapirs Soris, agreTve gorgolaWsa da mis RerZs Soris xaxuni umniSvneloa. a) erTeulovan-veqtorul CanawerSi ra aCqareba eqneba blokebis sistemis masis centrs? b) daadgineT masis centris siCqaris t droze damokidebulebis funqcia. g) gaavleT masis centris moZraobis traeqtoria. d) Tu traeqtoria arawrfivia, maRla da marjvniskenaa gadaxrili Tu dabla da marcxnisken? Tu traeqtoria wrfivia, ra kuTxes adgens is x RerZTan? rodesac t = 300 mwm? (am dros arc erTi qva miwa ze dacemuli ar aris). b) ra siCqariT moZraobs qvebis sistemis masis centri am dros? ILW ·10. 65 kg da 40 kg masis or moTxilamures 10 m sigrZis da umniSvnelo masis Woki uWiravs. moTxilamureebi Wokis saSualebiT erTmaneTisken moZraoben da xvdebian erTmaneTs. ra manZils gaivlis 40 kg masis mOoTxilamure? ·11. 1000 kg masis avtomobili SuqniSanTan uZravad dgas. mwvane Suqis anTebis dros avtomobili moZraobas 4m/wm2 aCqarebiT iwyebs. amave dros 8m/wm siCqariT moZravi 2000kg sur. 9.43. amocana 13. ··14. 9.44 suraTze gamosaxuli ori nivTieri wertili koordinatTa sistemis aTvlis saTavidan iwyebs moZraobas, rodesac t = 0. m1=5g masis pirveli nivTieri wertili x RerZis gaswvriv (xaxunisgan Tavisufal 309 zedapirze) 10m/ wm mudmivi si CqariT moZraobs. iqneba amis Semdeg ZaRli napiridan? m2=3g masis meore 9.46b. (miniSneba: ix. sur. uli kuTxiT da 20m/ ZaRli mar cxnisken moZraobs, xolo navi marjvni sken, magram moZraobs wm siCqariT maRla aisroles. meore nivTieri wertili zustad pirveli wertilis Tavze Tu ara navi+ZaRlis sistemis masis cen moZraobs. a) ra maqsimalur Hmax simaRles miaRwevs nivTieri wertilebis sistemis masis centri? erTeulovan-veqtorul CanawerSi ra iqneba masis centris b) siCqare da g) aCqareba tri? SSM WWW nivTieri wertili x RerZisadmi garkve sur. 9.44. amocana 14. Hmax simaRleze? ··15. saartilerio Wurvi =20m/wm sawyisi siCqariT da horizontalisadmi θ = 60° kuTxiT 0 isroles. traeqtoriis umaRles wertilSi Wurvi afeTqda da Tanabari masis or nawilad daiSala (sur. 9.45). erTi nawilis siCqare afeTqebis Semdeg nuli gaxda da vertikalurad Camovarda. gasrolis adgilidan ra manZilze Camovardeba Wurvis meore nawili, Tu teritoria sworia da haeris wevis Zala umniSvneloa? SSM (a) (b) sur. 9.46. amocana 17. paragrafi 9.5. nivTier wertilTa sis temis impulsi ·18. 0,7 kg masis burTi 5 m/wm siCqariT horizontalurad moZraobs da vertikalur kedels ejaxeba. burTi 2 m/wm siCqariT aisxliteba. rogor Seicvleba burTis im pulsi? ·19. 2100k g masis satvirTo manqana 41 km/sT siCqariT CrdiloeTisken moZraobs, Semdeg aRmosavleTisken 51 km/sT siCqariT ganagrZobs moZraobas. a) ramdeniT icvleba satvirTo manqanis kinetikuri energia? ra b) sididis da g) mimarTulebisaa manqanis impulsis cvli leba? ILW ··20. 9.47 suraTze 0,165 kg naCvenebia sur. 9.45. amocana 15. ···16. 80kg masis rikardo da ufro msubuqi karmelita 30kg masis kanoeSi sxedan da mersedis tbis xedebiT tkebian. rodesac kanoe mdore wyalSi uZravad gaCerda, rikardom da karmelitam 3m-iT dacilebuli da kanoes centrisadmi simetriulad ganlagebuli adgilebi gacvales. rikardom SeniSna, rom adgilebis Secvlis dros kanoe dambisadmi 40sm-iT gadaadgilda horizontalurad da karmelitas masa gamoTvala. ra masa aqvs karmelitas? masis burTis moZraobis traeqto ria, rodesac is biliardis magidis kididan aisxliteba. burTis sawyisi si Cqarea 2 m/wm, xolo θ1 kuTxea 30°. asxletis dros burTis y komponenti sapirispiro xdeba, xolo x komponenti ar icvleba. risi tolia a) θ2 kuTxe da b) ramdeniT icvleba burTis impulsi erTeulovan-veqtorul CanawerSi? (am amocanaSi burTis gorva ar ganixileba) ··21. t=0 dros burTs fexi daar ···17. 9.46a suraTze gamosaxuli 4,5kg masis ZaRli napiridan D=6,1m manZilze myof, 18kg tyes. 9.48 suraTze naCvenebia burTis masis navze dgas. ZaRli napiris mimarTulebiT p impulsis droze damokidebulebis grafiki frenis da navis gaswvriv 2,4m-s gadis. davuSvaT wyalsa da navs Soris xaxuni ar arsebobs, ra manZilze sur. 9.47. amocana 20. sur. 9.48. amocana 21. 310 Tavi 9 masis centri da impulsi dros. ra sawyisi kuTxiT daartyes burTs? ·25. 1955 wlis TebervalSi, paraSutisti 370m ··22. tafelasTan Sejaxebamde 0,3kg masis burTi 15m/wm siCqariT da horizontalidan qvemoT daxrili 35° kuTxiT moZraobs. ramdeniT simaRlidan gadmoxta da paraSutis gaxsna ver SeZlo da TovlSi Cavarda. raoden gasakviric ar unda iyos, paraSutistma mxolod mcire dazianebebi miiRo. davuSvaT, TovlSi Cavar icvleba burTis impulsi tafelasTan Sejaxebis Semdeg, Tu burTi moZraobs a) 20m/wm siCqariT da vertikalurad qvemoT eSveba da dnis dros misi siCqarea 56m/wm, masaa 85kg Zalis da Tovlidan masze moqmedi Zalaa 1,2×105n (maqsimaluri sidide, rodesac adamians gadarCenis Sansi aqvs). ra a) minimaluri siRrmis Tovli gadaarCenda paraSutists da b) ra Zalis impulsi moqmedebda masze Tovlidan? ·23. x RerZis gaswvriv 14m/wm siCqariT moZrav ·26. 1,2kg masis burTi Camovarda da zedapirze 25m/wm siCqariT daeca da Semdeg burTi 10m/wm b) 20m/wm siCqariT da horizontalurad ukan brundeba? paragrafi impulsi 9.6. Sejaxeba da 0,4kg masis burTze x RerZis uaryofiTi mimarTulebiT modebuli Zala 27mwm-is siCqariT axta. a) ra Zalis impulsi moqmedebs burTze zedapirTan kontaqtis dros? b) Tu ganmavlobaSi moqmedebs. Zalis sidide icvleba burTsa da zedapirs Soris kontaqti 0,02wmis ganmavlobaSi grZeldeba, ra saSualo Zala moqmedebs burTidan zedapirze? da Zalis impulsia 32,4n·wm. ra iqneba a) burTis siCqare da b) ra mimarTulebiT imoZravebs burTi Zalis modebis Semdeg? ra iqneba g) Zalis saSualo sidide da d) burTze moqmedi Zalis impulsis mimarTuleba? SSM ·24. 70-ian wlebSi, henri lamote 12m simaRlidan 30sm siRrmis wyalSi xteboda da auditorias aRafrTovanebda (sur. 9.49). davuSvaT lamote wylis fskerze Cerdeba. ra Zalis impulsi moqmedebs masze wylidan? lamotes masa Tqven airCieT. ·27. yvela icis, rom `zekacisaTvis~ (Superman)nasroli tyviebi misi mkerdidan aisxli teba. davuSvaT gangsteri 3 g masis tyviebs 100 tyvia/wT tempiT esvris „zekacs” da Ti Toeuli tyviis siCqarea 500m/wm. aseve davu SvaT, rom tyviebi sawyisi traeqtoriiT aisxliteba da maTi siCqare ar icvleba. ra saSualo Zala moqmedebs tyviebidan „zekacis” mkerdze? ··28. 5 kg saTamaSo masis manqana x RerZis gaswvriv g a d a a d g i l d e b a . 9.50 suraTze naCve nebia manqanaze sur. 9.50. amocana 28. moqmedi Fx Zalis cvlilebis grafiki. erTeulovan-veqtorul CanawerSi ra iqneba P a) t=4wm da b) t = 7 wm dros? ra iqneba g) t = 9 wm dros? ··29. 9.51 suraTze naCvenebia 0,3 kg masis beis bolis burTi tafelasTan Sejaxebamde da Se jaxebis Semdeg. Sejaxebamde burTis 1 siCqa- rea 12 m/wm da θ1 kuTxea 35°. Sejaxebis Semdeg burTi maRla mo Zraobs siCqariT. 2=10m/wm Sejaxeba 2 sur. 9.49. amocana 24. 30sm siRrmis wyalSi xtoma mwm-is ganmav lobaSi grZeldeba. ra a) sididis da b) mimarTulebis (x sur. 9.51. amocana 29. 311 RerZis dadebiTi mimarTulebisadmi) Zalis impulsi moqmedebs tafeladan burTze? ra g) sididis da d) mimarTulebis saSualo Zala moqmedebs tafeladan burTze? moqmedi saSualo Zala? g) ra saSualo Zala moqmedebs kedelze mravali Sejaxebis ganmavlobaSi? ··30. xvlik basiliskos wylis zedapirze sirbili SeuZlia (sur. 9.52). TiToeuli nabijis gadadgmis dros xvliki fexs wyals urtyams da swrafad awveba, raTa fexis garSemo haeris Rru gaCndes. nabijis dasrulebisTvis xvliki fexs manamde wevs, vidre wyali haeris RruSi SeaRwevs. Tu xvliki ar unda CaiZiros, maSin fexis dartymis, dawolis da awevis dros xvlikze moqmedi, maRla mimarTuli Zalis impulsi mizidulobis Zalis Semdegad gaCenili, qvemoT mimarTuli Zalis impulsis toli unda iyos. davuSvaT xvlikis masaa ··32. 9.54 suraTze naCvenebia 58g masis 90g, TiToeuli fexis masaa 3g, wyalze fexis dartymis siCqarea 1,5m/wm da nabijis gadadgmas aisxliteba. ra Fmax maqsimaluri Zala moqmedebs kedlidan burTze Sejaxebis dros? sur. 9.53. amocana 31. burTis kedelTan Sejaxebis dros moqmedi F Zalis t droze damokidebulebis funqciis grafiki. burTis sawyisi siCqarea 34m/wm da kedlidan igive siCqariT perpendikularulad 0,6wm sWirdeba. a) ra sididis Zalis impulsi moqmedebs xvlikze fexis dartymis dros? b) 0,6 wm-is ganmavlobaSi qvemoT mimarTuli ra Zalis impulsi moqmedebs xvlikze mizidulobis Zalis Sedegad? g) romeli moqmedeba (dartyma Tu fexis dawola) exmareba ufro metad xvliks, Tu maTi mniSvneloba Tanabaria? sur. 9.54. amocana 32. ··33. 9.55 300 g masis da ν = 6m/wm suraTze, siCqaris burTi θ = 30° kedels sur. 9.55. amocana 33. kuTxiT ejaxeba da igive siCqariTa da kuTxiT aisxliteba. sur. 9.52. amocana 30. wyalze morbenali xvliki ··31. ori saSualo Zala. 0,25kg masis Tovlis gundebs 4m/wm siCqariT esvrian kedels. TiToeuli gunda kedels ekroba. 9.53 suraTze naCvenebia kedelze moqmedi F Zalis t droze damokidebulebis funqciis grafiki Tovlis ori gundis SejaxebisTvis. Sejaxebebs Soris drois Sualedia Δtr=50mwm, xolo TiToeuli Sejaxeba Δtd=10mwm-s ganmavlobaSi grZeldeba. Sejaxebis Sedegad grafikze tolferda samkuTxedebi Cndeba da Zalis maqsimaluri sididea Fmax=200n. TiToeuli Sejaxebis dros risi tolia a) Zalis impulsi da b) kedelze burTsa da kedels Soris kontaqti 10 mwmis ganmavlobaSi grZeldeba. erTeulovanveqtorul CanawerSi risi tolia a) kedlidan burTze moqmedi Zalis impulsi da b) burTidan kedelze moqmedi saSualo Zala? ··34. 0,25kg masis hokeis Saiba umniSvnelo xaxunis mqone yinulis zedapirze uZravad   Zevs. t=0 dros, F =(12 – 3 t 2) Zala, sadac F niutonebSia, xolo t - wamebSi, iwyebs moqmedebas da manamde moqmedebs, vidre misi sidide nulis toli gaxdeba. a) ra Zalis impulsi moqmedebs Saibaze 0,5wm-dan 1,25wm-mde drois SualedSi? b) ramdeniT icvleba Saibis impulsi t = 0 drosa da im moments Soris, rodesac F = 0? 312 Tavi 9 masis centri da impulsi paragrafi 9.7. impulsis mudmivoba ·35. 91kg masis adamiani umniSvnelo xaxunis mqone zedapirze wevs da 68g masis qvas 4m/wm siCqariT isvris. ra siCqares SeiZens qvis msroleli? SSM ·36. saTamaSo xaxunisgan Tavisufal zedapirze x RerZis gaswvriv (–0,4m/wm) siCqariT ga daadgildeba, rodesac saTamaSos SigniT damagrebuli ori zambara mas sam nawilad yofs. ra siCqare eqneba A nawils? masis da damkvirveblisadmi ν siCqaris mqone sagani or nawilad iSleba. erTi nawilis masa meoreze samjer metia da afeTqeba kosmosSi xdeba. naklebi masis sagani damkvirveblis mimarT Cerdeba. ra kinetikuri energia emateba sistemas afeTqebis dros damkvirveblis aTvlis sistemidan? ··40. m ··41. xy koordinatTa sistemis aTvlis sa Taveze mdebare WurWeli afeTqebis Sedegad sam nawilad daiSala. m masis mqone erTi nawili (–30 m/wm) siCqariT moZraobs, aseve m nawili masa (kg) siCqare (m/wm) A 0,5 ? masis mqone meore nawili (–30 m/wm) siCqariT gadaadgildeba, xolo mesame nawilis masaa B 0,6 0,2 3m. afeTqebis Semdeg ra a) sididis da b) mimar C 0,2 0,3 Tulebis siCqare aqvs mesame nawils? ·37. kosmosuri xomaldi dedamiwis mimarT 4300km/sT siCqariT moZraobs, rodesac raketis daxarjuli Zrava raketas gamoeyofa da samarTavi modulis mimarT 82km/sT siCqariT moZraobs. Zravis masa modulis masaze oTxjer metia. ra siCqare aqvs samarTav moduls dedamiwis mimarT raketis gayofis Semdeg? ··38. 9.56 suraTze naCvenebia xaxunisgan Tavisufal zedapi rze mdebare `raketa~, ··42. 9.57 suraTze gamosaxuli uZravi bloki L da R nawilebad iSleba. nawilebi xaxunisgan Tavisufal zedapirze gadaadgildeba da Semdeg xaxunis mqone arealSi Sedian, sadac orive Cerdeba. 2 kg masis L nawilze μL = 0,4 kinetikuri xaxunis koeficienti moqmedebs da is dL = 0,15 m manZilis gavlis Semdeg Cerdeba. R nawilze μR = 0,5 kinetikuri xaxunis koeficienti moqmedebs da is dR=0,25m manZilis gavlis Semdeg Cerdeba. ra masisaa bloki? romlis centri x Rer Zis aTvlis saTaves sur. 9.56. amocana 38. emTxveva. „raketa“ C centraluri blokisgan (M = 6kg) da L da R blokebisgan (m = 2kg) Sedgeba. mcire afeTqebas gverdiT mdebare blokebis C blokisgan moSorebiT da x RerZis gaswvriv gadaadgileba SeuZlia gamoiwvios. 1) t = 0 dros, L bloki marcxnisken gada adgildeba afeTqebis mier miniWebuli siCqarisadmi raketisTvis 3 m/wm siCqariT. 2) t = 0,8 wm dros, R bloki marjvnisken gadaadgildeba C blokis siCqarisadmi 3 m/wm siCqariT. rodesac t = 2,8 wm, ra a) siCqare aqvs C bloks da b) sad mdebareobs misi centri? sur. 9.57. amocana 42. ··43. 20kg masis sxeuli 200m/wm siCqariT x RerZis dadebiTi mimarTulebiT gadaadgildeba da afeTqebis Sedegad sam nawilad iSleba. 10 kg masis erTi nawili afeTqebis wertilidan moSorebiT, y RerZis dadebiTi mimarTulebiT 100 m/wm siCqariT gadaadgildeba. 4 kg ma sis meore nawili x RerZis uaryofiTi mi marTulebiT 500 m/wm siCqariT moZraobs. a) erTeulovan-veqtorul CanawerSi ra siCqare aqvs mesame nawils? b) ra energia gamoiyofa afeTqebis dros? mizidulobis Zalis mo ··39. 4 kg masis sagani xaxunisgan Tavisufal zedapirze moZraobs da 2 kg masis or nawilad iSleba. erTi nawili CrdiloeTisken 3 m/wm siCqariT moZraobs, xolo meore nawili 5 m/wm siCqariT, 30°-iT Crdilo-aRmosavleTisken zambara erTad aCerebs. gaSvebis Semdeg zambara wertilebs sapirispiro mimarTulebiT ubiZ gadaadgildeba. sagans? metia da zambaraSi Senaxuli energiaa 60 j. ra sawyisi siCqare aqvs qmedeba ugulebelyaviT. SSM ILW WWW ···44. A da B nivTier wertilebs SekumSuli gebs. A nivTieri wertilis masa B-ze orjer 313 ··49. 9.59a suraTze, 3,5 g masis tyvia xaxunisgan davuSvaT zambaris masa umniSvneloa da misi energia mTlianad nivTier wertilebs gadaecema. energiis gadacemis Semdeg ra Tavisufal zedapirze mdebare or bloks horizontalurad esroles. tyvia pirvel kinetikuri energia eqneba a) A nivTier wertils blokSi (1,2 kg masis) gadis da meore blokSi da b) B nivTier wertils? (1,8 kg paragrafi 9.9. araelastiuri Sejaxebebi erT ganzomilebaSi ·45. 5,2g masis tyvia 672m/wm siCqariT moZraobs da xaxunisgan Tavisufal zedapirze uZravad mdebare 700g masis xis bloks ejaxeba. tyvia blokidan gamodis da 428m/wm siCqariT imave mimarTulebiT ganagrZobs moZraobas. a) ra siCqare aqvs bloks? b) ra siCqariT moZraobs tyvia-blokis masis centri? masis) rCeba. blokebis siCqareebi tyviis moxvedris Semdeg ν1 = 0,63 m/wm da ν2 = 1,4 m/wm xdeba (sur. 9.59b). tyviis mier pirveli blokidan gamotanili nivTiereba ugulebelyaviT da gamoTvaleT tyviis siCqare a) pirveli blokidan gamosvlisas da b) pirvel blokSi Sesvlisas. ·46. 10 g masis tyvia 2 kg masis balistikur qanqaras ejaxeba. qanqaris masis centri 12 smiT vertikalurad maRla gadaadgildeba. davuSvaT tyvia qanqaraSi CarCa, ra sawyisi siCqare hqonda mas? ··47. 1=(–4m/wm) +(–5m/wm) siCqariT moZravi 2kg masis nivTieri wertili da 2=(6m/wm) + (–2m/wm) siCqariT moZravi 4kg masis nivTieri wertili erTmaneTs ejaxeba. Sejaxebis Sedegad nivTieri wertilebi SeerTda. ra iqneba maTi siCqare a) erTeulovan-veqtorul CanawerSi da ra iqneba am siCqaris b) sidide da g) kuTxe? ··48. A manqana (1100 kg masis) SuqniSanze gaCerda da mas uknidan B manqana (1400kg masis) Seejaxa. Sejaxebis Semdeg orive manqana gasrialda da gzidan moqmedma xaxunis Zalam (μk = 0,13) manqanebi dA = 8,2 m da dB = 6,1 m manZilebze SeaCera. ra iqneba a) A manqanis da b) B manqanis siCqare srialis dawyebisas, Sejaxebis Semdeg? g) davuSvaT Sejaxebis dros impulsi inaxeba, ra siCqare eqneba B manqanas Sejaxebamde? d) axseniT, Tu ratom SeiZleba aRmoCndes amgvari daSveba araswori. sur. 9.59. amocana ··50. 9.60 suraTze, 10 g masis 1000 m/wm siCqariT ver tikalurad zeviT gasrolilma tyviam 5 kg masis blokis masis centrze ga iara. blokidan sur. 9.60. amocana 50. gamosuli tyvia igive mimarTulebiT 400 m/ wm siCqariT ganagrZobs moZraobas. sawyisi mdebareobidan ra maqsimalur simaRleze ava bloki tyviis moxvedris Sedegad? ··51. 3 m/wm siCqariT moZravi 5 kg masis bloki imave mimarTulebiT da 2 m/wm siCqariT moZrav 10 kg masis bloks ejaxeba. Sejaxebis Semdeg 10 kg masis bloki 2,5 m/wm siCqariT ganagrZobs moZraobas. a) ra iqneba 5kg masis blokis siCqare Sejaxebis Semdeg? b) ramdeniT Seicvleba blokebis sistemis jamuri kinetikuri ener gia Sejaxebis Sedegad? g) davuSvaT, 10 kg masis bloki Sejaxebis Semdeg 4 m/wm siCqariT ganagrZobs moZraobas. am SemTxvevaSi ramdeniT Seicvleba jamuri kinetikuri energia? ···52. sagozavis sveli burTebi erTmaneTisken sur. 9.58. amocana 48. moZraoben (vertikaluri RerZis gaswvriv) da sruliad araelastiurad ejaxebian. 314 masis centri da impulsi Tavi 9 Sejaxebamde 3 kg masis burTi 20 m/wm siCqariT bloki (1,6kg masis) xaxunisgan Tavisufal maRla adis, xolo 2 kg masis burTi 12 m/wm siCqariT dabla vardeba. Sejaxebis adgilidan ra simaRleze ava Sewebebuli burTebi? (haeris wevis Zala ugulvebelyaviT). zedapirze B blokisken (2,4kg masis) misrialebs. suraTze naCvenebia Sejaxebamde ···53. 9.61 suraTze wm, νBi=2,5m/wm da νBf=4,9m/wm. ra a) sidide da b) gamosaxuli pir veli bloki (2kg masis) 10 m/ wm siCqariT marjvniv moZraobs, sur. 9.61. amocanebi 53 da 124. 1120 n/m sixistis zambaraa damagrebuli. blokebis Sejaxebisas zambara maqsimalurad ikumSeba da blokebis siCqareebi Tanabrdeba. gamoTvaleT zambaris maqsimaluri SekumSvis manZili. ILW ···54. 9.62 suraTze (1 kg masis) zedapirze sur. 9.62. amocana 54. xaxunisgan uZravad Tavisufal Zevs da 200 n/m sixistis moSvebul zambaras exeba. zambaris meore bolo kedelzea mimagrebuli. ν1 = 4 m/wm siCqariT moZravi pirveli bloki (2 kg masis) meore bloks ejaxeba da blokebi erTmaneTs ekroba. blokebis wamieri gaCerebis momentSi ra manZiliT SeikumSeba zambara? paragrafi 9.10. elastiuri Sejaxebebi erT ganzomilebaSi ·55. 340 g masis Cemodani xaxunisgan Tavisufal zedapirze moZraobs da 1,2 m/wm sawyisi siCqariT uZrav meore Cemodans ejaxeba, romlis masa ucnobia. Sejaxebis Semdeg, pirveli Cemodani 0,66 m/wm siCqariT da sawyisi mimarTulebiT ganagrZobs moZraobas. a) ra masa aqvs meore Cemodans? b) ra siCqare aqvs meore Cemodans Sejaxebis Semdeg? g) ra siCqariT moZraobs Cemodnebis sis temis masis cen tri? SSM ·56. 9.63 suraTze gamosaxuli A mimarTuleba. Sesabamisia sidideebia νAi=5,5m/ mimarTuleba aqvs Af siCqares? g) aris Tu ara Sejaxeba elastiuri? xolo meore bloki (5 kg masis) 3 m/wm siCqariT aseve marjvniv gadaadgildeba. zedapiri xaxunisgan Tavisufali da meore blokze gamosaxuli meore bloki (i) da Sejaxebis Semdgomi (f) sami siCqaris sur. 9.63. amocana 56. ·57. Tanabari siCqariT moZravi titanis ori sfero erTmaneTs uaxlovdeba da elastiurad ejaxeba. Sejaxebis Semdeg 300g masis erT-erTi sfero Cerdeba. a) ra masa aqvs meore sferos? b) ra siCqariT moZraobs sferoebis masis centri, Tu TiToeuli sferos sawyisi siCqarea 2m/ wm? ··58. 9.64 suraTze gamosaxuli m1=0,3kg masis nivTieri sur. 9.64. amocana 58. wertili 2m/wm siCqariT x RerZis gas wvriv, marjvniv gadaadgildeba (xaxunisgan Tavisufal zedapirze). x = 0 koordinatze nivTieri wertili m2 = 0,4 kg masis uZrav meore nivTier wertils elastiurad ejaxeba (Sejaxeba erT ganzomilebaSi xdeba). meore nivTieri wertili xw = 70 sm koordinatze mdebare kedlidan aisxliteba da siCqares ar kargavs. x RerZis ra koordinatze Seejaxeba meore nivTieri wertili pirvels? ··59. 2kg masis sxeuli uZrav meore sxeuls elastiurad ejaxeba da moZraobas sawyisi siCqaris erTi meoTxedi siCqariT igive mi marTulebiT ganagrZobs. a) ra masa aqvs meore sxeuls? b) ra siCqariT moZraobs sxeulebis masis centri, Tu 2 kg masis sxeulis sawyisi siCqarea 4 m/wm? SSM ··60. 0,5 kg masis fo ladis burTi 70 sm sigrZis Tokze Ca mokides. Semdeg Toki horizontalur m d g o m a r e o b a S i moiyvanes da burTs gauSves xeli (sur. sur. 9.65. amocana 60. 9.65). burTi wonasworobis mdebareobaSi daejaxa xaxunisgan Tavisufal zedapirze uZravad mdebare bloks, romlis masaa 2,5kg. Sejaxeba elastiuria. gamoTvaleT a) burTis 315 siCqare da b) blokis siCqare Sejaxebis Semdeg. ··61. m1 masis pirveli bloki xaxunisgan Tavisufal zedapirze gadaadgildeba da m2=3m1 masis uZrav bloks elastiurad ejaxeba (Sejaxeba erT ganzomilebaSi xdeba). blokebis sistemis masis centri Sejaxebamde 3m/wm siCqariT moZraobs. ra siCqare eqneba a) masis centrs da b) meore bloks Sejaxebis Semdeg? ··62. 9.66 suraTze gamosaxuli m1 masis pirveli bloki xaxunisgan Tavisufal meore Saibas elastiurad ejaxeba (Sejaxeba erT ganzomilebaSi xdeba). meore Saiba zedapiridan vardeba da vardnis adgilidan d manZilze ecema. pirveli Saiba Sejaxebis Sedegad aisxliteba da sapirispiro mimarTulebiT moZraobs. Semdeg pirveli Saibac vardeba zedapiridan da vardnis adgilidan 2d manZilze ecema. ra masa aqvs meore Saibas? (miniSneba: niSnebs yuradReba miaqcieT). daRmarTze, h=2,5m simaRlidan srialdeba da m2=2m1 masis uZrav meore bloks ejaxeba. Sejaxebis Semdeg meore bloki μk=0,5 kinetikuri xaxunis koeficientis mqone zedapirze moZraobs da d manZilis gavlis Semdeg Cerdeba. risi tolia d, Tu Sejaxeba a) elastiuria da b) sruliad araelastiuria? sur. 9.68. amocana 64. paragrafi 9.11. zomilebaSi Sejaxebebi or gan ··65. 9.22 suraTze, pirveli nivTieri wertili alfa nawilakia, xolo samizne meore nivTieri wertili Jangbadis birTvia. alfa nawilaki sur. 9.66. amocana 62. ···63. θ1=64° kuTxiT moZraobs, xolo Jangbadis birTvi 1,2×105m siCqariT da θ2=51° kuTxiT gadaadgildeba. atomuri masis erTeulebSi alfa nawilakis masaa 4u, xolo Jangbadis horizon taluri zedapi birTvis masaa 16u. ra a) sawyisi da b) saboloo ridan h = 1,8 sima Rleze uWiravT siCqare aqvs alfa nawilaks? ILW ··66. 2kg masis A da B sxeuli erTmaneTs ejaxeba. Sejaxebamde A = (15 + 30 ) m/wm, xolo = (–10 + 5 ) m/wm. Sejaxebis Semdeg B Á = (–5 + 20 ) m/wm. ra iqneba a) B-s saboloo M = 0,63 kg masis burTi da mis ze viT m masis pa tara burTi ise, rom did burTs ar exeba (sur. siCqare da b) ramdeniT Seicvleba jamuri kinetikuri energia? 9.67a). ··67. burTebs erTdroulad gau Sves xeli (davu sur. 9.67. amocana 63. SvaT, rom burTebis radiusebi h simaRlesTan SedarebiT mcirea). a) Tu mozrdili burTi zedapiridan elastiurad aisxliteba da Semdeg patara burTi mozrdili burTidan aseve elastiurad aisxliteba, m-is ra mniSvnelobis dros gaCerdeba mozrdili burTi patara burTTan Sejaxebis dros? b) ra simaRleze ava am dros patara burTi (sur. 9.67b)? SSM WWW ···64. 9.68 suraTze gamosaxuli m1=0,2kg masis hokeis pirveli Saiba xaxunisgan Tavisufal zedapirze gadaadgildeba da hokeis uZrav 500m/wm siCqariT moZravi sasroli protoni uZrav samizne protons elas tiurad ejaxeba. Semdeg protonebi per pendikularuli traeqtoriebiT moZraoben da sasroli protonis traeqtoria sawyisi mimarTulebisadmi 60°-s adgens. Sejaxebis Semdeg ra siCqare aqvs a) samizne protons da b) sasrol protons? ··68. B burTi x RerZis dadebiTi mimarTulebiT ν siCqariT gadaadgildeba da aTvlis saTaveze mdebare A uZrav burTs ejaxeba. burTebis masebi sxvadasxvaa. Sejaxebis Semdeg B burTi y RerZis uaryofiTi mimarTulebiT ν/2 siCqariT ganagrZobs moZraobas. a) ra mimarTulebiT 316 masis centri da impulsi Tavi 9 moZraobs A burTi? b) daamtkiceT, rom A-s xaxunis Zalebi barJebis masebze damokidebuli siCqaris gamoTvla mocemuli informaciis ar aris. SSM daxmarebiT SeuZlebelia. ···69. erTi da igive masis da sawyisi siCqaris mqone ori sagani sruliad araelastiuri Sejaxebis Semdeg ganaxevrebuli Sordeba erTmaneTs. siCqariT gamoTvaleT kuTxe sagnebis sawyis siCqareebs Soris. SSM paragrafi 9.12. cvladi masis mqone sistemebi: raketa ·70. kosmosSi raketa aTvlis inerciuli sistemis mimarT uZravia. raketis Zravam garkveul drois SualedSi unda imuSaos. ra unda iyos raketis masebis Sefardeba (sawyisi da saboloo masebis Sefardeba) drois am SualedSi, raTa aTvlis inerciuli sistemisadmi a) raketis gamonabolqvis sawyisi siCqaris siCqare (gamonabolqvi produqtis siCqare raketisadmi) toli da b) gamonabolqvis siCqareze orjer meti iyos? ·71. kosmosSi, 2,55×105kg masis raketaSi, romelic aTvlis inerciuli sistemisadmi uZravia, aris 1,81×105kg sawvavi. Semdeg raketis Zrava 250 wamis ganmavlobaSi muSaobs da sawvavis moxmarebis tempia 480 kg/wm. raketis mimarT gamonabolqvi produqtis siCqarea 3,27 km/wm. a) risi tolia raketis Zalva? 250 wamis Semdeg ra iqneba raketis b) masa da g) siCqare? SSM ILW ·72. 6090 kg masis kosmosuri sakvlevi raketa mzis mimarT 105 m/wm siCqariT iupiterisken moZraobs. Zravis amuSavebis Semdeg 80 kg masis gamonabolqvi raketis mimarT 253 m/wm siCqariT moZraobs. ra iqneba raketis saboloo sur. 9.69. amocana 73. damatebiTi amocanebi 74. 9.70 suraTze gamosaxuli ori niv Tieri wertili xaxu nisgan Tavisufal zedapirze mudmivi siCqariT gadaadgil deba. nivTier wertilTa masebi Tanabaria da maTi sawyisi siCqarea ν = 4 m/wm. traeqtoriebis gadakveTis adgilze nivTieri wertilebi erTmaneTs ejaxeba. x RerZi traeqtoriebs Soris kuTxis biseqtrisaa da θ = 40°. Sejaxebis adgilidan marjvniv mdebare farTobs x RerZi da oTxi daStrixuli xazi oTx regionad yofs. romel regionSi an romeli wrfis gaswvriv imoZraveben nivTieri wertilebi, Tu Sejaxeba a) sruliad araelastiuria, b) elastiuria da g) araelastiuria? ra saboloo siCqareebi eqneba nivTier wertilebs, Tu Sejaxeba d) sruliad araelastiuria da e) elastiuria? moZraobs. erTis siCqarea 10 km/sT, xolo 75. siCqaris gama didebeli. 9.71 suraTze, m1 masis pirveli bloki x meoris siCqarea 20 km/sT. erTmaneTis Cavlis RerZis siCqare? ·73. 9.69 suraTze gamosaxuli ori grZeli barJa mdore wyalSi erTi da igive mimarTulebiT dros nela moZravi barJidan swrafad moZrav barJaze qvanaxSirs 1000 kg/wT tempiT yrian. ra damatebiTi Zala unda uzrunvelyos a) swrafad moZravi barJis da b) nela moZravi barJis Zravam, raTa siCqare ar Seicvalos? dauSviT, rom qvanaxSirs yovelTvis sworad yrian da barJebsa da wyals Soris moqmedi sur. 9.70. amocana 74. sur. 9.71. amocana 75. gaswvriv, xaxunisgan Tavisufal zedapirze ν1i=4m/wm siCqariT misrialebs da m2=0,5m1 masis uZrav meore bloks elastiurad ejaxeba (Sejaxeba erT ganzomilebaSi xdeba). Semdeg meore bloki m3=0,5m2 masis mesame uZrav bloks ejaxeba (kvlav erT ganzomilebaSi). a) ra iqneba mesame blokis siCqare? mesame blokis b) siCqare, 317 g) kinetikuri energia da d) impulsi pirveli blokis Sesabamis sawyis mniSvnelobaze metia, naklebia Tu misi tolia? 79. matareblis vagoni xorblis elevatoris qvemoT 3,2m/wm mudmivi siCqariT moZraobs. xorbali vagonze 540kg/wT tempiT iyreba. ra 76. siCqaris Sema mcirebeli. 9.72 suraTze, m1 masis pirveli bloki x sididis Zalaa saWiro vagonis mudmivi siCqaris SesanarCuneblad, Tu xaxuni umniSvneloa? RerZis sur. 9.72. amocana 76. gaswvriv, xaxunisgan Tavisufal zedapirze 4m/wm si CqariT gadaadgildeba da m2=2m1 masis uZrav meore bloks elastiurad ejaxeba (Sejaxeba erT ganzomilebaSi xdeba). Semdeg meore bloki m3=2m2 masis uZrav mesame bloks elastiurad ejaxeba (kvlav erT ganzomilebaSi). a) ra iqneba mesame blokis siCqare? mesame blokis b) siCqare, g) kinetikuri energia da d) impulsi pirveli blokis Sesabamis sawyis mniSvnelobaze metia, naklebia Tu misi tolia? 77. `fardobiTi~ mniSvnelovani sityvaa. 9.73 su raTze, ori bloki sur. 9.73. amocana 77. 80. 915 n wonis adamiani matareblis grZel vagonze (2415 n wonis) dgas da vagoni x RerZis dadebiTi mimarTulebiT 18,2m/wm siCqariT moZraobs. xaxuni umniSvneloa. Semdeg adamiani vagonze x RerZis uaryofiTi mimarTulebiT garbis da vagonis mimarT misi siCqarea 4m/wm. ramdeniT gaizrdeba vagonis siCqare? 81. 9.74 suraTze ga mosaxulia Saqris Ca sayreli ori WurWe li. erTmaneTTan um niSvnelo masisa da xaxunisagan Tavisufal gorgolaWze gada kiduli TokiT aris dakavSirebuli. Tokis masac umniSvneloa. sur. 9.74. amocana 81. _ mL=1kg masis L bloki da mR=0,5kg masis R bloki, romelTa Soris SekumSuli zambaraa damagrebuli, uZravia. xelis gaSvebis Semdeg zambara blokebs xaxunisgan Tavisufal zedapirze gadaaadgilebs (zambaris masa umniSvneloa da blokebis amoZravebis Semdeg Saqrisa da WurWlis erToblivi masaa 500g, zedapirze vardeba). a) Tu zambara L bloks b) pirveli WurWlidan meoreSi 20g Saqris gadayris Semdeg? Saqris gadayris Semdeg WurWlebs xeli gauSves. g) ra mimarTulebiT imoZravebs sistemis masis centri da d) ra aCqareba eqneba mas? zedapiris mimarT 1,2m/wm siCqares aniWebs, ra manZilze gadaadgildeba R bloki Semdgomi 0,8wm-is ganmavlobaSi? b) Tu zambara L bloks R blokis siCqaris mimarT 1,2m/wm siCqares aniWebs, ra manZilze gadaadgildeba R bloki Semdgomi 0,8wm-is ganmavlobaSi? 78. M masis raketa x RerZis gaswvriv νi=40m/wm mudmivi siCqariT moZraobs. mcire afeTqebis Sedegad raketa wina (m1 masis) da ukana nawilebad iSleba. orive nawili x RerZis gaswvriv moZraobas ganagrZobs. nawilebs Soris fardobiTi siCqarea 20m/wm. ra a) minimaluri SesaZlo νf saboloo siCqare SeiZleba hqondes raketis wina nawils da b) m1is ra zRvruli mniSvnelobis dros xdeba es? g) ra maqsimaluri SesaZlo νf siCqare SeiZleba hqondes raketis wina nawils da d) m1-is ra zRvruli mniSvnelobis dros xdeba es? WurWlebis centrebs Soris manZilia 50mm da isini erT simaRlezea gaCerebuli. ra horizontaluri manZilia pirveli WurWlis centrsa da ori WurWlisgan Semdgari sistemis masis centrs Soris a) Tavdapirvelad da 82. 9.75 suraTze 6d=6m naCvenebia gverdis sigrZis kvadratuli fir fita, romelsac 2d rZis gverdis sig kvadrati Ca moaWres. ra a) x da b) y koordinatze mdebareobs darCenili centri? sur. 9.75. amocana 82. firfitis masis 83. 2,65kg masis yuTi afeTqebis Sedegad sam nawilad daiSala da samive xaxunisgan Tavisufal zedapirze gasrialda. yuTi Tavdapirvelad koordinatTa sistemis aT 318 Tavi 9 masis centri da impulsi vlis saTaveze imyofeboda. pirveli nawilis masaa m1=0,5kg, xolo siCqarea (10 +12 )m/wm. meore nawilis masaa m2=0,75kg, siCqarea 14m/wm da 110° kuTxiT moZraobs (x RerZis dadebiTi mimarTulebidan saaTis isris moZraobis sawinaaRmdegod). a) ra siCqare aqvs mesame na wils da b) ra mimarTulebiT gadaadgildeba? 84. 3kg masis da (5m/wm) siCqaris pirveli nivTieri wertili 2kg masis da (–6m/wm) siCqaris meore nivTier wertils elastiurad ejaxeba (Sejaxeba erT ganzomilebaSi xdeba). Sejaxebis Semdeg ra siCqariT moZraobs a) pirveli nivTieri wertili da b) meore nivTieri wertili? 85. raketis ukanaskneli safexuri 7600m/wm siCqariT moZraobs da ori nawilisgan Sedgeba. erTi nawili 290kg masis raketis garsia, xolo meore nawili 150kg masis sasargeblo tvirTis kapsulaa. mogvianebiT SekumSuli zambara am or nawils ganacalkevebs da isini erTmaneTis mimarT 910m/wm siCqariT moZraoben. ra iqneba a) raketis garsis da b) sasargeblo tvirTis siCqare gancalkevebis Semdeg? davuSvaT yvela siCqare erTi da igive wrfis gaswvrivaa mimarTuli. gamoTvaleT orive nawilis jamuri kinetikuri energia g) gancalkevebamde da d) gancalkevebis Semdeg. 86. 200g masis da 3m/wm siCqariT moZravi pirveli nivTieri wertili 400g masis uZrav meore nivTier wertils erT ganzomilebaSi ejaxeba. ra Zalis impulsi moqmedebs pirvel nivTier wertilze, Tu Sejaxeba a) elastiuria da b) sruliad araelastiuria? 87. radiolokaciurma sadgurma sagani daafiqsira da aRmoCnda, rom misi mdebareobis  veqtoria r =(3500–160t) +2700 +300 , sadac  r metrebSia, xolo t wamebSi. sadguris x RerZi aRmosavleTisken aris mimarTuli, y RerZi CrdiloeTisken, xolo z RerZi - vertikalurad maRla. Tu sagani 250kg masis meteorologiuri ra simaRleze ava a) pirveli sfero da b) meore sfero? momdevno elastiuri Sejaxebis Semdeg ra simaRleze ava g) pirveli sfero da d) meore sfero? (miniSneba: damrgvalebul mniSvnelobebs nu gamoiyenebT). 89. 9.76 suraTze, 80kg masis adamiani 320kg masis sahaero burTze Camokidebul kibeze adis. burTi Tavdapirvelad deda miwisadmi uZravia. Tu adamiani kibisadmi 2,5m/wm siCqariT adis a) ra mimarTulebiT da b) ra siCqariT moZraobs burTi? g) Tu adamiani kibeze asvlas Sewyvets, ra siCqare eqneba burTs? sur. 9.76. amocana 89. 90. 0,15kg masis burTi kedels (5m/wm) + + (6,5m/wm) + (4m/wm) siCqariT ejaxeba. Semdeg burTi kedlidan (2m/wm) + (3,5m/wm) + (–3,2m/wm) siCqariT aisxliteba. a) ramdeniT icvleba burTis impulsi? ra Zalis impulsi moqmedebs b) burTze da g) kedelze? 91. cxrilSi mocemulia oTxi nivTieri wer tilis xy koordinatebi da siCqare garkveul momentSi. am momentSi ra a) x da b) y koordinatze mdebareobs maTi masis centri da g) ra siCqare aqvs masis centrs? nivTieri wertili masa (kg) mdebareoba (m) 1 2 0, 2 4 3, 0 6 3 3 0, –2 6 4 12 –1, 0 –2 3 siCqare (m/wm) –9 92. 0,55 kg masis burTi betonze 12m/wm siCqariT ecema da 3m/wm siCqariT maRla aisxliteba. raketaa, ra iqneba misi a) impulsi, b) moZraobis mimarTuleba da g) masze moqmedi jamuri Zala? y RerZi maRlaa mimarTuli. erTeulovan- 88. 9.10 amocanis amoxsnis nimuSSi mocemul b) burTze da g) betonze? ganlagebaze davuSvaT, rom pirveli sferos masaa 50g da sawyisi simaRlea h1=9sm, xolo meore sferos masaa 85g. mas Semdeg, rac pirveli sfero meore sferos elastiurad Seejaxeba, veqtorul CanawerSi a) ramdeniT icvleba burTi impulsi, ra Zalis impulsi moqmedebs 93. sxeuli x RerZis dadebiTi mimarTulebiT 2m/wm siCqariT moZraobs da masze jamuri Zala ar moqmedebs. Semdeg Sida afeTqebis gamo sxeuli erTi da igive 4 kg masis or 319 kinetikuri balistikur qanqaraze tyviis m masaa 8 g, energia 16j-iT izrdeba. saganis wina nawili moZraobas sawyisi mimarTulebiT ganagrZobs. ra siCqariT moZraobs saganis a) ukana nawili da b) wina nawili? blokis M masaa 7kg da blokis asvlis ver nawilad iSleba da jamuri 94. m1 masis pirveli bloki 4m/wm siCqariT x RerZis gaswvriv, xaxunisgan Tavisufal zedapirze gadaadgildeba da m2=0,4m1 masis tikaluri h simaRlea 5 sm. a) tyviis sawyisi kinetikuri energiis ra nawili darCeba tyviablokis qanqaris meqanikur energiad blokTan Sejaxebis Semdeg? b) Tu tyviis sawyis siCqares gavzrdiT, es nawili gaizrdeba, Semcirdeba Tu ar Seicvleba? ratom? uZrav meore bloks erT ganzomilebaSi elastiurad ejaxeba. Semdeg orive bloki 99. saTamaSo Tofi wamSi 2g masis 10 burTulas 500m/wm siCqariT isvris. burTulebi kedels 0,5 kinetikuri xaxunis koeficientis mqone xvdeba. risi tolia a) TiToeuli burTulis impulsi, b) kinetikuri energia da g) ra saSualo Zala moqmedebs kedlidan burTulebis nakadze? d) Tu burTulasa da kedels Soris monakveTSi Sedis da Cerdeba. ra manZils gaivlis am monakveTSi a) pirveli bloki da b) meore bloki? 95. t = 0 dros, 2×10-3kg masis uZrav nivTier wertilze 1 = (–4 + 5 )n Zala iwyebs moqme debas, xolo 4×10–3 kg masis uZrav nivTier wertilze = (2 – 4 )n Zala moqmedebs. 02 dan 2 mwm-mde drois SualedSi risi tolia nivTier wertilTa sistemis masis centris gadaadgilebis a) sidide da b) kuTxe (x RerZis dadebiTi mimarTulebisadmi)? g) ra kinetikuri energia aqvs masis centrs t = 2 mwm dros? 96. atomis birTvi xy koordinatTa sistemis aTvlis saTaveze Zevs da sam nawilakad gar daiqmneba. 16,7×10–27kg masis pirveli nawilaki aTvlis saTavidan moSorebiT (6×106 m/wm) siCqariT gadaadgildeba. 8,35×10–27kg masis meore nawilaki aTvlis saTavidan aseve moSorebiT (–8×106m/wm) siCqariT moZraobs. a) erTeulovan-veqtorul CanawerSi risi tolia 11,7×10–27kg masis mesame nawilakis impulsi? b) ra kinetikuri energia Cndeba gardaqmnis Sedegad? 97. 9.77 suraTze naCvenebia 22,7kg masis cigebi, romelTa Soris patara manZilia. 3,63kg masis kata pirveli cigidan meoreze xteba da Semdeg kvlav pirvel cigas ubrundeba. orive naxtomis siCqare yinulisadmi 3,05m/wm-ia. ra saboloo siCqare eqneba a) pirvel cigas da b) meore cigas? kontaqti 0,6 mwm-is ganmavlobaSi grZeldeba, ra saSualo Zala moqmedebs kedlidan bur Tulaze kontaqtis dros? e) ratom gansxvavdeba erTmaneTisgan g) da d) SekiTxvaze miRebuli pasuxebi? 100. kosmosuri xomaldi asafeTqebeli WanWikebis meSveobiT or nawilad daiSala. 1200 kg da nawilebis masebia WanWikebidan TiToeul nawilze 1800 kg. moqmedi Zalis impulsia 300 n·wm. ra fardobiTi siCqariT gancalkevdebian nawilebi afeTqebis Sedegad? 101. 1400 kg masis manqana y RerZis dadebiTi mimarTulebiT, CrdiloeTisken 5,3 m/wm si CqariT moZraobs. Semdeg manqana 4,6wamis ganmavlobaSi 90°-ian marjvena mosaxvevSi Sedis da xes ejaxeba, romelic manqanas 350 mwmis Semdeg aCerebs. erTeulovan-veqtorul CanawerSi ra Zalis impulsi moqmedebs manqanaze a) moxvevis Sedegad da b) Sejaxebis Sedegad? ra saSualo Zala moqmedebs manqanaze g) moxvevis dros da d) Sejaxebis dros? e) ra mimarTuleba aqvs moxvevis dros moqmed saSualo Zalas? 102. mxuTavi airis molekula x RerZis gaswvriv Zevs. naxSirbadis atomis koor x = 0, xolo Jangbadis atomi –10 x = 1,131×10 m koordinatzea ganlagebuli. ra x koordinatzea molekulis masis centri? dinatia 103. 150 g masis burTi kedels 5,2 m/wm siCqariT sur. 9.77. amocana 97. 98. 9.8 amocanis amoxsnis nimuSSi mocemul ejaxeba da asxletis Semdeg kinetikuri energiis naxevars kargavs. a) ra siCqare aqvs burTs asxletis Semdeg? b) ra Zalis impulsi moqmedebs kedlidan burTze? g) Tu burTsa da kedels Soris kontaqti 7,6 mwm-is ganmavlobaSi 320 Tavi 9 masis centri da impulsi grZeldeba, ra saSualo Zala moqmedebs kedlidan burTze drois am SualedSi? 104. 2400 kg masis kraisleri swor gzaze 80 km/sT siCqariT moZraobs. kraislers 1600 kg masis da 60 km/sT siCqariT moZravi fordi misdevs ukan. ra siCqariT moZraobs manqanebis masis centri? 105. fexburTeli 0,45kg masis burTs fexs artyams. fexsa da burTs Soris kontaqti 3×10–3 wm-is ganmavlobaSi grZeldeba, xolo 0 ≤ t ≤ 3×10–3 wm SualedSi (sadac t wamebSia) dartymis Zala Semdegi formuliT moicema: F(t)=[(6×106)t–(2×109)t2]n gamoTvaleT a) burTze moqmedi Zalis impulsi dartymis Sedegad, b) fexidan burTze moqmedi saSualo Zala kontaqtis ganmavlobaSi, g) fexidan burTze moqmedi maqsimaluri Zala kontaqtis ganmavlobaSi da d) burTis siCqare dartymis Semdeg. 106. 1 kg da 3 kg masis ori bloki zambariT erTmaneTze aris gadabmuli da xaxunisgan Tavisufal zedapirze uZravad Zevs. Semdeg blokebi erTmaneTisken moZraobs da 1 kg masis bloki uZravi masis centrisken 1,7 m/wm sawyisi siCqariT gadaadgildeba. ra sawyisi siCqare aqvs meore bloks? 107. 3,18×104kg masis satvirTo vagoni uZrav samuxruWe vagons ejaxeba. vagonebi Cerdeba da sawyisi kinetikuri energiis 27% siTbur energiaSi, bgeraSi, vibraciaSi da a.S. gadadis. gamoTvaleT samuxruWe vagonis masa. 108. 75kg masis adamiani 39kg masis uremze zis da 2,3m/wm siCqariT moZraobs. Semdeg adamiani miwisadmi nulovani horizontaluri siCqariT dabla xteba. ramdeniT Seicvleba urmis siCqare? 109. 9.78 suraTze, m=60g masis burTi νi=22m/wm siCqariT M=240g masis uZ sur. 9.78. amocana 109. rav zambariani Tofis lulaSi isroles. burTi lulaSi zambaris maqsimaluri SekumSvis adgilze Cerdeba. davuSvaT burTsa da Tofis lulas Soris xaxunis Sedegad siTburi energia umniSvnelod izrdeba. a) ra siCqare eqneba zambarian Tofs burTis gaCerebis Semdeg? b) burTis sawyisi kinetikuri energiis ra nawili inaxeba zambaraSi? 110. radioaqtiuri (mSobeli) birTvi sxva (`Svilobil~) birTvad gardaiqmneba da eleqtronsa da neitrinos gamoasxivebs. ~mSobeli~ birTvi xy koordinatTa sis temis aTvlis saTavezea ganlagebuli. eleqtroni aTvlis saTavidan moSore biT (–1,2×10–22kg·m/wm) impulsiT gada adgildeba, xolo neitrino agreTve aT vlis saTavidan moSorebiT moZraobs, oRond misi impulsia (–6,4×10–23kg·m/wm) . ra a) sididis da b) mimarTulebisaa `Svilobili~ birTvis impulsi? g) Tu Svilobili birTvis masaa 5,8×10-26kg, ra iqneba misi kinetikuri energia? 111. eleqtroni uZrav wyalbadis atoms elastiurad ejaxeba (Sejaxeba erT gan zomilebaSi xdeba). eleqtronis sawyisi kinetikuri energiis ra nawili (procentebSi) gadadis wyalbadis atomis kinetikur ener giaSi? (wyalbadis atomis masa eleqtronis masaze 1840-jer metia). 112. raketa mzis sistemidan moSorebiT 6×103m/wm siCqariT moZraobs. Semdeg raketis Zrava irTveba da gamonabolqvi raketisadmi 3×103m/wm siCqariT gamodis. am dros raketis masaa 4×104kg, xolo misi aCqarebaa 2m/wm2. a) risi tolia Zravis Zalva? b) ra tempiT gamodis gamonabolqvi (kg/wm-ebSi)? 113. 9.79 suraTze, m1=6,6kg masis pirveli bloki xaxunisgan Tavisufal magidaze uZravad Zevs. pirvel bloksa da kedels Soris moTavsebuli m2 masis meore bloki ν2i mudmivi siCqariT pirveli blokisken gadaadgildeba. gamoTvaleT m2-is mniSvneloba, rodesac meore blokis pirvelTan Sejaxebisa da, Semdeg, kedelTan Sejaxebis Semdeg orive bloki Tanabari siCqariT YimoZravebs. dauSviT, rom yvela Sejaxeba elastiuria (kedelTan Sejaxeba meore blokis siCqares ar cvlis). sur. 9.79. amocana 113. 114. mTvareze gadasvlis dros, mTvaris 321 mimarT 400m/wm siCqariT moZravi kosmosuri xomaldis siCqare 2,2m/wm-iT unda gaizardos. Zravidan gamosuli wvis produqti kosmosuri xomaldis mimarT 1000m/wm siCqariT moZraobs. kosmosuri xomaldis sawyisi masis ra nawili unda daiwvas da gamoiyos, raTa siCqare zemoT mocemuli sididiT gaizardos? 115. 200kn wonis da horizontalur liandagze 3m/wm siCqariT moZravi satvirTo vagoni 400kn wonis uZrav samuxruWe vagons Seejaxa. burTi kedels perpendikularulad ejaxeba da sapirispiro mimarTulebiT igive siCqariT aisxliteba. Sejaxeba 3,8mwm-is ganmavlobaSi grZeldeba. ra a) Zalis impulsi da b) saSualo Zala moqmedebs burTidan kedelze? 120. ori identuri moneta h=11m simaRlezea Camokidebuli. rodesac t=0, pirveli moneta vardeba da talaxian velze ecema. rodesac t=0,5wm, meore monetac vardeba da velze ecema.  ra acom aCqarebiT moZraobs monetebis masis centri a) 0-0,5wm SualedSi, b) t=0,5wm drosa t1 dros Soris, rodesac pirveli moneta velze Sejaxebis Sedegad vagonebi erTmaneTs gadaeba. ra raodenobis energia gadavida kinetikuri energiidan sxva saxeobis energiaSi Sejaxebis dros? ecema, g) t1 drosa da t2 dros Soris, rodesac meore monetac velze ecema? ra siCqariT 116. moZraobs masis centri, rodesac t aris saTavgadasa vlo filmis gadaRebis dros patara sportuli manqana d) 0,25wm, e) 0,75wm da v) 1,75wm? sur. 9.80. amocana 116. (1500kg masis da 3m sigrZis) brtyel navze (4000kg masis da 14m sigrZis) Cqardeba da navmisadgomis odnav ufro dabla ganlagebul zedapirze xteba. Tqven filmis teqnikuri mrCeveli xarT. navi Tavdapirvelad navmisadgoms 121. 3kg masis sagani 8m/wm siCqariT x RerZis dadebiTi mimarTulebiT moZraobs da M masis uZrav sagans elastiurad ejaxeba (Sejaxeba erT ganzomilebaSi xdeba). Sejaxebis Semdeg M masis sagani 6m/wm siCqariT RerZis dadebiTi mimarTulebiT moZraobs. risi tolia M? exeba (sur. 9.80) da wyalze winaRobis gareSe gadaadgildeba. garda amisa, navis da manqanis masa Tanabrad aris ganawilebuli. gamoTvaleT, ra iqneba navmisadgomsa da navs Soris xvrelis sigane, rodesac manqana naxtoms akeTebs. 122. 500kg masis moduli mTavari kosmosuri xomaldis mimarT 1000m/wm siCqariT moZrav da 400kg masis kosmosur xomaldzea da 117. 2kg masis nivTieri wertili = (4m/wm) siCqariT moZraobs da 4kg masis = (2m/wm) siCqariT moZrav nivTier wertils 100m/wm siCqariT moZraobs. mTavari kos ejaxeba. Sejaxebis Sedegad nivTieri wer tilebi erTiandeba. ra iqneba maTi siCqaris a) erTeulovan-veqtorul Canaweri, b) sidide da g) kuTxe? magrebuli. mcire afeTqebis Sedead moduli kosmosuri xomaldis axali siCqaris mimarT mosuri xomaldis TvalTaxedviT modulis da kosmosuri xomaldis kinetikuri energia ra nawiliT gaizarda afeTqebis Sedegad? 123. 6kg masis raketis modeli samxreTisken 20m/wm siCqariT moZraobs da afeTqebis Sedegad or nawilad iSleba. 2kg masis erTi 118. eleqtroni (m1=9,11×10–31kg) da protoni (m2=1,67×10–27kg) erTmaneTs eleqtruli ZaliT nawilis siCqarea: izidavs. davuSvaT, eleqtronsa da protons sadac Soris sawyisi manZilia d=3×10–6m. rodesac maT Soris manZili 1×10–6 m-mde mcirdeba, ra iqneba a) eleqtronis impulsis Sefardeba protonis impulsTan, b) eleqtronis siCqaris Sefardeba protonis siCqaresTan da g) eleqtronis kinetikuri energiis Sefardeba protonis kinetikur energiasTan? d) Tu nawilakebs Soris manZili kvlav mcirdeba, a)-g) pasuxebi izrdeba, mcirdeba Tu ar icvleba? 119. 140g masis da 7,8m/wm siCqariT moZravi 1 = (–12m/wm) + (30m/wm) – (15m/wm) mimarTulia aRmosavleTisken, - CrdiloeTisken da - maRla. a) ra impulsi aqvs meore nawils erTeulovan-veqtorul CanawerSi? b) ra kinetikuri energia aqvs meore nawils? g) ra kinetikuri energia warmoiqmneba afeTqebis Sedegad? 124. 9.61 suraTze, pirveli bloki (6kg masis) marjvniv 8m/wm siCqariT moZraobs da meore bloki (4kg masis) aseve marjvniv 2m/wm siCqariT gadaadgildeba. zedapiri xaxunisgan 322 Tavi 9 masis centri da impulsi Tavisufalia da meore blokze 8000 n/m sixistis zambaraa damagrebuli. saboloo jamSi, pirveli bloki meores uswrebs. rodesac pirveli bloki marjvniv 6,4 m/wm siCqariT gadaadgildeba a) ra siCqare aqvs meore bloks da b) risi tolia zambaris elastiuri potenciuri energia? 125. 2900kg masis motoriani, swrafmavali navi tbaSi 250m/wm siCqariT moZraobs. Nnavze damagrebuli xapia tbidan iRebs wyals da navze ganlagebul cariel WurWelSi gadaaqvs. impulsis Senaxvis principis gamoyenebiT navis siCqare gamoTvaleT mas Semdeg, rac 920kg wyali amoixapeba. 126. 9.81 suraTze, 3,2 kg masis yuTi hor izontalur, xaxunisgan Tavi sufal zedapirze srialdeba da sur. 9.81. amocana 126. magidis kideze, h = 0,4m simaRleze mdebare meore yuTs ejaxeba. 3,2 kg masis yuTis siCqare Sejaxebamde 3 m/wm-ia. Tu yuTebi Sewebdeba, ra iqneba maTi kinetikuri energia iatakze davardnamde? 127. 2140kg masis vagoni platformasTan uZravad dgas. 242kg masis sumoisti plat formis gaswvriv 5,3m/wm siCqariT garbis da vagonze xteba. ra iqneba vagonis siCqare, Tu sumoisti a) vagonze dgas, b) sawyisi 5,3m/wm mimarTulebiT Yda vagonisadmi siCqariT garbis da g) sawyisi mimarTulebis sapirispirod da vagonisadmi 5,3m/wm siCqariT garbis? 128. 2 kg masis saTamaSo t=0 vlis manqana dros aT saTaveze imyofeba da x RerZis dadebiTi sur. 9.82. amocana 128. mimarTulebiT gad aadgildeba. manqanaze moqmedi jamuri Zalis droze damokidebulebis funqciis 9.82 darboda, vinaidan fexis wamokvris SemTxvevaSi misi mokle winamxrebi vardnis Serbilebas ver SeZlebda. warmoidgineT, rom m masis dinozavrma fexi wamokra, daeca da misi masis centri 1,5 m manZilze Tavisuflad vardeba. Semdeg dinozavris masis centri kidev 0,3m-iT dabla iwevs, vinaidan sxeuli da miwa ikumSeba. a) daaxloebiT ra saSualo vertikaluri Zala moqmedebs dinozavrze miwasTan Sejaxebis dros (0,3m-iT vardnisas)? davuSvaT, dinozavri fexis wamokvris dros 19 m/wm siCqariT moZ raobs, srialdeba da 0,6 kinetikuri xaxunis koeficientis mqone zedapirze Cerdeba. aseve davuSvaT, rom srialis dros saSualo vertikaluri Zala amocanis a) nawilis iden turia. b) daaxloebiT ra saSualo jamuri Zala moqmedebs miwidan dinozavrze da g) ra manZilze srialdeba dinozavri? a) da b) nawilebSi miRebuli pasuxebi cxadyofs, rom dinozavri dacemis Sedegad sxeuls daizianebda, xolo ufro Soris davardnili Tavi kidev ufro met zians miiRebda. 130. 9.83 suraTze gamosaxuli samive burTi identuria. meore da mesame burTi erTmaneTs exeba da maTi mdebareoba pirveli burTis traeqtoriisadmi perpendikularulia. pir veli burTis siCqarea ν0=10m/wm da mimarTulia pirveli da meore burTis Sexebis wertilisken. Sejaxebis Semdeg ra a) sidide da b) mimarTuleba eqneba meore burTis siCqares, ra g) sidide da d) mimarTuleba eqneba mesame burTis siCqares da ra e) sidide da v) mimarTuleba eqneba pirveli burTis siCqares? (miniSneba: xaxuni ar arsebobs da TiToeulis Zalis impulsi Sejaxebis burTebis centrebze gamavali wrfis gaswvriv, Sejaxebuli zedapirebisadmi perpendikularulad aris mimarTuli). sur. 9.83. amocana 130. 131. 9.22 suraTze, biliardis pirveli burTi 2,2m/wm siCqariT moZraobs da identur, uZrav naCvenebi. meore burTs iribad ejaxeba. Sejaxebis Semdeg a) ra tempiT icvleba manqanis impulsi t = 3 wm meore burTi 1,1m/wm siCqariT da θ2=60° kuTxiT moZraobs. ra b) sidide da b) mimarTuleba aqvs pirveli burTis siCqares Sejaxebis Semdeg? grafiki suraTzea 129. tiranozavri savaraudod swrafad ver dros? b) ra impulsi aqvs manqanas t = 3 wm dros? 323 g) SeiZleba Tu ara arsebuli monacemebis saSualebiT Sejaxebis tipis gansazRvra? da Zalvam b) raketas 21 m/wm2 sawyisi aCqareba mianiWos? 132. biliardis TamaSis dros erTi burTi igive masis da uZravad mdebare meore burTs ejaxeba. Sejaxebis Semdeg pirveli burTi 136. 3000 kg masis bloki 6 m simaRlidan vertikalurad vardeba da 500 kg masis sagans ecema. sagani miwaSi 3 sm siRrmeze Cadis. 3,5m/wm siCqariT im wrfis gaswvriv moZraobs, mimarTulebisadmi davuSvaT bloksa da sagans Soris Sejaxeba sruliad araelastiuria. gamoTvaleT miwidan 22° kuTxes adgens. meore burTis siCqarea 2m/wm. gamoTvaleT a) kuTxe meore burTis saganze moqmedi saSualo Zala 3sm siRrmeze Casvlis ganmavlobaSi. romelic misi sawyisi moZraobis mimarTulebasa da pirveli burTis moZraobis sawyis mimarTulebas Soris da b) pirveli burTis sawyisi siCqare. g) inaxeba Tu ara kinetikuri energia? (saubaria masis centrebze, brunva ar ganixiloT). 133. a) ra manZilzea dedamiwa-mTvaris masis centri dedamiwis centridan? (C damatebaSi mocemulia dedamiwisa da mTvaris masebi da maT Soris manZili). b) dedamiwis radiusis ra procentia es manZili? 134. 9.84 suraTze gamosaxuli pirveli bloki xaxunisgan Tavisufal zedapi rze 0,75 m/wm siCqariT sur. 9.84. amocana 134. x RerZis gaswvriv gadaadgildeba. Semdeg pirveli bloki uZravad mdebare meore bloks elastiurad ejaxeba. cxrilSi mocemulia blokebis masebi da sigrZeebi, agreTve maTi centrebis mdebareobebi, rodesac t = 0. sad mdebareobs blokebis sistemis masis centri, rodesac a) t = 0, b) blokebi erTmaneTs Seexebian da g) t = 4 wm? bloki masa (kg) 1 2 0,25 0,5 sigrZe centri (sm) t=0 dros x=–1,5m x =0 5 6 135. 6100 kg masis raketa miwidan ver tikalurad gaisroles. Tu gamonabolqvis siCqarea 1200 m/wm, ra raodenobis airi unda gamovides TiToeul wamSi, raTa Zalva a) ra ketaze moqmedi mizidulobis Zalis toli iyos 137. 280 kn wonis 1,52 m/wm siCqariT moZravi vagoni 210 kn wonis da 0,914 m/wm siCqariT moZrav vagons eweva. Tu vagonebi dawyvildeba, a) ra siCqare eqneba mas Sejaxebis Semdeg da b) ra raodenobis kinetikuri energia dai kargeba Sejaxebis dros? Tu Sejaxeba elas tiuria (rac naklebad savaraudoa), ra iqneba g) msubuqi vagonis da d) mZime vagonis siCqare Sejaxebis Semdeg? 138. P da Q nawilakebs Soris manZilia 1 m. P-s masaa 0,1kg, xolo Q-s masa 0,3kg. P da Q nawilakebi erTmaneTs 1×10–2 n mudmivi ZaliT izidavs da sistemaze gare Zalebi ar moqmedebs. a) ra siCqariT moZraobs nawilakebis masis centri, rodesac maT Soris manZilia 0,5m? b) Ps sawyisi mdebareobidan ra manZilze Seejaxeba erTmaneTs nawilakebi? 139. 60 kg masis adamiani CrdiloeTisken 6 m/wm siCqariT misrialebs da 38 kg masis bavSvs ejaxeba. Sejaxebis Semdeg orive maTgani 3 m/wm siCqariT da 35° kuTxiT CrdiloaRmosavleTisken moZraobs. ra a) sididis da b) mimarTulebis siCqare hqonda bavSvs Sejaxebamde? 140. 0,2 kg masis hokeis Saiba xaxunisgan Tavisufal yinulze 10 m/wm siCqariT aRmo savleTisken misrialebs da hokeis yvanjs ejaxeba. rogor icvleba Saibis impulsis sidide da mimarTuleba yvanjTan Sexebis ganmavlobaSi, Tu Sexebis Semdeg Saibis siCqare a) 20 m/wm-ia da mimarTulia aRmosavleTisken, b) 5 m/wm-ia da mimarTulia aRmosavleTisken da g) 10 m/wm-ia da mimarTulia dasavleTisken? 10 brunva `rusuli mTebi~ saSiSroebis iluzias qmnis, magram sinamdvileSi inJinrebi yvelafers akeTeben, raTa atraqcioni sruliad usafrTxo iyos. miuxedavad maRali xarisxis usafrTxoebisa, milionobiT uiRblo adamians e.w. `rusuli mTebis Tavis tkivili~ emarTeba. am daavadebam SesaZloa ramdenime dRis ganmavlobaSi Tavi ar iGinos, Tumca Semdeg adamians Tavbrusxveva da Tavis tkivili awuxebs. ra iwvevs „rusuli mTebis Tavis tkivils“? pasuxs am TavSi SeityobT 10.1. ra aris fizika? rogorc ukve vTqviT, fizikis Seswavlis erT-erTi sagani moZraobaa. ganvixileT gadataniTi moZraoba, rodesac sagani 10.1a suraTis msgavsad wrfivad an mrudwirulad moZraobda. amjerad Cven brunviT moZraobas SevexebiT, rodesac sagani 10.1b suraTis msgavsad RerZis garSemo brunavs. brunvas yoveldRiur cxovrebaSi Zalian xSirad vawydebiT. manqanaSi, sasmelis WurWlis gaxsnisas Tu atraqcionebze garTobisas brunvis magaliTebs vexebiT. brunvas mravali gasarTobi saSualeba efuZneba. magaliTad golfi da beisbolic gamodgeba. brunva ufro seriozuli movlenebis mizezic SeiZleba gaxdes. magaliTad, moZvelebul TviTmfrinavze (a) liTonis cvdoma swored brunvasTan kavSirSia. brunvis ganxilvas moZraobis parametrebis gansazRvriT viwyebT, iseve rogorc me-2 paragrafSi gavakeTeT. 10.2. brunvis parametrebi Cven myari sxeulis brunvas fiqsirebuli RerZis garSemo ganvixilavT. myari sxeuli niSnavs, rom misi yvela nawili erTad moZraobs da sxeulis forma ar icvleba. fiqsirebuli RerZi niSnavs, rom brunva uZravi RerZis garSemo xdeba. maSasadame, sur. 10.1. mocigurave sara hiuzi a) fiqsirebuli mimarTulebiT gadataniTi moZraobisas da b) vertikaluri RerZis garSemo brunvisas 324 (b) 10.2. brunvis parametrebi 325 Cven ar SevexebiT mzis magvar sagnebs, vinaidan mzis (airis burTis) nawilebi erTad ar moZraoben. aseve ar SevexebiT boulingis burTis magvar sagnebs, vinaidan burTi moZravi RerZis garSemo brunavs (burTis moZraoba brunviTi da gadataniTi moZraobis nazavia). 10.2 suraTze naCvenebia fiqsirebuli RerZis garSemo mbrunavi, pirobiTi formis myari sxeuli. RerZs, romlis garSemo sxeuli brunavs, brunvis RerZi ewodeba. brunvis dros (kuTxuri moZraobis dros), sxeulis yvela wertili brunvis RerZze mdebare centris mqone wrewirze moZraobs da drois garkveul SualedSi yvela wertili Tanabar kuTxes Semowers. gadataniTi moZraobis dros (wrfivi moZraobisas), sxeulis yvela wertili wrfivad moZraobs da garkveul drois SualedSi yvela wertili Tanabar wrfiv manZils gadis (kuTxur da wrfiv moZraobas momavalSic SevadarebT erTmaneTs). modiT SevexoT wrfivi sidideebis - mdebareobis, gadaadgilebis, siCqaris da aCqarebis kuTxur eqvivalentebs. kuTxuri mdebareoba 10.2 suraTze naCvenebia sxeulze damagrebuli aTvlis wrfe, romelic brunvis RerZis perpendikularulia da sxeulTan erTad brunavs. am wrfis kuTxuri mdebareoba fiqsirebuli mimarTulebisadmi mdgeneli kuTxea. Tavad fiqsirebul mimarTulebas nulovan kuTxur mdebareobad miviCnevT. 10.3 suraTze, θ kuTxuri mdebareoba x RerZis dadebiTi mimarTulebisadmi izomeba. geometriidan gamomdinare: θ= s (radianebSi) , r (10.1) sadac s aris x RerZidan (nulovani kuTxuri mdebareobidan) aTvlis wrfemde gadaWimuli wriuli rkalis sigrZe, xolo r - wris radiusi. 10.1 formuliT gansazRvruli kuTxe radianebSi izomeba da ara brunebSi an gradusebSi. radiani ori sigrZis Sefardebaa, ricxvia da maSasadame ganzomileba ar gaaCnia. vinaidan r radiusis wris sigrZea 2πr, srul wreSi 2π radiania da gveqneba: 2πr 1 bruni = 360° = = 2π rad (10.2) r aqedan gamomdinare: 1 radiani = 57,3° = 0,159 bruni sur. 10.2. pirobiTi formis myari sxeuli koordinatTa sistemis z RerZis garSemo brunavs. myari sxeulis aTvlis wrfis mdebareoba agreTve pirobiTadaa arCeuli, magram is yovelTvis brunvis RerZis perpendikularulia. aTvlis wrfe sxeulzea fiqsirebuli da sxeulTan erTad brunavs. (10.3) sur. 10.3. 10.2 suraTze gamosaxuli mbrunavi, myari sxeulis zemodan danaxuli ganivi kveTi. ganivi kveTis sibrtye saxelmZRvanelos gverdidan moSorebiT, Tqvensken mimarTuli brunvis RerZis perpendikularulia. sxeulis amgvari mdebareobis dros, aTvlis wrfe x RerZTan θ kuTxes adgens. 326 Tavi 10 brunva brunvis RerZis garSemo aTvlis wrfis srulad Semobrunebis Semdeg θ kuTxe kvlav nuli ar xdeba. Tu aTvlis wrfem nulovani kuTxuri mdebareobidan ori sruli bruni Seasrula, maSin wrfis θ mdebareobaa 4π radiani. x RerZis gaswvriv gadataniTi moZraobis dros yvelafers gavigebT, Tu moZravi sxeulis mdebareobis droze damokidebulebis x(t) funqcia viciT. brunviTi moZraobis drosac yvelafers gavigebT, Tu mbrunavi sxeulis aTvlis wrfis kuTxuri mdebareobis droze damokidebulebis θ(t) funqcia viciT. kuTxuri gadaadgildeba Tu 10.3 suraTze gamosaxuli sxeuli 10.4 suraTis msgavsad brunvis RerZis garSemo brunavs da aTvlis wrfis kuTxuri mdebareoba θ1-dan θ2-ze icvleba, maSin sxeuli Δθ kuTxur gadaadgilebas gadis: Δθ = θ2 – θ1 . (10.4) kuTxuri gadaadgilebis es gansazRvreba ara mxolod myar sxeuls miesadageba, aramed am sxeulis yvela wertils. Tu sxeuli x RerZis gaswvriv gadataniT moZraobas asrulebs, misi Δx gadaadgildeba dadebiTi an uaryofiTia, rac imazea damokidebuli sxeuli RerZis dadebiTi mimarTulebiT moZraobs Tu uaryofiTi mimarTulebiT. mbrunavi sxeulis Δθ kuTxuri gadaadgilebac dadebiTia an uaryofiTi. saaTis isris moZraobis sawinaaRmdego mimarTulebiT kuTxuri gadaadgileba dadebiTia, xolo saaTis isris moZraobis mimarTulebiT kuTxuri gadaadgileba uaryofiTi. kuTxuri siCqare davuSvaT mbrunavi sxeuli t1 droSi θ1 kuTxur mdebareobaze imyofeba, xolo t2 droSi θ2 kuTxur mdebareobazea ganlagebuli (sur. 10.4). Δt drois SualedSi sxeulis saSualo kuTxuri siCqarea: Δθ θ –θ ωsaS = t 2 – t 1 = , (10.5) Δt 2 1 sadac Δθ kuTxuri gadaadgileba Δt drois SualedSi xdeba. ω myisi kuTxuri siCqare 10.5 formulis zRvaria, rodesac Δt nulisken miiswrafvis: ω= Δθ dθ = . Δt dt (10.6) Tu θ(t) viciT, ω kuTxur siCqares diferencirebis Sedegad gamovTvliT. 10.5 da 10.6 formulebi ara mxolod mbrunav myar sxeulebs miesadageba, aramed am sxeulebis yvela wertils, vinaidan yvela sur. 10.4. 10.2 da 10.3 suraTebze wertili erTad moZraobs. kuTxuri siCqaris erTeulia radiani/ gamosaxuli myari sxeulis aT wm an bruni/wm. es ukanaskneli rok-musikis ganviTarebis pirvel vlis wrfe t1 droSi θ1 kuTxur ganlagebuli, sam aTwleulSi gamoiyeneboda. vinilis firfitebze aRniSvna “33 mdebareobazea xolo t2 droSi θ2 kuTxur mdeba 1/3 rpm” an “45rpm” niSnavs, rom maTi gamoyeneba firsakravze reobaze imyofeba. Δθ(=θ – θ ) 2 1 SeiZleba wuTSi 33 1/3 brunis (33 1/3 br/wT) an 45 brunis (45br/wT) sxeulis kuTxuri gadaadgilebaa Δt(=t2 – t1) drois SualedSi. Tavad Sesrulebis SemTxvevaSi. sxeuli naCvenebi ar aris. 10.2. brunvis parametrebi 327 Tu nivTieri wertili x RerZis gaswvriv gadataniT moZraobas asrulebs, misi ν wrfivi siCqare dadebiTi an uaryofiTia, rac imazea damokidebuli, wertili RerZis dadebiTi mimarTulebiT moZraobs Tu uaryofiTi mimarTulebiT. mbrunavi myari sxeulis ω kuTxuri siCqarec dadebiTi an uaryofiTia, rac Tavis mxriv imazea damokidebuli, sxeuli saaTis isris moZraobis sawinaaRmdego mimarTulebiT brunavs (siCqare dadebiTia), Tu saaTis isris moZraobis mimarTulebiT (siCqare uaryofiTia). kuTxuri aCqareba Tu mbrunavi sxeulis kuTxuri siCqare icvleba, maSin sxeuls kuTxuri aCqareba aqvs. davuSvaT, ω2 da ω1 aris sxeulis siCqareebi t2 da t1 droSi. drois am SualedSi mbrunavi sxeulis saSualo kuTxuri aCqarebaa: Δω ω –ω αsaS = t2 – t 1 = (10.7) Δt 2 1 sadac Δω aris Δt drois SualedSi momxdari kuTxuri siCqaris cvlileba. α myisi kuTxuri aCqareba am sididis zRvaria, rodesac Δt nulisken miiswrafvis: Δω dω α= = . (10.8) Δt dt 10.7 da 10.8 formulebi sxeulis yvela wertils miesadageba. kuTxuri aCqarebis erTeulia radiani/wm2 an br/wm2 amocanis amoxsnis nimuSi 10.1 10.5a suraTze gamosaxuli disko centraluri wrfe, rodesac t = –2 wm, 0 wm, 4 wm da rodesac RerZis garSemo brunavs. diskos aTvlis wrfis grafiki t RerZs kveTs. θ(t) kuTxuri mdebareobaa: amoxsna: diskos kuTxuri mdebareoba aT θ = –1 – 0,6t + 0,25t , (10.9) vlis wrfis θ(t) kuTxuri mdebareobaa, rom sadac t wamebSia, θ - radianebSi, xolo nulovani lis droze damokidebulebis funqcia 10.9 2 kuTxuri mdebareoba suraTzea naCvenebi. a) aageT diskos kuTxuri mdebareobis droze damokidebulebis funqciis grafiki -3wm- dan 5,4wm-mde drois SualedSi. gamosaxeT disko da misi kuTxuri mdebareobis aTvlis (a) (b) formuliTaa mocemuli. 10.9 formulis gra fiki 10.5b suraTze naCvenebia. diskos da aTvlis wrfis garkveul droSi gamosaxvisTvis θ gvWirdeba. amisTvis 10.9 for mulaSi dro unda CavsvaT. rodesac t = –2wm: (g) sur. 10.5. a) mbrunavi disko. b) diskos θ(t) kuTxuri mdebareobis funqciis grafiki. diskos xuTi gamosaxuleba grafikis xuTi wertilisTvis aTvlis wrfis kuTxur mdebareobas asaxavs. g) diskos ω(t) kuTxuri siCqaris funqciis grafiki. ω-s dadebiTi mniSvnelobebi saaTis isris moZraobis sawinaaRmdego mimarTulebiT brunvas Seesabameba, xolo uaryofiTi mniSvnelobebi saaTis isris moZraobis mimarTulebiT brunvas asaxavs. 328 brunva Tavi 10 θ = –1 – (0,6)(–2) + (0,25)(–2)2 360° = 1,2 rad = 1,2 rad = 69°. 2πr rad es niSnavs, rom t = –2wm dros diskos aTvlis wrfe nulovani mdebareobidan saaTis isris moZraobis sawinaaRmdego mimarTulebiT 1,2 radianiT (69°) Semobrunda. 10.5b suraTis amoxsna: 10.6 formulidan gamomdinare, ω kuTxuri siCqare dθ/dt-s tolia, amitom 10.10 formulidan ω = –0,6 + 0,5t (10.11) ω(t) funqciis grafiki 10.5g suraTzea na Cvenebi. rodesac t = 0, θ = –1 radiani = –57° da es niS t = –2 wm dros diskos gamosaxvisTvis es mniS vneloba 10.11 formulaSi CavsvaT: ω = –1,6 radiani/wm (pasuxi) minus niSani gvauwyebs, rom t = –2 wm dros navs, rom aTvlis wrfe nulovani kuTxuri disko saaTis isris moZraobis mimarTulebiT mdebareobidan brunavs da disko 10.5g suraTze sul qvemoTaa gamosaxuleba 1 aTvlis wrfis am mdebareobas asaxavs. saaTis isris moZraobis mimarTulebiT 1 radianiT anu 57°-iT Semo gamosaxuli. brunda (gamosaxuleba 3). rodesac t=4wm, t = 4 wm drois 10.11 formulaSi Casma gvaZ θ=0,6 radiani=34° (gamosaxuleba 5). diskos gamosaxva, rodesac grafiki t RerZs kveTs, advilia, vinaidan θ=0 da aTvlis wrfe nulovan kuTxur mdebareobazea (gamosaxulebebi 2 da 4). levs: b) ra tmin dros gaxdeba θ(t)-s mniSvneloba ω=1,4 radiani/wm (pasuxi) plius niSani gvauwyebs, rom t = 4 wm dros disko saaTis isris moZraobis sawinaaRmdego mimarTulebiT brunavs (10.5g suraTze disko gamosaxulia sul maRla). mniSvneloba (am SemTxvevaSi minimumi) unda tmin drois SemTxvevaSi dθ/dt=0 da maSasa dame, ω=0. es niSnavs, rom 10.5b suraTze θ-s minimaluri mniSvnelobis dros disko wamierad Cerdeba (10.5g suraTze gamosaxuli gamovTvaloT. amisTvis funqcia gavawarmooT Sua disko). umciresi da ra iqneba es mniSvneloba? amoxsna: amjerad funqcis eqstremaluri da nuls gavutoloT. θ(t)-s pirveli rigis warmoebulia: d) gamoiyeneT amocanis a)-g) SekiTxvebi da -3wm- aRwereT. amoxsna mogvcems: amoxsna: rodesac t = –3wm, diskos dadebiTi kuTxuri mdebareoba aqvs da saaTis isris dθ – 0,6 + 0,5t (10.10) dt = am formulis 0-Tan gatoleba da t-sTvis tmin=1,2 wm (pasuxi) θ-s umciresi mniSvnelobis gamoTvlisTvis tmin 10.9 formulaSi CavsvaT da miviRebT: θ(t) θ = –1,36 radiani ≈ 77,9° (pasuxi) (10.5b suraTze funqciis minimumi gamosaxuli grafikis fskeri) saaTis isris moZraobis mimarTulebiT diskos maqsimalur mobrunebas asaxavs nulovani kuTxuri mde bareobidan. g) aageT diskos ω kuTxuri siCqaris droze damokidebulebis funqciis grafiki -3wm- dan 6wm-mde SualedSi. gamosaxeT disko da mobrunebis mimarTuleba da daadgineT ω-s niSani, rodesac t = –2wm, 4wm da tmin. dan 6wm-mde drois SualedSi diskos moZraoba moZraobis mimarTulebiT brunavs. θ = –1,36 Senelebulad radiani kuTxur mdeba reobaze disko wamierad Cerdeba da saaTis isris moZraobis sawinaaRmdego mimarTulebiT iwyebs brunvas. Sesabamisad, misi kuTxuri mdebareoba kvlav dadebiTi xdeba. sakontrolo wertili 1. disko (sur. 10.5a) centraluri RerZis garSemo brunavs. sawyisi da saboloo kuTxuri mdebareobebis romeli wyvili gvaZlevs uaryofiT kuTxur gadaadgilebas: a) –3 radiani, + 5radiani, b) –3 radiani, – 7 radiani, g) 7 radiani, -3 radiani? 10.3. aris Tu ara kuTxuri sidide veqtori? sur. 10.6. a) firfita vertikaluri RerZis garSemo brunavs, romelic lilvs emTxveva. b) mbrunavi firfitis kuTxuri siCqare RerZis gaswvriv gavlebuli da qvemoT mimarTuli veqtoris saSualebiT SeiZleba gamoisaxos. g) kuTxuri siCqaris veqtoris mimarTulebas marjvena xelis wesiT ganvsazRvravT. Tu firfitas marjvena xels moZraobis mimar TulebiT SemovavlebT, gaSlili cera TiTi -s mimarTulebas gviCvenebs. (a) (b) 329 (g) 10.3. aris Tu ara kuTxuri sidide veqtori? nivTieri wertilis mdebareoba, siCqare da aCqareba veqtorebis saSualebiT SegviZlia aRvweroT. Tu nivTieri wertili wrfivad moZraobs, veqtorebi ar gvWirdeba. aseT dros nivTier wertils mxolod ori SesaZlo mimarTuleba aqvs da maTi asaxva minus-plius niSnebiT SeiZleba. fiqsirebuli RerZis garSemo mbrunav sxeulsac mxolod ori mimarTulebiT SeuZlia moZraoba (saaTis isris moZraobis da saaTis isris moZraobis sawinaaRmdego mimarTulebiT) da amjeradac plius-minus niSnebi SegviZlia gamoviyenoT. Cndeba SekiTxva: `SesaZlebelia Tu ara mbrunavi sxeulis kuTxuri gadaadgileba, siCqare da aCqareba veqtorebad ganvixiloT?~. am SekiTxvis pasuxia, rom SesaZlebelia. ganvixiloT kuTxuri siCqare. 10.6a suraTze naCvenebia firsakravze mrbunavi firfita. firfita mudmivi ω (= 33 1/3 br/wT) kuTxuri siCqariT saaTis isris moZraobis mimarTulebiT brunavs. kuTxuri siCqare brunvis RerZis gaswvriv mimarTuli veqtoris saSualebiT SegviZlia warmovadginoT, rogorc es naCvenebia 10.6b suraTze. veqtoris sigrZis dasadgenad raime mosaxerxebel skalas viyenebT, magaliTad 1sm-s 10br/wT-s SevusabamebT. veqtoris mimarTulebas marjvena xelis wesiT vadgenT (sur. 10.6g). mbrunav firfitas marjvena xeli SemoavleT da TiTebi brunvis mimarTulebiT gaiSvireT. am SemTxvevaSi, cera TiTis mimarTuleba kuTxuri siCqaris veqtoris mimarTulebas emTxveva. Tu firfita sawinaaRmdego mimarTulebiT brunavs, maSin marjvena xelis wesi kuTxuri siCqaris veqtoris sapirispiro mimarTulebas mogvcems. kuTxuri sidideebis veqtorebad warmodgena martivi ar aris. instinqti gvkarnaxobs, rom veqtoris moZraobis mimarTulebiT raRac moZraobs, magram es ase ar aris. myari sxeuli (anu, es raRaca) veqtoris mimarTulebis garSemo brunavs. veqtori brunvis RerZs gansazRvravs da ara raRacis moZraobis mimarTulebas. garda amisa, veqtorebi me-3 paragrafSi mocemul wesebs emorCileba.  a kuTxuri aCqareba kidev erTi veqtoria da isic am wesebs emorCileba. (a) (b) a) wigni 90°-iT jer x RerZis garSemo, xolo Semdeg y RerZis garSemo orjer brundeba. b) igive wigni sawinaaRmdego TanmimdevrobiT brundeba. sur. 10.7. mocemul TavSi mxolod fiqsirebuli RerZis garSemo brunvas ganvixilavT. am dros veqtorebi ar gvWirdeba, vinaidan kuTxur siCqares ω simboloTi warmovadgenT, kuTxur aCqarebas – α simboloTi da maTi mimarTulebis asaxvisTvis plius da minus niSans viyenebT. 330 brunva Tavi 10 gaxsovdeT, rom kuTxur gadaadgilebas veqtorad ver ganvixilavT, Tu is Zalian mcire ar aris. ratom? gadaadgilebas 10.6 suraTze gamosaxuli kuTxuri siCqaris veqtoris msgavsad sidide da mimarTuleba SegviZlia mivaniWoT, magram veqtoruli sidide veqtorTa Sekrebis kanonsac unda emorCilebodes, romlis Tanaxmad veqtorebis Sekrebis Tanmimdevrobas mniSvneloba ar aqvs. gadaadgilebis SemTxvevaSi ki Tanmimdevroba namdvilad mniSvnelovania. 10.7 suraTze naCvenebia magaliTi. Tavdapirvelad horizontaluri wigni 90°-ian or kuTxur gadaadgilebas jer 10.7a suraTze gamosaxuli TanmimdevrobiT, xolo Semdeg 10.7b suraTze gamosaxuli TanmimdevrobiT ganicdis. orive kuTxuri gadaadgileba identuria, magram maTi Tanmimdevroba gansxvavdeba da wignis saboloo mimarTulebac gansxvavebulia. aqedan gamomdinare, ori kuTxuri gadaadgilebis Sekreba maT Tanmimdevrobazea damokidebuli da gadaadgilebebs veqtorebad ver ganvixilavT. cxrili 10.1 moZraobis formulebi Tanabari wrfivi aCqarebis da Tanabari kuTxuri aCqarebis SemTxvevaSi formulis nomeri wrfivi formula (2.11) ν = ν0 + at (2.15) cvladi, romelic ar Cans kuTxuri formula x – x0 θ – θ0 x – x0 = νot + 2 at2 ν ω (2.16) ν2 = ν02 + 2a(x – x0) t t ω2 = ω02 + 2α(θ – θ0) (2.17) x – x0 = 1 (ν0+ν)t a α θ – θ0 = 2 (ω0 + ω)t (2.18) x – x0= νt – ν0 ω0 θ – θ0 = ωt – 2 αt2 1 2 1 2 at2 ω = ω0 + αt θ – θ0 = ω0t + 1 2 nomeri (10.12) αt 1 1 formulis (10.13) (10.14) (10.15) (10.16) 10.4. brunva Tanabari kuTxuri aCqarebiT gadataniTi moZraoba mudmivi wrfivi aCqarebiT mniSvnelovani SemTxvevaa. 2.1 cxrilSi mocemulia amgvari moZraobis amsaxveli formulebi. brunva mudmivi kuTxuri aCqarebiT aseve mniSvnelovani SemTxvevaa da amgvar moZraobasac garkveuli formulebi Seesabameba. am formulebs mocemul paragrafSi ar gamoviyvaneT da ubralod Sesabamis wrfiv formulebSi eqvivalenturi kuTxuri sidideebi SevitaneT. swored ase miviReT 10.1 cxrilSi mocemuli formulebi. gaixseneT, rom 2.11 da 2.15 Tanabarwrfivi aCqarebiT moZraobis ZiriTadi formulebia. 10.12 da 10.13 aseve Tanabari kuTxuri aCqarebiT brunvis ZiriTadi formulebia da danarCeni formulebi maTgan miiReba. Tanabar kuTxur aCqarebasTan dakavSirebuli amocanis amoxsnisTvis swored es formulebi unda gamoiyenoT. amocanis amoxsnis dros SeecadeT is formula airCioT, romelSic Semavali erTaderTi ucnobi cvladi amocanis SekiTxvaSia dasmuli. magaliTi 10.3 amocanis amoxsnis nimuSSia mocemulia. sakontrolo wertili 2. mbrunavi sxeulis θ(t) kuTxuri mdebareobaa, a) θ = 3t – 4, b) θ = –5t3 + 4t2 + 6, g) θ = 2/t2 – 4/t da d) θ = 5t2 – 3. romel SemTxvevaSi gamodgeba 10.1 cxrilSi moyvanili formulebi? 331 10.4. brunva Tanabari kuTxuri aCqarebiT amocanis amoxsnis nimuSi salesi qva (sur. 10.2 α=0,35radiani/wm2 10.8) Tanabari kuTxuri aCqarebiT brunavs. rodesac t=0, misi kuTxuri siCqarea ω0 = –4,6radiani/wm da aTvlis wrfe horizontaluria (θ0=0). a) ra dros iqneba aTvlis wrfis kuTxuri mdebareoba θ=5br? amoxsna: kuTxuri aCqareba Tanabaria, amitom 10.1 cxrilSi mocemuli formulebi SegviZlia gamoviyenoT. CvenTvis sasurveli formula sur. 10.8. salesi qva. t = 0 dros aTvlis wrfe horizontaluria. 10.13-ia, vinaidan is CvenTvis saintereso erTaderT ucnobs, t dros Seicavs: miuTiTebs, rom qva brunvis dros neldeba, θ – θ0 = ω0t + 1 αt2 2 monacemebis da θ0=0, θ = 5 br = 10π rad Sedegis Cerdeba da Semdeg sapirispiro mimarTulebiT Casma mogvcems: droSi qva kidev xuTjer brundeba. 10π rad = –(4,6 rad/wm)t + 1 (0,35rad/wm2)t2 2 gantolebis amonaxsnia: t = 32 wm (pasuxi) b) aRwereT salesi qvis brunva 0-dan 32wm-mde drois SualedSi. ω0 = –4,6 radiani/wm kuTxuri siCqariT brunavs, magram misi α kuTxuri aCqareba dadebiTia. kuTxuri siCqaris da kuTxuri aCqarebis sapirispiro niSani amocanis amoxsnis nimuSi rotoris (6.8 θ = 0 sawyis mdebareobas daubrundeba, t = 32 wm g) ra t dros SeCerdeba wamierad qva? amoxsna: kvlav vubrundebiT gvWirdeba, zemoT da mocemul amjeradac romelic is mxolod cxrils formula t ucnobs Seicavs. garda amisa, es formula ω cvladsac amoxsna: qva saaTis isris moZraobis mimar TulebiT iwyebs brunvas. mas Semdeg rac aTvlis wrfe amocanis amoxsnis nimuSSi mocemuli mbrunavi cilindri) marTvis dros mgzavrs jer mwvave tkivils agrZnobinebT, xolo Semdeg cilindris kuTxur siCqares 20 brunis ganmavlobaSi mudmivi kuTxuri aCqarebiT 3,4radiani/wm-dan 2radiani/wm-mde unda Seicavdes, romelsac 0-ad miviCnevT da Sesabamisi t droisTvis amovxsniT. 10.12 formula mogvcems: t= ω – ω0 0 – (–4,6 rad/wm) = 0,35 rad/wm2 = 13 wm (pasuxi) α 10.3 davukavSiroT. sawyisi kuTxuri siCqarea ω0=3,4radiani/wm, kuTxuri gadaadgilebaa θ – θ0 = 20 br da gadaadgilebis bolos kuT xuri siCqarea ω = 2 radiani/wm. α kuTxuri aCqareba da t dro ar viciT, magram orive sidide ZiriTad formulebSi Sedis. amcirebT (mgzavri cxadia gadataniT moZraobas ucnobi t-s gabaTilebisTvis 10.12 formula asrulebs da ara brunviT moZraobas). gamoviyenoT da davweroT: a) risi tolia Tanabari kuTxuri aCqareba kuTxuri siCqaris Semcirebis dros? amoxsna: moZraobis davuSvaT brunva sawinaaRmdego saaTis isris mimarTulebiT xdeba da aCqareba t = 0 dros, θ0 kuTxuri mdebareobisas iwyeba. vinaidan cilindris kuTxuri aCqareba Tanabaria, is kuTxur siCqares da kuTxur gadaadgilebas 10.12-10.13 ZiriTadi formulebis saSualebiT SegviZlia ω – ω0 . α Semdeg miRebuli Sedegi 10.13 formulaSi t= CavsvaT: θ – θ 0 = ω0 ( ω – ω0 α ) ( + 1α 2 ω – ω0 α ) 2 α-sTvis amoxsna, monacemebis Setana da 20 bris 125,7 radianSi gadayvana mogvcems: ω2 – ω02 (2 rad/wm)2 – (3,4 rad/wm)2 α = 2(θ – θ ) = 2(125,7 rad) 0 = –0,0301 rad/wm2 (pasuxi) 332 Tavi 10 brunva amoxsna: amjerad α viciT da 10.12 formula t- = 46,5 wm (pasuxi) es amocana 2.5 amocanis amoxsnis nimuSis sTvis amovxsnaT: kuTxuri versiaa. monacemebi da simboloebi b) ra dro dasWirda siCqaris Semcirebas? t= ω – ω0 = α 2 rad/wm – 3,4 rad/wm –0,0301 rad/wm2 gansxvavdeba, magram amoxsnis meTodi identuria. 10.5. wrfivi da kuTxuri cvladebis erTmaneTTan dakavSireba 4.7 paragrafSi Tanabari wriuli moZraoba ganvixileT, rodesac nivTieri wertili wrewirze da brunvis RerZis garSemo ν mudmivi wrfivi siCqariT moZraobda. rodesac myari sxeuli RerZis garSemo brunavs, sxeulis TiToeuli wertili RerZis garSemo sakuTar wrewirze moZraobs. vinaidan sxeuli myaria, wertilebi erT bruns drois erTi da igive SualedSi asruleben da anu maTi ω kuTxuri siCqare Tanabaria. rac ufro Sors aris RerZidan wertili, miT ufro metia wrewiris sigrZe da miT ufro meti unda iyos ν wrfivi siCqare. es faqti kargad SeimCneva karuselze. miuxedavad karuselis centridan daSorebisa, ω kuTxuri siCqare ar icvleba, magram ν wrfivi siCqare mniSvnelovnad izrdeba, Tu karuselis kideze dajdebiT. sxeulis garkveuli wertilisTvis s, ν da a wrfivi cvladebis θ, ω da α kuTxur cvladebTan dakavSireba xSirad saWiroa. cvladTa wyvilebis erTmaneTTan dakavSireba brunvis RerZidan am wertilamde r perpendikularuli manZilis saSualebiT xdeba. perpendikularuli manZili wertilsa da brunvis RerZs Soris aris manZili, romelic RerZis perpendikularis gaswvriv izomeba. igive sididisaa wertilis mier brunvis RerZis garSemo gavlili wrewiris r radiusi. mdebareoba Tu myari sxeulis aTvlis wrfe θ kuTxiT brunavs, sxeulis brunvis RerZidan r mdebareobaze ganlagebuli wertili wriuli rkalis gaswvriv s manZils gadis, sadac s 10.1 formuliT gamoiTvleba: s=θr (radianebSi) (10.17) miRebuli formula oTxidan erT-erTi kavSiria wrfiv da kuTxur cvladebs Soris. gafrTxileba: θ kuTxe radianebSi unda gaizomos, vinaidan 10.17 formula kuTxis radianebSi gazomvis gansazRvrebaa. siCqare 10.17 formulis droiT diferencireba (rodesac r mudmivia) gvaZlevs: ds dθ = r dt dt ds/dt wertili wrfivi siCqarea, xolo dθ/dt mbrunavi sxeulis ω kuTxur siCqares asaxavs. aqedan gamomdinare: ν = ωr (radianebSi) gafrTxileba: ω kuTxuri siCqare radianebSi unda gamoisaxos. (10.18) 333 10.5. wrfivi da kuTxuri cvladebis erTmaneTTan dakavSireba 10.18 formula gvauwyebs, rom vinaidan myari sxeulis yvela wertilis ω kuTxuri siCqare erTi da igivea, meti r radiusis mqone wertilebis ν wrfivi siCqare Sesabamisad metia. 10.9a suraTis mixedviT wrfivi siCqare yovelTvis wertilis traeqtoriis mxebia. Tu myari sxeulis ω kuTxuri siCqare ucvlelia, maSin 10.18 formulis Tanaxmad misi nebismieri wertilis ν wrfivi siCqarec ar icvleba. aqedan gamomdinare, sxeulis TiToeuli wertili Tanabarwriulad moZraobs. yoveli wertilis da, Tavad myari sxeulis, brunvis T periodi 4.35 formuliT (a) gamoiTvleba: T= 2πr v (10.19) 10.19 formulis mixedviT, erTi SemobrunebisTvis saWiro dro Semobrunebis ganmavlobaSi ganvlili 2πr manZilis am manZilis gavlis siCqaresTan Sefardebaa. 10.18 formulidan ν-s Casma da r-is gabaTileba mogvcems: T= 2π ω (10.20) (radianebSi) eqvivalenturi formulis Tanaxmad, erTi SemobrunebisTvis saWiro dro Semobrunebis ganmavlobaSi ganvlili 2π radiani manZilis fardobaa am kuTxis Semoweris kuTxur siCqaresTan. aCqareba 10.18 formulis droiT diferencireba (rodesac r mudmivia) mogvcems: dv dω = r . dt dt (10.21) (b) sur. 10.9. 10.2 suraTze gamosaxuli, mbrunavi, myari sxeulis ganivi kveTis zedxedi. sxeulis yvela wertili (magaliTad, PP) brunvis RerZis garSemo wriulad moZraobs. a) yoveli wertilis wrfivi siCqare wertilis wriuli traeqtoriis  mxebia. b) wertilis a aCqareba at mxebi komponentisa da ar radialuri komponentisgan Sedgeba. 10.21 formulaSi, dν/dt wrfivi aCqarebis mxolod im nawils asaxavs, romelic wrfivi siCqaris cvlilebazea pasuxismgebeli. -s msgavsad, wrfivi aCqarebis es nawilic wertilis traeqtoriis mxebia. am aCqarebas wertilis wrfivi aCqarebis at mxeb komponents vuwodebT da vwerT: at = αr (radianebSi), (10.22) sadac α=dω/dt. gafrTxileba: 10.22 formulaSi α kuTxuri aCqareba radianebSi unda gamoisaxos. garda amisa, 4.34 formulidan gamomdinare, wriuli traeqtoriiT moZrav wertils wrfivi aCqarebis radialuri komponenti aqvs (ar=ν2/r), romelic radialurad wrewiris SigniT, centriskenaa mimarTuli da wrfivi siCqaris mimarTulebis cvlilebazea pasuxismgebeli. 10.18 formulidan ν-s Casmis Semdeg es komponenti Semdegi saxiT SeiZleba Caiweros: ar = ν2 = ω2 r r (radianebSi). (10.23) 10.9b suraTze kargad Cans, rom mbrunavi myari sxeulis wertilis wrfivi aCqareba ori komponentisgan Sedgeba. radialurad centrisken mimarTuli ar komponenti arsebobs, Tu sxeulis kuTxuri siCqare nuli ar aris. at mxebi komponenti maSin arsebobs, rodesac kuTxuri aCqareba nuli ar aris. 334 brunva Tavi 10 sakontrolo wertili 3. xoWo mbrunav karuselze zis. Tu sistemis (karuseli+xoWo) kuTxuri siCqare Tanabaria, aqvs Tu ara xoWos a) radialuri da b) mxebi aCqareba? Tu ω mcirdeba, aqvs Tu ara xoWos g) radialuri da d) mxebi aCqareba? 10.4 amocanis amoxsnis nimuSi Cven induqciuri `rusuli mTebi~ unda davaproeqtoT, romelic magnituri Zalebis saSualebiT Cqardeba. Tavdapirvelad, mgzavri horizontalur trasaze g aCqarebiT moZraobs, xolo trasis pirveli monakveTi wriuli rkalis formisaa (sur. 10.10), raTa sur. 10.10. `rusuli mTebis~ horizontaluri trasis zedxedi. trasa jer wriuli rkalia, xolo P wertilidan moyolebuli rkalis mxebi wrfea. mgzavrma centriskenuli aCqarebac SeigrZnos. rkalis Semoweris dros centriskenuli aCqareba mkveTrad izrdeba. rodesac rkalis θP kuTxeze ganlagebul raime P wertilSi jamuri aCqarebis a sidide 4g gaxdeba, mgzavrma wrfivi traeqtoriiT unda imoZraos, romelic rkalis mxebia. a) ra θP kuTxe unda Semoweros rkalma, raTa P wertilSi a iyos 4g-is toli?  amoxsna: mgzavris a jamuri aCqareba nebismier  dros trasis gaswvrivi at mxebi aCqarebis da rkalis simrudis centrisken mimarTuli  ar radialuri aCqarebis veqtoruli  jamia (10.9b suraTis msgavsad). at urTierTperpendikularuli  da a r veqtorebia, amitom maTi jamia: (10.24) . mdebareobis ar = ω2r , sadac r aris wriuli rkalis radiusi. (10.25) gaSvebis adgilze mgzavris sawyisi kuTxuri ω0=0, xolo sawyisi kuTxuri mdebareobaa θ0=0. mgzavris aCqarebis dros ω-s zrdis tempi α kuTxuri aCqarebaa. 10.22 formulidan gamomdinare, α-sa da at-s Soris siCqarea Semdegi kavSiria: α= at . r (10.26) vinaidan at da r mudmivebia, α-c mudmivia. aqedan gamomdinare, 10.1 cxrilis formulebi SegviZlia gamoviyenoT. Cven θ kuTxuri gvWirdeba, amitom 10.14 formulas virCevT: ω2 = ω02 + 2α(θ – θ0) (10.27) CavsvaT ω0 = 0, θ0 = 0, α = at / r da miviRebT: 2at θ ω2 = (10.28) r miRebuli Sedegi 10.25 formulaSi CavsvaT da gveqneba: Semdeg ar = 2at θ . (10.29) ar-sTvis miRebuli θ . 10.24 Sedegi formulaSi gadavitanoT da θ-sTvis amov xsnaT: (10.30) rodesac a gaxdeba 4g, θ kuTxe CvenTvis sain tereso θP kuTxis tolia. 10.30 formulaSi a = 4g, θ = θP , at = g CavsvaT da miviRebT: ar-is mniSvneloba nebismier droSi ω kuTxur siCqarezea damokidebuli da 10.23 formulis Tanaxmad: gamoTvla = 1,94 rad = 111°. (pasuxi) b) ra iqneba mgzavris a jamuri aCqareba P wertilSi da Semdeg? amoxsna: P wertilSi a = 4g. amis Semdeg mgzavri wrfivad moZraobs da centriskenuli aCqareba aRar aqvs. maSasadame: P wertilSi a = 4g da amis Semdeg a = g (pasuxi) `rusuli mTebis Tavis tkivili~ maSin Cndeba, rodesac mgzavris Tavi aCqarebis mkveTr cvlilebas ganicdis. aCqarebis cvlileba Tavis qalaSi tvinis moZraobas iwvevs da tvinisa da Tavis qalis damakavSirebeli venebi wydeba. aCqarebis g-dan 4g-mde zrda savaraudod swored Tavis tkivils gamoiwvevs. 10.6. brunvis kinetikuri energia 335 amocanis amoxsnis taqtika taqtika 1: kuTxuri cvladebis erTeu romlebic mxolod kuTxur cvladebs Sei cavs (magaliTad 10.1 cxrilSi moyvanili lebi rodesac kuTxuri da wrfivi formulebi). am SemTxvevaSi, kuTxuri cvla cvladebis Semcvel formulebs viyenebT, yvela kuTxur debis cvlads damokidebuli. radianebSi vzomavT. maSasadame, formulebSi, kuTxur gadaadgilebas radianebSi gamovsa xavT, kuTxur siCqares - erTeulebis radiani/wm-ebSi, sadac gansazRvra Tqvenzea mxolod radianebi gamoiyeneba, “radiani”-s sityvierad Cawera xolo kuTxur aCqarebas - radiani/wm -ebSi da 2 aucilebeli arc aris. radiani/wT2-ebSi. gamonaklisia is formulebi, 10.6. brunvis kinetikuri energia saWreli Carxis xerxis swrafad brunvis dros mas namdvilad aqvs kinetikuri energia. rogor 1 2 gamovsaxoT es energia? K= mν2 formulas ver gamoviyenebT, vinaidan is mxolod xerxis masis centris kinetikur energias anu nuls mogvcems. amis nacvlad xerxi (an nebismieri swrafad mbrunavi myari sxeuli) sxvadasxva siCqareebis mqone nivTieri wertilebis nakrebad unda warmovidginoT. Semdeg yvela nivTieri wertilis kinetikur energiebs SevkrebT da mTeli sxeulis kinetikur energias gamovTvliT. maSasadame, mbrunavi sxeulis kinetikuri energiaa: K = 12 m1v12 + 12 m2v22 + 12 m3v32 + . . . = (10.31) , sadac mi aris i rigis nivTieri wertilis masa, xolo νi misi siCqarea. dajameba sxeulis yvela nivTieri wertilisTvis xdeba. 10.31 formulis problema is aris, rom νi ar aris yvela nivTieri wertilis siCqare. am problemis gadaWrisTvis ν 10.18 formulidan (ν=ωr) aviRoT da gveqneba: K= , (10.32) sadac ω yvela nivTieri wertilisTvis erTi da igivea. 10.32 formulis marjvena mxares frCxilebSi moqceuli sidide mbrunavi sxeulis masis brunvis RerZis garSemo ganawilebas gviCvenebs. am sidides brunvis RerZisadmi (a) sxeulis I brunviT inercias anu inerciis moments vuwodebT. es sidide raime myari sxeulisTvis da brunvis RerZisTvis ucvlelia. amjerad SegviZlia davweroT: I= (brunviTi inercia) (10.33) es formula 10.32 formulaSi CavsvaT da miviRebT: K = 12 Iω2 (radianebSi) (10.34) miviReT swored is gamosaxuleba, rac gvainteresebda. 10.34 formulis gamoyvanis dros ν = ωr formula gamoviyeneT da amitom ω radianebSi unda gamoisaxos. SI sistemaSi I-s erTeulia kg·m2. 10.34 formula mbrunavi myari sxeulis kinetikur (b) sur. 10.11. grZeli Zelakis brunva a) centraluri (gaswvrivi) RerZis garSemo ufro martivia, vidre b) mis centrze gamavali perpendikularuli RerZis garSemo. mizezi is aris, rom a) SemTxvevaSi masa brunvis RerZTan ufro axlos aris gavrcelebuli. 336 Tavi 10 brunva energias gvaZlevs da myari sxeulis gadataniTi moZraobis dros kinetikuri energiis momcemi K=12Mν2com formulis kuTxuri eqvivalentia. orive formulaSi 12 koeficienti gvxvdeba. erT formulaSi M masa Sedis, xolo meoreSi - I (rac masas da mis ganawilebas aerTianebs). garda amisa, orive formulaSi siCqaris kvadrati Sedis. gadataniTi moZraobis da brunvis kinetikuri energiebi sxvadasxva saxis energia ar aris – orive maTgani kinetikuri energiaa da ubralod moZraobis saxeobis mixedviT gamoisaxeba. mbrunavi sxeulis brunviTi inercia ara mxolod sxeulis masas, aramed masis ganawilebasac moicavs. magaliTisTvis grZeli, mZime Zelaki ganvixiloT. Tavdapirvelad es Zelaki misi centraluri (gaswvrivi) RerZis garSemo vabrunoT (sur. 10.11a), xolo Semdeg mis centrze gamavali perpendikularuli RerZis garSemo (sur. 10.11b). orive SemTxvevaSi masa Tanabaria, magram pirvel SemTxvevaSi brunva ufro iolia. amis mizezi is aris, rom masa pirvel SemTxvevaSi brunvis RerZTan ufro axlos aris ganawilebuli. Sedegad, 10.11a suraTze Zelakis brunviTi inercia bevrad naklebia, vidre 10.11b suraTze. naklebi brunviTi inercia ufro martivad brunvas niSnavs. sakontrolo wertili 4. suraTze naCvenebi sami sfero vertikaluri RerZis garSemo brunavs. RerZsa da TiToeuli sferos centrs Soris perpendikularuli manZili naCvenebia. daalageT sferoebi RerZis garSemo brunviTi inerciebis mixedviT dididan mcireze. 10.7. brunviTi inerciis gamoTvla Tu myari sxeuli ramdenime nivTieri wertilisgan Sedgeba, brunvis RerZis garSemo misi brunviTi inercia 10.33 formuliT (I= ∑m r 2 ) gamoiTvleba anu Cven SegviZlia TiToeuli nivTieri wertilis mr namravli vipovoT da Semdeg es namravlebi SevkriboT (gaixseneT, rom r 2 i i aris mocemuli brunvis RerZidan nivTieri wertilis daSoreba). Tu myari sxeuli mravali mimdebare nivTieri wertilisgan Sedgeba, maSin 10.33 formulis gamoyeneba kompiuters saWiroebs. amitom 10.33 formulaSi mocemul jams integraliT vcvliT da sxeulis brunviT inercias ase ganvsazRvravT: I= r2 dm (brunviTi inercia, mTliani sxeuli) (10.35) 10.2 cxrilSi integrebis Sedegebia mocemuli cxra sxeulisTvis. paraleluri RerZebis Teorema modiT, gamovTvaloT M masis sxeulis I brunviTi inercia mocemuli RerZis garSemo. I-s gamoTvla 10.35 formulaSi mocemuli integralis saSualebiT SeiZleba, magram mdgomareoba ufro gamartivdeba, Tu sxeulis masis centrze gamavali paraleluri RerZis garSemo sxeulis Icom brunviTi inercia viciT. davuSvaT, h aris perpendikularuli manZili mocemul RerZsa da masis centrze gamaval RerZs Soris (gaxsovdeT, rom es ori RerZi urTierTparaleluri unda iyos). mocemuli RerZis garSemo I brunviTi inerciaa: I = Icom + Mh2 (paraleluri RerZebis Teorema) (10.36) 10.7. brunviTi inerciis gamoTvla cxrili 10.2 brunviTi inerciebi fersos brunva centraluri RerZis garSemo (a) (b) I = MR2 (i) I = 12MR2 I = 25MR2 firfitis brunva mis centrze gamavali perpendikularuli RerZis garSemo fersos brunva nebismieri diametris garSemo (T) I = 23MR2 I = 12MR2 myari sferos brunva nebismieri diametris garSemo (v) 1 I = 12 ML2 sferuli garsis brunva nebismieri diametris garSemo (z) (g) I = 12M(R12 + R22) Zelakis brunva mis centrze gamavali perpen dikularuli RerZis garSemo (e) 1 I = 14MR2 + 12 ML2 myari cilin dris (diskos) brunva cen traluri Rer Zis garSemo rgoluri ci lindris brunva centraluri RerZis garSemo myari cilin dris (diskos) brunva cen traluri diametris garSemo (d) 337 1 I = 12 M(a2 + b2) 10.36 formulas paraleluri RerZebis Teorema ewodeba. momdevno paragrafSi am formulas davamtkicebT da 10.5 amocanis amoxsnis nimuSSi gamoviyenebT. paraleluri RerZebis Teoremis damtkiceba davuSvaT, O aris 10.12 suraTze gamosaxuli pirobiTi formis sxeulis masis centri. koordinatTa sistemis aTvlis saTave O wertils davamTxvioT. ganvixiloT suraTis sibrtyis perpendikularuli, O wertilze gamavali RerZi da pirveli RerZis paraleluri, P wertilze gamavali meore RerZi. P wertilis x da y koordinatebi a da b iyos. dm aris zogadad x da y koordinatebis mqone masis elementi. P wertilze gamavali RerZis garSemo sxeuli brunviTi inercia 10.35 formulidan gamomdinare aris: I= r2 dm = [(x – a)2 + (y – b)2] dm , gardaqmnebis Sedegad miviRebT: I = (x2+y2)dm – 2a xdm – 2b ydm + (a2+b2)dm. masis centris gansazRvrebidan (10.37) gamomdinare (9.9 formula), 10.37 formulis Sua ori integrali masis sur. 10.12. O masis centris mqone myari sxeulis ganivi kveTi. paraleluri RerZebis Teorema (10.36 formula) O wertilze gamavali RerZis garSemo sxeulis brunviT inercias sxeulis masis centridan h manZiliT daSorebul P wertilSi gamavali RerZis garSemo sxeulis brunviT inerciasTan akavSirebs. orive RerZi suraTis sibrtyis perpendikularulia. 338 brunva Tavi 10 centris koordinatebs gvaZlevs da maSasadame orive nulis tolia. vinaidan x2+y2=R2, sadac R aris manZili O-dan dm-mde, pirveli integrali Icom-ia, rac masis centrze gamavali RerZis garSemo sxeulis brunviTi inerciaa. 10.12 suraTze dakvirveba cxadyofs, rom 10.37 formulis ukanaskneli wevria Mh2, sadac M sxeulis jamuri masaa. yovelive zemoTqmulidan gamomdinare, 10.37 formula 10.36 formulamde daiyvaneba da Teorema damtkicebulia. sakontrolo wertili 5. suraTze naCvenebia wignis magvari sagani (erTi gverdi meoreze grZelia) da sagnis zedapiris perpendikularuli brunvis RerZis oTxi arCevani. daalageT RerZebi maT garSemo sagnis brunviTi inerciis mixedviT dididan mcireze. amocanis amoxsnis nimuSi 10.5 10.13a suraTze naCvenebia m masis ori nivTieri wertilisgan Semdgari sxeuli. wertilebi erTmaneTTan L sigrZis da umniSvnelo masis ZelakiT arian dakavSirebuli. a) risi tolia sxeulis Icom brunviTi iner (a) cia masis centrze gamavali, Zelakis per pendikularuli RerZis garSemo? amoxsna: vinaidan mxolod ori nivTieri wertili gvaqvs, sxeulis Icom brunviTi iner ciis gamoTvla integrebis nacvlad 10.33 (b) formuliT SegviZlia. nivTier wertilebs da brunvis RerZs Soris manZilia 1 L da gveqneba: 2 2 1 2 I= mi ri = (m)( L) + (m)( 1 L)2 2 2 = 1 mL2 . (pasuxi) 2 ∑ b) risi tolia sxeulis I brunviTi inercia sur. 10.13. myari sxeuli umniSvnelo masis ZelakiT gaerTianebuli, m masis ori nivTieri wertilisgan Sedgeba. manZili L-is tolia. 10.33 formula gvaZlevs: I = m(0)2 + mL2 = mL2 Zelakis marcxena boloze gamavali da pirveli garda amisa, vinaidan masis centrze gamavali RerZis paraleluri RerZis garSemo? amoxsna: brunviTi inercia kvlav amocanis a) nawilis msgavsad SegviZlia gamovTvaloT, magram amjerad wertilisTvis ri marcxniv manZilia ganlagebuli nulia, (pasuxi) xolo marjvniv ganlagebuli wertilisTvis igive amocanis amoxsnis nimuSi RerZis garSemo Icom viciT da SekiTxvaSi naxsenebi RerZi am RerZis paraleluria, Cven SegviZlia paraleluri RerZebis Teorema (10.36 formula) gamoviyenoT: I = Icom + Mh2 = 1 mL2 + (2m)( 1 L)2=mL2 (pasuxi) 2 2 10.6 10.14 suraTze naCvenebia viwro, erTgvarovani, M masis da L sigrZis Rero, romelic x RerZze Zevs da am RerZis aTvlis saTave Zelakis centrs emTxveva. a) risi tolia Zelakis brunviTi inercia centrze gamavali perpendikularuli brunvis RerZis garSemo? sur. 10.14. L sigrZis da M masis erTgvarovani Zelaki. naCvenebia dm masis da dx sigrZis elementi. 10.7. brunviTi inerciis gamoTvla amoxsna: vinaidan Zelaki erTgvarovania, misi masis centri Reros centrzea. Zelaki mTliani sagania, amitom brunviTi inerciis 10.35 gamoTvlisTvis formula unda gamoviyenoT: I= r2 dm, (10.38) = 1 ML2. 12 339 (pasuxi) miRebuli pasuxi 10.2e cxrilSi mocemul SedegTan SesabamisobaSia. b) risi tolia Zelakis I brunviTi inercia Zelakis perpendikularuli Cven x koordinatiT integreba gvsurs (da boloze gamavali ara miTiTebuli dm masiT), amitom Zelakis garSemo? elementis dm masa Zelakis dx sigrZes unda amoxsna: amjerad x RerZis aTvlis saTave axali da marcxena brunvis RerZis 10.14 Zelakis marcxena bolos unda davamTxvioT suraTzea naCvenebi). Zelaki erTgvarovania, da Semdeg 0-dan L-mde vaintegroT. Sedegad amitom masis Sefardeba sigrZesTan yvela miviRebT: elementisTvis da ZelakisTvis Tanabaria. davukavSiroT (amgvari elementi aqedan gamomdinare: elementis dm masa elementis dx sigrZe anu dm = amjerad = Zelakis M masa Zelakis L sigrZe M dx. L 1 3 ML2 (pasuxi) ufro martivi iqneba, Tu Zelakis bolos RerZs ise gavavlebT, rom is masis centrze gamavali RerZis paraleluri iyos. am Sem TxvevaSi 10.38 formulaSi dm-s nacvlad I= paraleluri gamoviyenebT (10.36 RerZebis Teoremas formula). amocanis 1 a) nawilidan viciT, rom Icom= 12 ML2. 10.14 miRebul Sedegs SevitanT, xolo r-s x-iT suraTze, brunvis axal RerZsa da masis centrs SevcvliT. Semdeg Zelakis mTeli sigrZisTvis Soris h perpendikularuli manZilia 2 L. 10.36 (-L/2-dan L/2-mde) vaintegrebT da miviRebT: 1 formula gvaZlevs: 1 1 1 I = Icom+Mh2= 12 ML2+(M)( 2 L)2= 3 ML2 (pasuxi) I= es Sedegi Zelakis orive boloze gamaval perpendikularul RerZs Seesabameba, miuxe = davad imisa, es RerZi 10.14 suraTze naCvenebi RerZis paraleluria Tu ara. amocanis amoxsnis nimuSi 10.7 meqanizmebis did nawilebs, romlebic xan grZlivad da maRal siCqareebze brunavs, sacdeli brunvis sistemiT amowmeben. es sistema hermetuli sadebiTa da tyviis agu rebiT damzadebuli cilindria, romelic sarqveliTaa daxuruli. nawils am cilindrSi aTavseben da maRali siCqariT abruneben. Tu brunvis Sedegad nawili ziandeba, tyviis msubuqi agurebi namsxvrevebs iWers da maTi Semdgomi analizi xdeba. 1985 wels, kompania Test Devices, Inc. M = 272 kg masis da R = 38 sm radiusis foladis myar rotors brunvaze amowmebda. rodesac sur. 10.15. swrafad mbrunavi foladis diskos afeTqebis Sedegi diskos ω kuTxuri siCqare 14000 br/wT gaxda, sacdeli sistemis oTaxSi mimaval koridorSi inJinrebma erTi sarTulidan qvemoT (sadac iyo gabneuli, oTaxis kari manqanebis mimdebare sacdeli sistema iyo ganlagebuli) dartymis sadgomSi egdo, tyviis erTi aguri mezoblis xma gaiges. aRmoCnda, rom tyviis agurebi samzareulos kedels moxvda, Senobis sayrdeni 340 Tavi 10 brunva I-s gamosaxuleba 10.2g cxrilSia mocemuli 1 (I= 2 MR2): Zelebi dazianda, sacdeli oTaxis betonis iatakma 0,5sm-iT dabla Caiwia, xolo 900kg 1 masis sarqveli Wers moxvda da Semdeg sacdel aRWurvilobas daeca (sur. 10.15). namsxvrevebma rotoris kuTxuri siCqarea: inJinrebis oTaxSi SemTxveviT ar SeaRwies. ra energia gamoiyo rotoris 1 I = 2 MR2 = 2 (272kg)(0,38m)2 = 19,64kg·m2. afeTqebis (160 ) wT wm ω = (14000 br/wT)(2π rad/br) Sedegad? =1,466 × 103 radiani/wm amoxsna: gamoyofili energia rotoris K amis Semdeg 10.34 formula mogvcems: brunviTi kinetikuri energiis tolia, rodesac 1 misi kuTxuri siCqare 14000br/wT gaxda. K-s 1 K = 2 Iω2 = 2 (19,64 kg·m2)(1,466 × 103rad/wm)2 gamoTvla 10.34 formulis (K=1Iω2) saSualebiT = 2,1 × 107j (pasuxi) 2 SeiZleba, magram Tavdapirvelad I brunviTi inerciis gamosaxuleba unda miviRoT. rotori pasuxidan gamomdinare, rotoris afeTqeba diskoa da karuselis msgavsad brunavs, amitom bombis afeTqebis msgavsi iyo. 10.8. Zalis momenti karis saxeluri anjamidan maqsimalurad φ Sorsaa ganlagebuli da es Wkvianuria. mZime karis gaRebisas Zalis modeba saWiroa, magram es sakmarisi ar aris. Zalis modebis adgils da biZgis mimarTulebas Zalian didi mniSvneloba aqvs. Tu Zalas anjamasTan ufro axlos mosdebT, vidre karis saxelurTan an Zala karis sibrtyisadmi 90°-isgan (a) gansxvavebul kuTxes adgens, maSin karis gaRebisTvis ufro meti Zalaa saWiro, vidre im SemTxvevaSi, rodesac Zala saxelurzea modebuli da karis sibrtyis perpendikularulia. 10.16a suraTze naCvenebia O wertilze gamavali da ganivi φ kveTis perpendikularuli RerZis garSemo Tavisuflad  mbrunavi sxeuli. P wertilis mdebareoba O-sadmi r mdebareobis veqtoris saSualebiT ganisazRvreba da am    wertilze F Zalaa modebuli. F da r veqtorebs Soris kuTxea φ (simartivisTvis mxolod im Zalebs ganvixilavT, romelTac brunvis RerZis paraleluri komponenti ar  gaaCniaT anu F Zala saxelmZRvanelos gverdis sibrtyeSia (b) modebuli).  F Zala or komponentad davSaloT (sur. 10.16b), raTa gavarkvioT, Tu ra gavlenas axdens es Zala brunvis RerZis  garSemo sxeulis brunvaze. Fr radialuri komponenti r φ veqtoris gaswvrivaa mimarTuli da brunvas ar iwvevs, φ  sur. 10.16. a) F Zala myari sxeulis P wertilzea modebuli da sxeuli O wertilze gamavali RerZis garSemo Tavisuflad brunavs. brunvis RerZi suraTze naCvenebi ganivi kveTis sibrtyis perpendikularulia. b) Zalis Sedegad miRebuli Zalis momentia (r)(Fsin φ).  Zalis momenti rFt saxiTac SeiZleba gamoisaxos, sadac Ft aris F Zalis mxebi komponenti.  g) Zalis momenti r┴F saxiTac SeiZleba Caiweros, sadac r┴ aris F Zalis momentis mxari.  F -is moqmedebis wrfe  F -is mxari (g) 10.8. Zalis momenti 341 vinaidan is O-ze gamavali wrfis gaswvriv moqmedebs (Tu karebs misi sibrtyisadmi paralelurad  ubiZgebT, kari ar gaiReba). Ft mxebi komponenti r veqtoris perpendikularulia da misi sididea Ft = F sin φ. mxebi komponenti brunvas iwvevs (Tu karebs misi sibrtyisadmi perpendikularulad ubiZgebT, kari SesaZloa gaiRos).  F Zalis mier sxeulis Semobrunebis SesaZlebloba ara mxolod Ft mxebi komponentis sididezea damokidebuli, aramed O wertilidan Zalis modebis manZilze. am ori faqtoris mxedvelobaSi misaRebad τ Zalis momenti Semogvaqvs, rac xsenebuli faqtorebis namravlia: τ = (r)(F sin φ) (10.39) Zalis momentis gamosaTvleli eqvivalenturi formulebia: τ = (r)(F sin φ) = rFt (10.40) da τ = (r sin φ)(F) = r┴F (10.41)  sadac r┴ aris brunvis RerZis O wertilsa da F veqtoris ganlagebis wrfes Soris  perpendikularuli manZili (sur. 10.16g). am wrfes F -is moqmedebis wrfe ewodeba, xolo r┴ aris  F -is momentis mxari. 10.16b suraTze kargad Cans, rom r SegviZlia aRvweroT, rogorc Zalis Ft komponentis momentis mxari.  sityva `Zalis momenti~ laTinuri sityvidan Torque (`mabruni momenti~) modis da F Zalis mabrunebel Tvisebas gamoxatavs. rodesac saganze ZaliT moqmedebT da misi mobruneba gsurT, Tqven Zalis moments iyenebT. SI sistemaSi Zalis momentis erTeulia n·m. gafrTxileba: n·m muSaobis erTeulicaa, Tumca muSaoba da Zalis momenti srulad gansxvavebuli sidideebia da isini erTmaneTSi ar unda agverios. muSaoba xSirad joulebSi gamoisaxeba, xolo Zalis momenti _ arasdros. momdevno paragrafSi Zalis moments ganvixilavT, rogorc veqtorul sidides. amjerad Cven mxolod erTi RerZis garSemo brunvas vexebiT da veqtori saWiro ar aris. Zalis moments dadebiTi an uaryofiTi mniSvneloba aqvs da es mniSvneloba sxeulis brunvis mimarTulebazea damokidebuli. Tu sxeuli saaTis isris moZraobis sawinaaRmdego mimarTulebiT brunavs, Zalis momenti dadebiTia, xolo Tu sxeuli saaTis isris moZraobis mimarTulebiT brunavs, maSin Zalis momenti uaryofiTia. Zalis momenti me-5 paragrafSi ganxilul superpoziciis princips emorCileba. rodesac sxeulze ramdenime Zalis momenti moqmedebs, jamuri Zalis momenti calkeuli Zalis momentebis jamia. jamuri Zalis momentis simboloa τt . sakontrolo wertili 6. suraTze naCvenebia metrebSi dagra duirebuli saxazavi, romelic 20 sm niSnulze mdebare wertilis garSemo brunavs. saxazavze moqmedi xuTive Zala horizontaluria da maTi sidideebi Tanabaria. daalageT Zalebi maT mier warmoqmnili Zalis momentis mixedviT dididan mcireze. 342 Tavi 10 brunva sur. 10.17. martivi myari sxeuli r sigrZis da umniSvnelo masis Zelakis boloze mibmuli m masis nivTieri wertilisgan Sedgeba. sxeuli O  wertilze gamavali RerZis garSemo Tavisuflad brunavs. brunvas F modebuli Zala iwvevs. 10.9. niutonis meore kanoni brunvisTvis Zalis momenti myari sxeulis, magaliTad, karebis brunvas iwvevs. amjerad Cven gvsurs myar sxeulze moqmed τt jamur Zalis momentsa da brunvis RerZis garSemo am Zalis momentis mier gamowveul α kuTxur aCqarebas Soris kavSiri davadginoT. amisTvis niutonis meore kanons (Ft = ma) viyenebT. Ft-s τt-iT vcvliT, m-s I-iT, xolo a-s α-iT da vwerT: τt = Iα (niutonis meore kanoni brunvisTvis) , (10.42) sadac α radianebSi unda iyos gazomili. 10.42 formulis damtkiceba 10.42 formulis dasamtkiceblad 10.17 suraTze gamosaxuli mdgomareoba ganvixiloT. myari sxeuli r sigrZis da umniSvnelo masis Zelakis boloze mibmuli m masis nivTieri wertilisgan Sedgeba. Zelaki saxelmZRvanelos gverdis sibrtyisadmi perpendikularuli brunvis RerZis garSemo brunavs. maSasadame, nivTieri wertili mxolod wriuli traeqtoriiT moZraobs da brunvis wrewiris centrze gadis.  nivTier wertilze F Zala moqmedebs. vinaidan nivTieri wertili wriuli traeqtoriiT moZraobs, Zalis mxolod Ft mxeb komponents SeuZlia traeqtoriis gaswvriv nivTieri wertilis aCqareba. Ft-sa da traeqtoriis gaswvriv nivTieri wertilis at mxeb aCqarebas niutonis meore kanoni akavSirebs: Ft = mat nivTier wertilze moqmedi Zalis momenti 10.40 formulidan gamomdinare aris: τ = Ft r = mat r 10.22 formulis (at = αr) daxmarebiT, es formula Semdegi saxiT SeiZleba Caiweros: τ = m(αr)r = (mr2)α (10.43) formulis marjvena mxares frCxilebSi moqceuli sidide aris nivTieri wertilis brunvis RerZis garSemo brunviTi inercia anu inerciis momenti I (ix. 10.33 formula). maSasadame, 10.43 formula Semdeg saxes iRebs: τ = Iα (radianebSi) rodesac nivTier wertilze erTze meti Zala moqmedebs, 10.44 formulis zogadi saxea: (10.44) (10.45) τt = Iα (radianebSi) formula damtkicebulia da is fiqsirebuli RerZis garSemo mrbunav nebismier myar sxeuls SegviZlia mivusadagoT, vinaidan yvela aseTi sxeuli nivTier wertilTa erTobliobaa. 10.9. niutonis meore kanoni brunvisTvis 343 sakontrolo wertili 7. suraTze naCvenebia metrebSi dagraduirebuli saxazavi, romelic Sua wertilidan marcxniv    horizontaluri Zalebi moqmedebs. naCvenebia mxolod F1  Zala. F2 Zala saxazavis perpendikularulia da mis marjvena ganlagebuli wertilis garSemo brunavs. saxazavze F1 da F2 boloze moqmedebs. Tu saxazavi ar brunavs, a) ra mimarTuleba  aqvs F2 Zalas da b) F2 Zala F1-ze metia, naklebia Tu misi tolia? amocanis amoxsnis nimuSi 10.8 10.18a suraTze naCvenebia uZrav horizonta lur RerZze damagrebuli R=20sm radiusis da M=2,5kg masis erTgvarovani disko. m=1,2kg masis bloki umniSvnelo masis Tokzea (g) Camokidebuli, xolo Toki diskos fersozea gadaxveuli. gamoTvaleT blokis vardnis aCqareba, diskos kuTxuri aCqareba da Tokis daWimuloba. Toki ar wydeba da RerZze xaxuni ar arsebobs. amoxsna: bloki sistemad miviCnioT da misi a aCqareba masze moqmed Zalebs niutonis meore kanonis saSualebiT ( t (a) (b) = m ) davukavSiroT. es Zalebi 10.18b suraTze naCveneb Tavisufali sxeulis grafikze aris gamosaxuli. Tokidan   T Zala moqmedebs, xolo Fg mizidulobis Zalis sididea mg. vertikaluri y RerZis sur. 10.18. a) blokis vardna diskos brunvas iwvevs. b) Tavisufali sxeulis grafiki blokisTvis. g) Tavisufali sxeulis arasruli grafiki disko sTvis. gaswvrivi komponentebisTvis niutonis meore kanoni (Ft,y = may) aseTi saxiT Caiwereba: T – mg = ma da diskos fersos mxebia. aqedan gamomdinare, (10.46) am formulas a-sTvis ver amovxsniT, vinaidan is T ucnobsac Seicavs. modiT disko sistemad miviCnioT da misi α kuTxuri aCqareba masze moqmed Zalis moments brunvisTvis niutonis meore kanonis misi Zalis momentia –RT. disko saaTis isris moZraobis mimarTulebiT brunavs da amitom Zalis momenti uaryofiTia. 10.2g cxrilidan gamomdinare, diskos I 1 MR2. 2 brunviTi inerciaa maSasadame, τt = Iα aseTi saxiT SeiZleba Caiweros: 1 2 (τt = Iα) saSualebiT davukavSiroT. jamuri TvlisTvis CavTvaloT, rom brunvis RerZi 10.47 formula erTi SexedviT gamousadegaria, vinaidan α da T ucnobebs Seicavs da arc erTi maTgani ar aris CvenTvis saintereso a. Zalis momentis da I inerciis momentis gamo diskos perpendikularulia da mis O centrze gadis (sur. 10.18g). amis Semdeg Zalis momenti 10.40 formulis (τ = rFt ) saSualebiT gamoiTvleba. mizidu –RT = MR2α (10.47) miuxedavad amisa, gavixsenoT, rom Toki ar wydeba da blokis a wrfivi aCqareba da diskos fersos at wrfivi (mxebi) aCqareba erTmaneTis centrze anu r = 0 manZilze moqmedebs da tolia. 10.22 formulidan (at = αr) Cans, rom α = a/R. es Sedegi 10.47 formulaSi CavsvaT da maTi Zalis momenti nulia. Tokidan diskoze miviRebT: lobis Zala da RerZidan moqmedi Zala diskos  modebuli T Zala r = R manZilze moqmedebs 1 2 T = – Ma (10.48) 344 brunva Tavi 10 Semdeg 10.46 da 10.48 formulebi gavaerTia daWimuloba (6 n) Camokidul blokze moqmed noT: mizidulobis a = –g 2m (2)(1,2 kg) = –(9,8 m/wm2) M + 2m 2,5 kg + (2)(1,2 kg) = 4,8 m/wm2 (pasuxi) T-s gamoTvlisTvis 10.48 formula gamovi yenoT: 1 1 T = – Ma = – (2,5kg)(–4,8 m/wm2) 2 2 = 6 n rogorc mosalodneli Zalaze (mg = 11,8 n) mcirea. blokis aCqareba da Tokis daWimuloba diskos masazea damokidebuli da ara mis radiusze. zemoT miRebuli formulebi masis armqone diskosTvis (M = 0) a = –g da T = 0 pasuxebs gvaZlevs. es logikuricaa, vinaidan bloki Tavisufali sxeulis msgavsad vardeba. (pasuxi) iyo, 10.22 formulidan diskos kuTxuri aCqareba: –4,8 m/wm2 α= a = = –24 rad/wm2. (pasuxi) R 0,2 m vardnili blokis aCqareba g-ze naklebia, xolo Tokis amocanis amoxsnis nimuSi 10.9  masis davuSvaT, rom Fg Zala Tqveni mowinaaRmdegis mowinaaRmdegeze F ZaliT moqmedebT da Tqvens masis centrze moqmedebs. masis centris sa ZiudoSi Sejibris dros  80kg marjvena TeZoze mdebare sayrdeni wertilidan (brunvis RerZidan) momentis mxaria d1 = 0,3 m. mowinaaRmdege sayrdeni wertilis garSemo α = –6 radiani/wm 2 kuTxuri aCqarebiT unda daatrialoT da es kuTxuri aCqareba saaTis  yrden wertilze mdebareobis dros, Fg Zalis  momentis mxaria r┴ = 0 da Fg Zalis momenti nulis tolia. maSasadame, Tqvens oponentze  moqmed Zalis moments mxolod F Zala iwvevs da τt = Iα formula ase SeiZleba Caiweros: –d1F = Iα isris moZraobis mimarTulebisaa. davuSvaT, sayrdeni wertilisadmi I brunviTi inerciaa 15 kg·m . 2  a) ra unda iyos F Zala, raTa datrialebamde mowinaaRmdegis masis centri Tqvens TeZoze aRmoCndes (sur. 10.19a)?  amoxsna: mowinaaRmdegeze modebuli F Zala da misi α kuTxuri aCqareba brunvisTvis Semdeg gamovTvliT: –Iα –(15 kg·m2)(–6 rad/wm2) F= = d 0,3 m = 300 n. (pasuxi)  b) rogori unda iyos F Zala, Tu mowinaaRmdege  ileTis Catarebamde sworad dgas da Fg Zalis momentis mxari sayrdeni wertilidan aris xebi iataks ascdeba, maTze mxolod sami d2 = 0,12 m (sur. 10.19b)?  amoxsna: vinaidan Fg Zalis momentis mxari nulis toli aRar aris, am Zalis momentia d2mg Zala moqmedebs: F Zala, sayrden wertilze da dadebiTia, vinaidan brunva saaTis isris niutonis meore kanoniT (τt = Iα) SegviZlia davakavSiroT. rodesac mowinaaRmdegis fe   Tqvenidan moqmedi N Zala (10.19 suraTze es  Zala gamosaxuli ar aris) da Fg mizidulobis Zala. τt = Iα formulis gamoyenebisTvis sa yrdeni wertilis garSemo Sesabamisi sami Zalis momenti gvWirdeba.  10.41 formulidan (τ = r┴F) gamomdinare, F Zalis Sedegad miRebuli Zalis momentia –d1F, sadac d1 aris r┴ mxari da minus niSani saaTis isris moZraobis mimarTulebiT brunvas  asaxavs. N ZaliT gamowveuli Zalis momenti nulis tolia, vinaidan es Zala sayrden wertilze moqmedebs da misi momentis mxari r┴ = 0.  Fg Zalis momentis gamosaTvlelad. SegviZlia (a) (b) sur. 10.19. Ziudos a) sworad da b) arasworad Sesrulebuli ileTi 10.10 muSaoba da brunvis kinetikuri energia moZraobis sawinaaRmdego mimarTulebiT tolia. am da sxva monacemebis Casmis Sedegad xdeba. amjerad τt = Iα formula aseT saxes mii miviRebT: Rebs: F = 300 n + –d1F + d2mg = Iα rac gvaZlevs: F=– Iα d2mg + d1 d1 mxaris pirveli (0,12 m2)(80 kg)(9,8 m/wm2) 0,3 m = 613,6 n ≈ 610 n (pasuxi) pasuxi cxadyofs, rom Tu mowinaaRmdegis masis centrs Tqvens TeZos ar daamTxvevT, ileTis amocanis a) nawilidan viciT, rom formulis marjvena 345 wevri 300 n-is CatarebisTvis bevrad meti Zala iqneba saWiro. karg Ziudoistebs es yovelTvis axsovT. 10.10 muSaoba da brunvis kinetikuri energia me-7 TavSi ganvixileT, rom rodesac F Zala m masis myar sxeuls koordinatTa RerZis gaswvriv 1 aCqarebs, Zala sxeulze W muSaobas asrulebs. maSasadame, sxeulis kinetikuri energia (K= mν2) 2 SesaZloa Seicvalos. davuSvaT, mxolod sxeulis kinetikuri energia icvleba. am SemTxvevaSi, kinetikuri energiis ΔK cvlileba W muSaobas muSaoba-kinetikuri energiis Teoremis (7.10 formula) saSualebiT ukavSirdeba: 1 1 ΔK = Kf – Ki= mνf2 – mνi2=W (muSaoba-kinetikuri energiis Teorema). 2 2 (10.49) x RerZis gaswvriv moZraobis SemTxvevaSi, muSaoba 7.32 formuliT gamoiTvleba: W= (muSaoba, moZraoba erT ganzomilebaSi) (10.50) rodesac F mudmivia da sxeulis gadaadgilebaa d, 10.50 formula W=Fd saxiT Caiwereba. muSaobis Sesrulebis tempi simZlavrea da is 7.43 da 7.48 formulebiT gamoiTvleba: dW P = dt = Fv (simZlavre, moZraoba erT ganzomilebaSi) (10.51) modiT, amjerad, brunviTi moZraoba ganvixiloT. rodesac fiqsirebuli RerZis garSemo mbrunav myar sxeuls Zalis momenti aCqarebs, Zalis momenti sxeulze W muSaobas asrulebs. aqedan 1 2 gamomdinare, sxeulis brunvis kinetikuri energia (K= Iω2) SesaZloa Seicvalos. davuSvaT, sxeulis mxolod kinetikuri energia icvleba. am SemTxvevaSi, kinetikuri energiis ΔK cvlileba W muSaobas muSaoba-kinetikuri energiis Teoremis saSualebiT SegviZlia davukavSiroT, oRond, amjerad, kinetikuri energia brunvis kinetikuri energiaa: 1 1 ΔK = Kf – Ki= Iωf2 – Iωi2 = W (muSaoba-kinetikuri energiis Teorema) 2 2 (10.52) am formulaSi I aris fiqsirebuli RerZis garSemo sxeulis brunviTi inercia, xolo ωi da ωf sxeulis kuTxuri siCqareebi muSaobis Sesrulebamde da Sesrulebis Semdeg. 10.50 formulis brunviTi eqvivalentis saSualebiT muSaoba SegviZlia gamovTvaloT: W= (muSaoba, brunva fiqsirebuli RerZis garSemo), (10.53) sadac τ aris Zalis momenti, xolo θi da θf - sxeulis kuTxuri mdebareobebi muSaobis Sesrulebamde da Sesrulebis Semdeg. rodesac τ mudmivia, 10.53 formula aseT saxes iRebs: W = τ(θ f – θ i) (muSaoba, mudmivi Zalis momenti) (10.54) muSaobis Sesrulebis tempi simZlavrea da is 10.51 formulis brunviTi eqvivalentiT gamoiTvleba: dW P = dt = τω (simZlavre, brunva fiqsirebuli RerZis garSemo) (10.55) 10.3 cxrilSi Tavmoyrilia fiqsirebuli RerZis garSemo mbrunavi myari sxeulis brunvis da gadataniTi moZraobis formulebi. 346 Tavi 10 brunva 10.3 cxrili kavSirebi gadataniT moZraobasa da brunvas Soris gadataniTi moZraoba (fiqsirebuli mimarTuleba) brunva (fiqsirebuli RerZi) mdebareoba x kuTxuri mdebareoba θ siCqare ν = dx/dt kuTxuri siCqare ω = dθ/dt aCqareba a = dν/dt kuTxuri aCqareba α = dω/dt masa m brunviTi inercia I niutonis meore kanoni Ft = ma niutonis meore kanoni τt = Iα muSaoba W= ∫ Fdx 1 W= muSaoba Fdx ∫ τdθ 1 kinetikuri energia K = 2 mν2 kinetikuri energia K = 2 Iω2 simZlavre (mudmivi Zala) P = Fν simZlavre (mudmivi Zalis momenti) P = τω muSaoba-kinetikuri energiis Teorema W = ΔK muSaoba-kinetikuri energiis Teorema W = ΔK 10.52-10.55 formulebis damtkiceba  modiT, kvlav 10.17 suraTze gamosaxuli mdgomareoba ganvixiloT, rodesac F Zala masis armqone Zelakze damagrebuli m masis nivTieri wertilisgan Semdgar myar sxeuls abrunebs. brunvis dros   F Zala sxeulze muSaobas asrulebs. davuSvaT, F Zala mxolod sxeulis kinetikur energias cvlis. am SemTxvevaSi, 10.49 formuliT gamosaxuli muSaoba-kinetikuri energiis Teorema Semdeg saxes miiRebs: ΔK = Kf – Ki = W 1 2 (10.56) K= mν2 da 10.18 formulis (ν=ωr) gamoyenebiT 10.56 formula ase Caiwereba: ΔK = 1 2 2 1 2 2 mr ωf – mr ωi = W 2 2 (10.57) 10.33 formulidan gamomdinare, erTi nivTieri wertilisgan Semdgari sxeulis brunviTi inerciaa I=mr2. es toloba 10.57 formulaSi CavsvaT da miviRebT: ΔK = 1 2 1 Iω – Iωi2 = W 2 f 2 10.52 formula damtkicebulia. formula erTi nivTieri wertilisgan Semdgari myari sxeulisTvis gamoviyvaneT, magram is fiqsirebuli RerZis garSemo mbrunav nebismier myar sxeuls miesadageba.  amjerad 10.17 suraTze gamosaxul sxeulze Sesrulebul W muSaobasa da F Zalis Sedegad sxeulze moqmed τ Zalis moments Soris kavSiri davadginoT. rodesac nivTieri wertili wriuli traeqtoriis gaswvriv ds manZils gaivlis, mas Zalis mxolod Ft mxebi komponenti aCqarebs. aqedan gamomdinare, nivTier wertilze muSaobas mxolod Ft asrulebs. Sesrulebuli dW muSaoba Ft ds saxiT CavweroT da Semdeg ds SevcvaloT rdθ-iT, sadac dθ aris nivTieri wertilis mier Semowerili kuTxe: dW = Ft r dθ (10.58) 347 10.10 muSaoba da brunvis kinetikuri energia 10.40 formulidan gamomdinare, Ft r namravli τ Zalis momentis tolia da 10.58 formula asec SeiZleba Caiweros: dW = τdθ (10.59) θi-dan θf-mde sasruli kuTxuri gadaadgilebis ganmavlobaSi Sesrulebuli muSaobaa: W= , 10.53 formula damtkicebulia. is fiqsirebuli RerZis garSemo mrbunav nebismier myar sxeuls miesadageba. 10.54 formula pirdapir 10.53 formulidan miiReba. brunviTi moZraobisas P simZlavre 10.59 formuliT SegviZlia gamovTvaloT: 10.55 formulac damtkicebulia. dW dθ P = dt = τ dt = τω, amocanis amoxsnis nimuSi 10.8 amocanis amoxsnis nimuSSi moyvanili da 10.18 suraTze gamosaxuli disko moZraobas t = 0 dros iwyebs. ra iqneba misi K brunvis kinetikuri energia t = 2,5 wm dros? amoxsna: K-s gamoTvla 10.34 formuliT (K = 1 Iω2) SeiZleba. ukve viciT, rom I = 12 MR2, magram 2 ar viciT ω kuTxuri siCqare t = 2,5 wm dros. α kuTxuri aCqarebis mudmivi mniSvnelobaa –24radiani/wm2 da 10.1 cxrilSi mocemuli formulebi SegviZlia gamoviyenoT. vinaidan α da ω0(= 0) viciT da ω-s gamoTvla gvsurs, 10.12 formulas viyenebT: ω = ω0 + αt = 0 + αt = αt miviRebT: K = Ki + W = 0 + W = W (10.60) amis Semdeg W muSaoba unda gamovTvaloT. W muSaobasa da diskoze moqmed Zalis momentebs Soris kavSiri 10.53 an 10.54 formuliT SegviZlia gamovsaxoT. kuTxuri aCqarebis gamomwvevi da muSaobis Semsrulebuli erTad erTi Zalis momenti Tokidan diskoze moqmedi  T ZaliTaa gamowveuli. 10.8 amocanis amoxsnis nimuSidan gavixsenoT, rom es Zalis momentia –TR. vinaidan α mudmivia, Zalis momentic mudmivia da 10.54 formulis daxmarebiT miviRebT: 1 10.34 formulaSi ω = αt da I = MR2 CavsvaT da 2 miviRebT: 1 1 1 1 2 2 2 4 1 = (2,5kg)[(0,2m)(–24 radiani/wm2)(2,5wm)]2 4 K = Iω2 = ( MR2)(αt)2 = M(Rαt)2 = 90 j 10.10 W = τ(θf-θi) = –TR(θf – θi) (10.61) vinaidan α mudmivia, 10.13 formulis daxma rebiT θf – θi SegviZlia gamovTvaloT. ωi=0 da gveqneba: 1 2 1 2 1 2 θf – θi = ωit + αt2 = 0 + αt2 = αt2 (pasuxi) diskos kinetikuri energia masze Sesrulebuli miRebuli Sedegi 10.61 formulaSi CavsvaT da muSaobis gamov pasuxi10.60 formulaSi gadavitanoT. T = 6 n, TvaloT. Tavdapirvelad, muSaoba-kinetikuri α = –24 radiani/wm2 (10.8 amocanis amoxsnis saSualebiTac SegviZlia energiis Teoremis (Kf – Ki = W) daxmarebiT diskos kinetikuri energiis cvlileba da diskoze Sesrulebuli jamuri W muSaoba erTmaneTTan davakavSiroT. Kf -s nacvlad K CavsvaT, xolo Ki-s nacvlad 0 SevitanoT da amocanis amoxsnis nimuSi nimuSidan) da gveqneba: 1 1 K = W = –TR(θf – θi) = –TR( αt2) = – TRαt2 2 2 1 = – 2 (6n)(0,2m)(–24radiani/wm2)(2,5wm)2 = 90j (pasuxi) 10.11 myari qandakeba fersosgan (m masis da R=0,15m fersos sibrtyeSi mdebare da mis centrze radiusis) da radialuri Zelakisgan (m masis da gamavali horizontaluri RerZis garSemo L = 2R sigrZis) Sedgeba (sur. 10.20). qandakeba brunavs. 348 brunva Tavi 10 1 (K= Iω2) saSualebiT ukavSirdeba erT 2 maneTs. 2. K kinetikuri energia qandakebis U mizi dulobis potenciur energias brunvis dros qandakebis E meqanikuri energiis Senaxvis principis saSualebiT ukavSirdeba. brunvis dros E ar icvleba, vinaidan energia U-dan K-Si gadadis. 3. mizidulobis potenciuri energiisTvis qandakeba sur. 10.20. fersosa da Zelakisgan Semdgari myari qandakeba horizontaluri RerZis garSemo brunavs a) risi tolia qandakebis I brunviTi inercia brunvis RerZis garSemo m-sa da R-is masis centrze ganlagebul nivTier wertilad SegviZlia ganvixiloT, sadac 2m masaa koncentrirebuli. meqanikuri energiis Senaxvis faqti ase Cai wereba: ΔK + ΔU = 0 (10.62) saSualebiT? amoxsna: Cven SegviZlia fersos da Zelakis sawyisi mdgomareobidan Sebrunebul mdgo brunviTi inerciebi cal-calke gamovTvaloT mareobamde, sadac qandakebas ω da Semdeg maTi Sekrebis gziT qandakebis I siCqare aqvs, kinetikuri energiis ΔK cvli brunviTi inercia miviRoT. 10.2T cxrilidan lebaa: gamomdinare, 1 2 1 Iω – 0 = Iω2 2 2 brunviTi inercia 1 diametris garSemo aris Iferso= mR2. 10.2e 2 cxrilis mixedviT, Zelakis brunviTi 8.7 formulidan (ΔU = mgΔy) gamomdinare, inercia misi masis centrze gamavali da mizidulobis potenciuri energiis Sesabamisi qandakebis fersos kuTxuri brunvis RerZis paraleluri RerZis garSemo aris Icom=mL2/12. garSemo Zelakis paraleluri RerZebis Teoremas viyenebT: IZelaki = Icom + mh 2 = com ( mL2 L +m R+ 12 2 ) 2 = 4,33mR2 , sadac L=2R da Zelakis masis centrsa da brunvis RerZs Soris perpendikularuli manZilia h=R+L/2. maSasadame, qandakebis I brunviTi inercia brunvis RerZis garSemo aris: 1 2 = 4,83mR2 ≈ 4,8mR2 ΔU = (2m)gΔycom , (10.64) sadac 2m aris qandakebis masa, xolo Δycom – masis centris vertikaluri gadaadgildeba brunvis dros. Δycom-is gamoTvlisTvis gamosaxuli masis 10.20 centris suraTze sawyisi ycom mdebareoba unda davadginoT. fersos (m masis) centria y=0. Zelakis (m masis) centria y=R+L/2. maSasadame, 9.5 formulidan gamomdinare, qandakebis masis centris mdebareobaa: ycom = I = Iferso + IZelaki = mR2 + 4,33mR2 (10.63) ΔU cvlilebaa: brunvis IZelaki brunviTi inerciis gamoTvlisTvis 10.36 formulas da RerZis ΔK = Kf – Ki = m(0) + m(R + L/2) 0 + m(R + 2R/2) = = R. 2m 2m rodesac qandakeba Sebrundeba, masis centri (pasuxi) b) qandakeba swordeba da brunvis RerZis garSemo brunvas iwyebs. ra iqneba misi ω kuTxuri siCqare RerZis garSemo, rodesac qandakeba amotrialdeba? amoxsna: 1. qandakebis ω siCqare da misi brunvis K kinetikuri energia 10.34 formulis brunvis RerZidan igive R manZilzea. aqedan gamomdinare, masis centris vertikaluri gadaadgileba sawyisi mdebareobidan Sebru nebul mdebareobamde aris Δycom = –2R. modiT, miRebul Sedegebs Tavi erTad movuyaroT. 10.63 da 10.64 formulebi 10.62 formulaSi CavsvaT da miviRebT: 1 2 Iω + (2m)gΔycom = 0 2 349 10.10 muSaoba da brunvis kinetikuri energia amocanis a) nawilidan I = 4,83mR2 CavsvaT, gavixsenoT, rom Δycom = –2R da zemoT miRebuli formula ω-sTvis amovxsnaT: ω= = 10 rad/wm. (pasuxi) mimoxilva da Sejameba kuTxuri mdebareoba. fiqsirebuli anu ω2-ze icvleba, misi αsaS saSualo kuTxuri brunvis aCqarebaa: RerZis garSemo myari sxeulis Δω ω –ω αsaS = t2 – t 1 = . Δt 2 1 brunvis aRwerisTvis vuSvebT, rom aTvlis wrfe sxeulze gadis, perpendikularulia da brunvis RerZis sxeulTan erTad xolo α myisi aCqareba moZraobs. am wrfis θ kuTxur mdebareobas fiqsirebuli mimarTulebisadmi vzomavT. θ radianebSi izomeba da gvaqvs: θ= s r (radianebSi), (10.1) sadac s aris r radiusis da θ kuTxis mqone wriu li traeqtoriis rkalis sigrZe. radianebis, brunebis da gradusebis erTeulebs Soris Semdegi kavSiria: 1 br = 360° = 2π radiani (10.7) α= dω . dt (10.8) αsaS da α veqtorebia. kinetikuri formulebi mudmivi kuT xuri aCqarebisTvis. Tanabari kuTxuri aCqareba brunvis gansakuTrebuli SemTxveva. 10.1 cxrilSi Sesabamisi kinetikuri formu lebia mocemuli: ω = ω0 + αt (10.2) (10.12) 1 2 θ – θ0 = ωot + αt2 (10.13) brunvis RerZis garSemo brunavs da misi ω2=ω02 + 2α(θ – θ0) (10.14) kuTxuri mdebareoba θ1-dan θ2-ze icvleba, θ – θ0 = (ω0 + ω)t kuTxuri gadaadgileba. rodesac sxeuli 1 2 kuTxuri gadaadgilebaa: Δθ = θ2 – θ1 , (10.4) sadac Δθ dadebiTia, Tu sxeuli saaTis isris moZraobis sawinaaRmdego mimarTulebiT moZraobs da uaryofiTia, Tu sxeuli saaTis isris moZraobis mimarTulebiT moZraobs. kuTxuri siCqare. Tu sxeuli Δt drois SualedSi Δθ kuTxur gadaadgilebas ganicdis, misi ωsaS saSualo kuTxuri siCqarea: Δθ θ –θ ωsaS = t 2 – t 1 = Δt 2 1 sxeulis ω myisi kuTxuri siCqarea: Δθ dθ ω= = . Δt dt (10.5) 1 2 θ – θ0 = ωt – αt2 (10.15) (10.16) wrfiv da kuTxur cvladebs Soris kavSiri. mrbunavi myari sxeulis brunvis RerZidan r perpendikularuli manZiliT daSorebuli wertili r radiusis wrewirze moZraobs. Tu sxeuli θ kuTxiT brunavs, wertili rkalze moZraobs, romlis s sigrZea: s = θr (radianebSi), (10.17) sadac θ radianebSia mocemuli. wertilis wrfivi siCqare wrewiris mxebia da Semdegi formuliT gamoiTvleba: (10.6) ωsaS da ω veqtorebia da maTi mimarTulebebi marjvena xelis wesiT (sur. 10.6) ganisaz Rvreba. saaTis isris moZraobis sawinaaRmdego mimarTulebiT brunvisas kiTxuri siCqare dadebiTia, xolo saaTis isris moZraobis mimarTulebiT brunvisas – uaryofiTi. kuTxuri aCqareba. Tu Δt = t2 – t1 drois SualedSi sxeulis kuTxuri siCqare ω1-dan ν = ωr (radianebSi), (10.18) sadac ω sxeulis kuTxuri siCqarea.  wertilis a wrfiv aCqarebas mxebi da radia luri komponenti aqvs. mxebi komponentia: at = αr (radianebSi), (10.22)  sadac α sxeulis kuTxuri aCqarebaa. a aCqa rebis radialuri komponentia: ar = Tu wertili ν2 = ω2r r (radianebSi) Tanabarwriulad (10.23) moZraobs, 350 brunva Tavi 10 wertilis da sxeulis moZraobis T periodia: 2πr 2π = (radianebSi) (10.19, 10.20) v ω brunvis kinetikuri energia da brun viTi inercia. fiqsirebuli RerZis garSemo mbrunavi myari sxeulis K kinetikuri T= energiaa: 1 (10.34) 2 sadac I aris sxeulis brunviTi inercia anu K= Iω2 (radianebSi) ,   da F -s Soris. r┴ sidide brunvis RerZsa da kularuli komponenti, xolo φ - kuTxe r -sa im wrfes Soris perpendikularuli manZilia,  romelzec F veqtoria mimarTuli. am wrfes  F -is moqmedebis wrfe ewodeba, xolo r┴  sidides F -is momentis mxars uwodeben. r aris Ft-s momentis mxari. SI sistemaSi Zalis momentis erTeulia n·m. Zalis momenti dadebiTia, Tu is sxeuls saaTis inerciis momenti, gansazRvruli wyvetil isris sawinaaRmdego mimarTulebiT abrunebs wertilTa sistemisTvis: da uaryofiTia, Tu is sxeuls saaTis isris I= (10.33) moZraobis mimarTulebiT abrunebs. uwyveti ganawilebis masis mqone sxeulis I niutonis meore kanoni kuTxur formaSi. brunviTi inerciaa: niutonis meore kanonis brunviTi analogia I= r2 dm τt = Iα, (10.35) am formulebSi r da ri sidideebi brunvis RerZidan sxeulis TiToeul masis elementamde perpendikularul manZils asaxavs. (10.45) sadac τt aris myar sxeulze an nivTier wertilze moqmedi jamuri Zalis momenti, I - nivTieri wertilis an sxeulis brunviTi inercia brunvis RerZis garSemo, xolo α – paraleluri RerZebis Teorema. para Sedegad miRebuli kuTxuri aCqareba brunvis leluri RerZebis Teorema nebismieri RerZis RerZis garSemo. garSemo mbrunavi sxeulis brunviT inercias masis centrze gamavali paraleluri RerZis garSemo mbrunavi igive sxeulis brunviT inerciasTan akavSirebs: I=Icom + Mh2 , sadac h (10.36) muSaoba da brunvis kinetikuri energia. brunviTi simZlavris gadataniTi paralelur RerZebs Soris. da is sxeuls brunvis RerZis garSemo abrunebs.   Tu F Zala r mdebareobis veqtoriT mocemul wertilzea modebuli, maSin Zalis momentis sididea: τ = rFt = r┴F = rF sin φ , (10.40, 10.41, 10.39)   sadac Ft aris F -is r -isadmi perpendi dros muSaobis gamosaTvleli moZraobis dros formulebi gamoyenebul da W= (10.53) dW P = dt = τω (10.55) rodesac τ mudmivia, 10.53 formula Semdeg saxes iRebs: W = τ(θf – θi) (10.54) muSaoba-kinetikuri energiis Teorema mbru navi sxeulebisTvis ase Caiwereba: 1 1 ΔK = Kf – Ki= Iωf2 – Iωi2 = W 2 2 SekiTxvebi 1. 10.21b suraTze naCvenebia 10.21a suraTze gamosaxuli mbrunavi diskos kuTxuri mde bareobis funqciis grafiki. rodesac a) t = 1 wm, b) t = 2 wm da g) t = 3 wm, diskos kuTxuri siCqare dadebiTia, uaryofiTia Tu nulis toli? d) kuTxuri aCqareba dadebiTia Tu uar yofiTi? da formulebs Seesabameba: aris perpendikularuli manZili  Zalis momenti. Zalis moments F Zala iwvevs moZraobis sur. 10.21. SekiTxva 1. (10.52) 351 2. 10.22 suraTze naCvenebia 10.21a 90°-idan mcirdeba da Zelakma kvlav ar unda imoZraos, F2 Zala unda gaizardis, Semcirdes suraTze Tu igive darCes? gamosa xuli mrbunavi diskos kuTxuri siCqaris sur. 10.22. SekiTxva 2. droze 6. 10.25b suraTze gamosaxul horizontalur   Zelakze F1 da F2 horizontaluri Zalebi moqmedebs da Zelaks sayrdeni wertilis wertilisTvis a, b, c da d momentebi a) mxebi da  garSemo brunva SeuZlia. F2 Zala ZelakTan φ kuTxes adgens. φ-s Semdegi mniSvnelobebi b) radialuri aCqarebis mixedviT daalageT Zelakis kuTxuri aCqarebis mixedviT dididan dididan mcireze. mcireze daalageT: 90°, 70° da 110°. 3. 10.23 suraTze 7. 10.26 suraTze naCvenebia gamosaxul ucnaur damokidebulebis grafiki. diskos fersos navi mbru θ diskos karuselze Tana kuTxuri mdeba reobis t bari sididis xuTi droze Zala damokidebulebis sami grafiki. TiToeul Sem karuseli kvadra sur. 10.23. SekiTxva 3. TxvevaSi, garkveul θ kuTxur mdebareobaze brunvis mimarTuleba moqmedebs. icvleba. a) samive SemTxvevaSi gansazRvreT θ, θ0-dan moyole tia, romelsac misi erT-erTi gverdis sur. 10.26. SekiTxva 7 SuaSi ganlagebuli P wertilis gar Semo SeuZlia brunva. Zalebi P wertilis buli, saaTis isris moZraobis mimarTulebisaa garSemo maT mier warmoqmnili Zalis momentis Tu saaTis isris moZraobis sawinaaRmdego mixedviT dididan mcireze daalageT. mimarTulebis? samive SemTxvevaSi b) ω nulis toli t=0 dromdea, am drois Semdegaa da Tu zustad am drosaa? g) α dadebiTia, uaryofiTia Tu nulis tolia? 4. 8. 10.27 suraTze, sa kuTari centris gar Semo mrbunav diskoze   F1 da F2 Zala moq 10.24 suraTze medebs. disko saaTis naCvenebia liTonis isris moZraobis sa erTgvarovani firfi winaaRmdego ta, romelic kvadrati TulebiT da mudmivi iyo, vidre mis 25%s amoWridnen. sur. 10.24. SekiTxva 4. aseve naCvenebia sami wertili. daalageT isini maTze gamavali perpendikularuli RerZis garSemo firfitis brunviTi inerciis mixedviT tempiT mimar brunavs, sur. 10.27. SekiTxva 8. xolo TiToeuli Zala   Zalis θ kuTxe ise unda SevamciroT, rom F1 suraTze miTiTebul kuTxes adgens. Cven F1 -is sidide ar Seicvalos. a) kuTxuri siCqaris mudmivobis SenarCunebisTvis  F2 unda 5. 10.25a suraTze mbrunavi, horizontaluri   da g) F2 Zala diskos saaTis isris moZraobis Zelakis zedxedia gamosaxuli. Zelakze ori mimarTulebiT horizontaluri Zala moqmedebs, magram is saaTis dididan mcireze.  uZravia. Tu Zelaksa da F2 Zalas Soris kuTxe gaizardos, Semcirdes Tu igive darCes? b) F1 isris Sebrunebas moZraobis cdilobs Tu sawinaaRmdego mimarTulebiT? 9. 10.28 suraTze naCvenebia erTi da igive radiusis sakuTari sami brtyeli centris disko. garSemo diskoebs SeuZliaT brunva. TiToeuli disko ori, erTi da igive sur. 10.25. SekiTxvebi 5 da 6 nivTierebisgan Sedgeba. erTi nivTiereba 352 Tavi 10 brunva meoreze ufro mkvrivia. pirvel da mesame diskoebi a) maTi centris garSemo Zalis diskoSi ufro mkvrivi nivTiereba diskos gare momentis, b) maTi centris garSemo brunviTi naxevarzea. meore diskos SemTxvevaSi, mkvrivi inerciis da g) kuTxuri aCqarebis mixedviT nivTiereba diskos Sida naxevars Seadgens. dididan mcireze. Tanabari sididis Zalebi diskos kides an 10. nivTierebebs Soris zedapirs exeba. ganalageT disko karuselis msgavsad brunavs, kuTxur siCqares icvlis da mis fersoze Zalaa modebuli. diskos sawyisi da saboloo kuTxuri siCqareebia: a) –2radiani/wm, 5radiani/wm, b) 2radiani/wm, 5radiani/wm, g) –2 radiani/wm, –5radiani/wm da d) 2radiani/wm, –5radiani/wm. es SemTxvevebi Zalis momentis mier Sesrulebuli muSaobis mixedviT dididan mcireze daalageT. sur. 10.28. SekiTxva 9. amocanebi SSM amoxsnebi mocemulia Student Solutions Manual-Si. www amoxsnebi mocemulia veb-gverdze http://www.wiley.com/college/halliday ILW Interactive LearningWare amoxsnebi mocemulia veb-gverdze http://www.wiley. com/college/halliday. • – ••• wertilebis raodenoba amocanis sirTules gviCvenebs paragrafi 10.2. brunviTi cvladebi ·1. ra kuTxuri siCqare aqvs saaTis a) wamebis maCvenebel isars, b) wuTebis maCvenebel isars da g) saaTebis maCvenebel isars? pasuxi radiani/wm-ebSi gaeciT. ·2. karg momwodebels beisbolis burTis srola `saxlisken~ 85mili/sT siCqariT da 1800br/wTiT SeuZlia. ramden bruns Seasrulebs burTi saxlisken moZraobis dros? simartivisTvis davuSvaT, rom 60 futi sigrZis traeqtoria wrfea. ·3. sinaTlis stroboskopuli wyaro. disko saaTis isris moZraobis mimarTulebiT 10π radiani/wm kuTxuri siCqariT brunavs da stroboskopuli sinaTliT naTdeba. ganaTeba diskos fersoze patara Sav wertils aCens. pirveli ganaTebis dros wertili 12:00 mdebareobazea (rogorc saaTze). Semdeg xuT ganaTebas f1, f2 da a.S. uwodeT. Tu ganaTebebs Soris Δt drois Sualedia 0,2wm, sad iqneba wertili a) f1 da b) f2 ganaTebis dros? Tu Δt=0,05wm, sad iqneba wertili g) f1, d) f2, e) f3 da v) f4 ganaTebis dros? Tu Δt = 40 mwm, sad iqneba wertili z) f1, T) f2, i) f3, k) f4 da l) f5 ganaTebis dros? ··4. mrbunavi borblis fersoze ganlagebuli wertilis kuTxuri mdebareobaa θ = 4t – 3t2 + t3, sadac θ mocemulia radianebSi, xolo t – wamebSi. ra iqneba wertilis kuTxuri siCqare a) t = 2 wm da b) t = 4 wm dros? g) ra iqneba wertilis saSualo kuTxuri aCqareba 2 wm-dan 4wm-mde drois SualedSi? ra iqneba wertilis myisi kuTxuri aCqareba drois am Sualedis d) dasawyissa da e) dasasrulSi? ··5. wyalSi mxtomeli 10m simaRlidan xteba da gzad 2,5-jer Semobrunebas aswrebs. davuSvaT sawyisi siCqare nulis tolia, ra iqneba mxtomelis saSualo kuTxuri siCqare? ILW ··6. mbrunavi borblis wertilis kuTxuri mdebareobaa θ=2+4t2+2t3, sadac θ mocemulia radianebSi, xolo t - wamebSi. rodesac t=0, ra a) kuTxuri mdebareoba da b) kuTxuri siCqare aqvs wertils? g) ra kuTxuri siCqare aqvs wertils, rodesac t=4wm? d) gamoTvaleT wertilis kuTxuri aCqareba, rodesac t=2wm. e) aris Tu ara wertilis kuTxuri aCqareba mudmivi? ···7. 10.29 suraTze gamosaxul borbals Tanabari sididis da 30 sm radiusis sur. 10.29. amocana 7. 353 rva mana aqvs. borbali fiqsirebul RerZzea iZens da gaCerebamde –α1 mudmivi tempiT damagrebuli da 2,5br/wT siCqariT brunavs. kuTxur siCqares kargavs. diskos nebismieri 20sm sigrZis isari RerZis paralelurad nawilis centriskenulma aCqarebam 400 m/wm2-s borbalSi ise unda gaataroT, rom is arc ar unda gadaaWarbos. a) ra umciresi dro erT manas ar Seexos. davuSvaT isari da manebi sWirdeba brunvas? b) risi tolia α1? Zalian wvrilia. a) ra minimaluri siCqare unda ··15. borbali 40 bruns akeTebs da 1,5 radiani/wm kuTxuri siCqaridan gaCerebamde neldeba. hqondes isars? b) aqvs Tu ara mniSvneloba fersosa da RerZs Soris sad daumiznebT isars? Tu mniSvneloba aqvs, sad iqneba isris gatarebis saukeTeso Sansi? ra drois Semdeg gaCerdeba da b) ra kuTxiT brunavs, rodesac Cerdeba? ·9. avtomobilis Zravis kuTxuri siCqare 12 wamis ganmavlobaSi 1200br-wT-idan 3000br/wTmde mudmivi tempiT gaizarda. a) ra iqneba misi kuTxuri aCqareba br/wT2-ebSi? b) ramden bruns Seasrulebs Zrava 12 wamis ganmavlobaSi? ·10. 120radiani/wm siCqariT mbrunavi disko 4radiani/wm2 mudmivi kuTxuri aCqarebiT a) ra dro dasWirdeba diskos gaCerebas? b) ra kuTxiT brunavs disko am drois ganmavlobaSi? ·11. davuSvaT kuTxuri aCqareba mudmivia. ra dro dasWirdeba borbals gaCerebamde? b) risi tolia kuTxuri aCqareba? g) ra dro paragrafi 10.4. brunva Tanabari kuTxuri aCqarebiT ·8. doli centraluri RerZis garSemo 12,6radiani/wm kuTxuri siCqariT brunavs. Tu doli 4,2radiani/wm2 Tanabrad neldeba, a) neldeba. a) disko centraluri RerZis garSemo mudmivi kuTxuri aCqarebiT brunavs. 5 wamis ganmavlobaSi disko 25 radians Semowers. risi dasWirdeba bolo 20 brunis Sesrulebas? ILW ··16. disko centraluri RerZis garSemo brunavs da mudmivi kuTxuri aCqareba aqvs. garkveul momentSi diskos siCqarea 10 br/wm. 60 brunis Semdeg misi kuTxuri siCqare 15 br/wm xdeba. gamoTvaleT a) kuTxuri aCqareba, b) 60 brunis SesrulebisTvis saWiro dro, g) 10 br/wm kuTxuri siCqaris misaRwevad sa Wiro dro da d) 10 br/wm kuTxuri siCqaris miRwevamde Sesrulebuli brunebis ricxvi. ··17. rodesac t = 0, borblis kuTxuri siCqarea 4,7 radiani/wm, mudmivi kuTxuri aCqarebaa –0,25 radiani/wm2 da aTvlis wrfis mdebareobaa θ0 = 0. a) ra θmax maqsimalur kuTxes Semowers aTvlis wrfe dadebiTi mimarTulebiT? ra dros iqneba aTvlis wrfis mdebareoba θ = 12 θmax b) pirvelad da g) meored? ra d) uaryofiT da e) dadebiT dros iqneba aTvlis wrfis mdebareoba θ = –10,5 radiani? v) aageT θ mdebareobis t droze damokidebu tolia diskos a) kuTxuri aCqareba da b) saSualo lebis funqciis grafiki da amocanis SekiT kuTxuri siCqare am drois ganmavlobaSi? g) ra xvebs grafikis mixedviT gaeciT pasuxi. myisi kuTxuri siCqare aqvs diskos 5 wamiani Sualedis miwuruls? d) Tu kuTxuri aCqareba ar icvleba, ra damatebiT kuTxes Semowers disko Semdegi 5 wamis ganmavlobaSi? ··12. karuseli 1,5radiani/wm2 kuTxuri aCqare biT brunavs. ra droSi Seasrulebs karuseli a) pirvel 2 bruns da b) Semdgom 2 bruns? ··13. borbali 3radiani/wm2 mudmivi kuTxuri aCqarebiT brunavs. 4 wamis ganmavlobaSi borbali 120 radian kuTxes Semowers. ra drois ganmavlobaSi moZraobda borbali 4 wamiani Sualedis dawyebisas? SSM ··14. 0,25m radiusis disko 800 radian kuTxes Semowers, pirveli 400 radianis ganmavlobaSi α1 mudmivi tempiT garkveul kuTxur siCqares ··18. borbali 2 radiani/wm2 mudmivi kuTxuri aCqarebiT moZraobs. 3 wamis ganmavlobaSi borbali 90 radian kuTxes Semowers. a) ra drois ganmavlobaSi moZraobda borbali 3wamiani Sualedis dawyebamde? b) ra kuTxuri siCqare aqvs borbals 3-wamiani Sualedis dawyebisas? paragrafi 10.5. wrfiv cvladebs Soris kavSiri da kuTxur ·19. 1911-1990 wlebSi pizis koSki 1,2 mm/iardi tempiT samxreTisken gadaixara. koSki 55 m si maRlisaa. ra saSualo kuTxuri siCqare aqvs koSkis wvers koSkis fskeris mimarT radiani/ wm-ebSi? 354 Tavi 10 brunva ·20. vinilis firfitis daaxloebiT wriuli aqvs mqnevaras fersoze ganlagebul wertils? formis zedapiri firsakravis nemsis qveS b) ra radialuri aCqareba aqvs am wertils, srialebs. zedapiris rodesac mqnevara sruli siCqariT brunavs? g) borcvebs ejaxeba da irxeva. Semdeg mowyo ra manZils gaivlis es wertili brunvis dros? biloba am rxevebs eleqtrul signalebad ··27. a) ra ω kuTxuri siCqare aqvs dedamiwis da bgerad gardaqmnis. davuSvaT firfita zedapiris wertils polaruli RerZis garSemo nemsi periodulad 33 13 br/wT tempiT brunavs, zedapiris radiusi 10 sm-ia da borcvebs Soris dacilebaa 1,75 mm. 400N simaRleze? (dedamiwa polaruli RerZis garSemo brunavs). b) ra ν wrfivi siCqare aqvs am ra tempiT ejaxeba nemsi borcvebs? wertils? risi tolia ekvatorze ganlagebuli ·21. 1,2 m diametris borbali 200 br/wT kuTxu wertilis g) ω da d) ν? SSM ri siCqariT brunavs. a) ra kuTxuri siCqare ··28. 10.30 suraTze, rA=10sm radiusis A borbali Rvedis saSualebiT rC=25 sm radiusis C bor aqvs borbals radiani/wm-ebSi? b) ra wrfivi siCqare aqvs borblis fersoze ganlagebul wertils? g) ra mudmivi kuTxuri aCqareba (br/wT2-ebSi) gazrdis borblis siCqares 1000 br/wT-mde 60 wamSi? d) ramden bruns Seasrulebs borbali am 60 wamis ganmavlobaSi? SSM WWW ·22. kosmonavti centrifugaSi Semowmebas gadis. 10 m radiusis centrifuga θ = 0,3t2 for mulis mixedviT brunavs, sadac t mocemulia wamebSi, xolo θ – radianebSi. rodesac t = 5 wm, ra b) kuTxuri siCqare, b) wrfivi siCqare, balzea gadabmuli. sur. 10.30. amocana 28. A borblis kuTxuri siCqare 1,6radiani/wm2 mudmivi tempiT izrdeba. ra dro dasWirdeba C borbals 100br/wT kuTxuri siCqaris ganviTarebisTvis, Tu Rvedi ar gawydeba? (miniSneba: Tu Rvedi ar wydeba, orive borblis wrfivi siCqare Tanabaria). g) mxebi aCqareba da d) radialuri aCqareba aqvs ··29. sinaTlis siCqaris gazomvisTvis mbrunavi kosmonavts? dakbiluli borbali gamoiyeneboda. sinaTlis sxivi borblis kideze ganlagebul kbilebs ·23. kosmosuri xomaldi 29000km/sT siCqariT 3220km radiusis wres Semowers. ra a) kuTxuri siCqare, b) radialuri aCqareba da g) mxebi aCqareba aqvs kosmosur xomalds? ·24. TviTmfrinavis propeleri 2000 br/wT siCqariT brunavs da TviTmfrinavi deda miwisadmi 480 km/sT siCqariT mifrinavs. ra wrfivi siCqare aqvs propeleris kideze, 1,5 m radiusze ganlagebul wertils a) TviT mfrinavis pilotis da b) dedamiwaze myofi damkvirveblis TvalTaxedviT? TviTmfrinavis Soris gadioda (sur. 10.31), Semdeg moSorebiT mdebare sarkes daecemoda da borblisken brundeboda, ris Semdegac momdevno kbilebs Soris gadioda. borblis radiusia 5sm da mas kideze 500 kbili aqvs. rodesac sarke borblidan L=500m manZiliTaa daSorebuli, borbali 3×105km/wm sinaTlis siCqares aCvenebs. a) ra mudmivi kuTxuri siCqariT moZraobs borbali? b) ra wrfivi siCqariT moZraobs borblis kideze ganlagebuli wertili? SSM siCqare propeleris brunvis RerZis parale luria. ·25. sagani fiqsirebuli RerZis garSemo brunavs da misi aTvlis wrfis mdebareobaa θ = 0,4e2t, sadac θ mocemulia radianebSi, xolo t - wamebSi. ganvixiloT sagnis brunvis RerZidan 4 sm-iT daSorebuli wertili. rodesac t = 0, risi tolia wertilis aCqarebis a) mxebi komponenti da b) radialuri komponenti? ··26. giroskopis 2,83 sm radiusis mqnevara 14,2 radiani/wm2-iT Cqardeba da misi kuTxuri siCqare 2760 br/wT xdeba. a) ra mxebi aCqareba sur. 10.31. amocana 29. 355 ··30. orTqlis Zravis mqnevara 150br/wT kuTxuri aCqareba (br/wT2-ebSi) aqvs mqnevaras Senelebis dros? b) ramden bruns Seasrulebs mqnevara gaCerebamde? g) rodesac mqnevara gar-Semo mbru navi Zelakis kuT xuri siCqaris droze damoki debulebis grafiki. a) ra kuTxuri aCqareba aqvs 75br/wT kuTxuri siCqariT brunavs, risi tolia brunvis RerZidan 50sm-iT daSorebuli kinetikuri energiaa 1,6 j. ra kinetikuri mudmivi kuTxuri siCqariT brunavs. orTqlis miwodebis Sewyvetis SemTxvevaSi, sayrdeni wertilebisa da haerisgan moqmedi xaxuni mqnevaras 2,2sT-is Semdeg aCerebs. a) ra mudmivi nivTieri wertilis wrfivi aCqarebis mxebi komponenti? d) ra jamuri wrfivi aCqareba aqvs am wertils? ··31. firsakravi 33 13 br/wT kuTxuri siCqariT brunavs. firsakravis brunvis RerZidan 6sm manZilze sazamTros kurka gdia. a) gamoTvaleT kurkis aCqareba, Tu is firsakravidan ar vardeba. b) statikuri xaxunis ra minimaluri koeficienti unda arsebobdes kurkasa da firsakravs Soris, raTa kurka ar Camovardes? g) davuSvaT, firsakravi Tavis kuTxur siCqares 0,25 wamis ganmavlobaSi mudmivi kuTxuri aCqarebiT aRwevs. gamoTvaleT statikuri xaxunis minimaluri koeficienti kurkasa da firsakravs Soris 0,25 wamis ganmavlobaSi, romlis arsebobis drosac kurka ar Camo vardeba. ···32. pulsari swrafad mbrunavi neitronis varskvlavia, romelic radiosxivs iseve asxivebs, rogorc Suqura sinaTlis sxivs. varskvlavis TiToeuli Semobrunebis dros Cven radiopulss (radiosignals) viRebT. varskvlavis brunvis T periodi pulsebs Soris drois gamoTvlis Sedegad dgindeba. pulsaris sur. 10.32. amocana 34. Zelaks? b) rodesac t = 4 wm, Zelakis brunvis energia aqvs Zelaks, rodesac t = 0? paragrafi 10.7. brunviTi inerciis gamo Tvla. ·35. 1,25kg masis ori myari, erTgvarovani cilindri centraluri RerZis garSemo 235radiani/wm kuTxuri siCqariT brunavs da maTi radiusebi gansxvavebulia. risi tolia a) 0,25m radiusis mqone mcire cilindris da b) 0,75m radiusis mozrdili cilindris brunvis kinetikuri energiebi? SSM ·36. 10.33 suraTze n a C v e n e b i a 0,01 kg masis sa mi nivTieri wer tili, romelic L = 6 sm sur. 10.33. amocanebi 36 sigrZis da 62. da umniSvnelo masis Zelakzea damagrebuli. Zelaki (nivTier wertilebTan erTad) O wertilze gamavali perpendikularuli RerZis garSemo brunavs. Semdeg erTi wertili (anu masis 33%) mov xseniT. ra procentiT Semcirdeba Zelakis brunviTi inercia, Tu es nivTieri wertili brunvis T periodia 0,033wm da is 1,26×19–5wm/ iardi tempiT izrdeba. a) risi tolia pulsaris a) O wertilTan yvelaze axlosaa da b) Zelakis gare kidezea? α kuTxuri aCqareba? b) Tu α mudmivia, ramdeni ·37. 0,56 kg masis da erTi metri sigrZis saxa zavi 20 sm niSnulze gamavali da saxazavis wlis Semdeg Sewyvets pulsari brunvas? g) pulsari 1054 wels momxdari afeTqebis Sedegad warmoiSva. α mudmivia. gamoTvaleT perpendikularuli RerZis garSemo brunavs. gamoTvaleT saxazavis brunviTi inercia am sawyisi T periodi. RerZis garSemo. SSM paragrafi 10.6. brunvis kinetikuri energia ·33. borbali 602br/wT siCqariT brunavs da misi kinetikuri energiaa 24400j. gamoTvaleT borblis brun-viTi iner-cia. SSM ·34. 10.32 suraT-ze naCvenebia er-Ti bolos ·38. 10.34a suraTze naCvenebia zedapiris perpendikularuli da diskos centridan h radialuri manZiliT daSorebuli RerZis garSemo mbrunavi disko. 10.34b suraTze gamosaxulia RerZis garSemo diskos I brun viTi inerciis h manZilze damokidebulebis grafiki. ra masa aqvs diskos? 356 Tavi 10 brunva kuTxeze gamavali da didi zedapirebis per pendikularuli RerZis garSemo. SSM WWW ··42. 0,5kg masis oTxi identuri nivTieri 2m×2m kvadratis wveroebzea wertili (a) (b) sur. 10.34. amocana 38. ··39. 10.35 suraT ze, m = 0,85 kg masis ori nawila ki d=5,6m sigrZis da M=1,2kg ori ZelakiT erT maneTTan da 0 wertilTan aris ganlagebuli. wertilebs erTmaneTTan oTxi masis armqone Zelaki akavSirebs, romlebic kvadratis gverdebs qmnis. risi tolia miRebuli sxeulis brunviTi inercia a) mopirdapire gverdebis Sua wertilze gamavali da kvadratis sibrtyebSi mdebare RerZis garSemo, b) erT-erTi gverdis Sua wertilze gamavali da kvadratis sibrtyis perpendikularuli RerZis garSemo da g) diagonalurad urTierTsapirispiro wertilebze gamavali da kvadratis sibrtyeSi mdebare RerZis garSemo? ··43. oTxi nivTieri wertilis masebi da sur. 10.35. amocana 39. dakavSirebuli da O wertilze gamavali brunvis RerZis garSemo ω=0,3radiani/wm kuTxuri siCqariT brunavs. ra iqneba kombinaciis a) brunviTi inercia da b) kinetikuri energia? ··40. 10.36 suraTze naCvenebia 15 erTmaneTze Sewebebuli disko, romelic L=1m sigrZis da M=100mg masis sistemas qmnis. sistema centraluri diskos O wertilze gamavali, koordinatebia: 50 g, x = 2 sm, y = 2 sm; 25 g, x = 0, y = 4 sm; 25 g, x = –3 sm, y = –3 sm; 30 g, x = –2 sm, y = 4 sm. risi tolia miRebuli sis temis brunviTi inercia a) x, b) y da g) z Rer Zis garSemo. davuSvaT a) da b) SekiTxvebis pasuxebia A da B. am SemTxvevaSi, ra iqneba g)-s pasuxi A-s da B-s TermebSi? ··44. evropuli satvirTo manqanebis nawili mbrunav mqnevaraSi Senaxuli energiis saSua lebiT moZraobs. eleqtroZrava mqnevaras 200π radiani/wm kuTxur siCqares aniWebs. erT- perpendikularuli RerZis garSemo brunavs. a) risi tolia sistemis brunviTi inercia am erTi mqnevara myari, erTgvarovani cilindria RerZis garSemo? b) Tu sistemas M masis da L sigrZis erTgvarovan Zelakad ganvixilavT, ra procentul Secdomas miviRebT brunviTi energia aqvs mqnevaras damuxtvis Semdeg? g) Tu 10.2e cxrilis inerciis gamoTvlisas formulis gamoyenebis SemTxvevaSi? 500 kg masiT da 1 m radiusiT. a) ra kinetikuri satvirTo manqana saSualod 8kvt simZlavres moixmars, ramdeni wuTis ganmavlobaSi SeuZlia mas moZraoba damuxtvebs Soris? paragrafi 10.8. Zalis momenti ·45. 0,75 kg masis burTi 1,25 m sigrZis da masis sur. 10.36. amocana 40. ·41. 10.37 suraTze qanqara vertikalidan 30°-iT gadaixreba, ra iqneba sayrdeni wertilis garSemo moqmedi gamosaxuli blo kis masaa 0,172kg da misi gverdebis Zalis momenti? SSM ·46. velosipedis pedlis sigrZea 0,152 m da sigrZeebia a = 3,5sm, b = 8,4 sm da c = 1,4 sm. gamoTvaleT blokis brunviTi inercia erT armqone Zelakis erT bolozea Camokidebuli, xolo Zelakis meore bolo sayrden wertilzea damagrebuli. rodesac Sedegad miRebuli sur. 10.37. amocana 41. mgzavri pedalze 111 n qvemoT mimarTuli ZaliT moqmedebs. ra Zalis momenti moqmedebs pedlis garSemo, rodesac pedali vertikali sadmi a) 30°, b) 90° da g) 180° kuTxes adgens? 357 ·47. ··52. 10.38 s u r a T z e gamosaxuli sxe uli O wertils sur. 10.38. amocana 47. eyrdnoba da mas ze ori Zala naCvenebi mimarTulebiT moqmedebs. Tu r1=1,3m, r2=2,15m, F1=4,2n, F2=4,9n, θ1=75° da θ2=60°, ra jamuri Zalis momenti moqmedebs sayrdeni wertilis garSemo? SSM ILW ·48. 10.39 suraTze gamosaxuli ecema. 10.41 su raTze naCvenebia mowinaaRmdegis ga martivebuli gamo sxe uli O wertils eyrdnoba da mas ze sami Zala mo qmedebs. ZiudoSi Se jibris dros Tqven mowinaaRmdegis marcxena fexs maR la swevT da amave dros mis kimonos qaCavT. Sedegad mowinaaRmdege mar jvena fexze tria ldeba da xaliCaze sur. 10.41. amocana 52. saxuleba. brunvis RerZi O wertilze gadis. O-dan 8m manZiliT da sur. 10.39. amocana 48 Sorebul A we rtilze FA=10n Zalaa modebuli, O-dan 4m manZiliT daSorebul B wertilze FB=16n Zala moqmedebs, xolo O-dan 3m-iT daSorebul C wertilze FC=19n Zala moqmedebs. risi tolia jamuri Zalis momenti O wertilis garSemo? paragrafi 10.9. niutonis meore kanoni brunvisTvis ·49. vardnis dros wyalSi mxtomelis kuTxuri siCqare masis centrisadmi nulidan 6,2 radiani/wm-mde 220mwm-is ganmavlobaSi icvleba. mxtomelis brunviTi inercia masis centris garSemo aris 12kg·m . risi tolia mxtomelis a) saSualo kuTxuri aCqareba da b) trampli nidan moqmedi saSualo gare Zalis momenti? 2 mowinaaRmdegis masis centri O wertilidan  d=28sm horizontalur manZilzea da masze Fg mizidulobis Zala moqmedebs. mowinaaRmdegis masaa 70kg da O wertilis garSemo misi brunviTi inerciaa 65kg·m2. ra sawyis kuTxur aCqarebas SeiZens mowinaaRmdege O wertilis  garSemo, Tu mis kimonoze moqmedi Fa biZgi a) umniSvneloa, b) horizontaluria, 300n sididisa da h=1,4m simaRlezea modebuli? ··53. 10.42 suraTze gamosaxuli 2kg masis cilindri O wertilze gamavali centraluri RerZis garSemo brunavs. modebuli Zalebia: F1=6n, F2=4n, F3=2n da F4=5n. r=5sm da R=12sm. gamoTvaleT cilindris kuTxuri aCqarebis a) sidide da b) mimarTuleba. SSM ILW ·50. Tu borbalze moqmedi 32 n·m Zalis momenti 25radiani/wm2 kuTxur aCqarebas iwvevs, ra iqneba borblis brunviTi inercia? ··51. 10.40 suraTze naCvenebia centris garSemo mrbunavi, erTgvarovani disko. diskos radiusia 2 sm, da masaa 20 grami Tavdapirvelad gaCerebulia. t = 0 dro idan moyolebuli, dis sur. 10.42. amocana 53. sur. 10.40. amocana 51. koze ori Zala iwyebs moqmedebas da t = 1,25 wm dros disko 250 radiani/wm kuTxuri siCqariT saaTis isris moZraobis sawinaaRmdego  mimarTulebiT brunavs. F1 Zalis sididea 0,1 n. risi tolia F2 Zala? ··54. 10.43 suraTze, m masis pirveli da meore nivTieri wertili L1+L2 sigrZis da ma sis armqone Zelakis sur. 10.43. amocana 54. 358 Tavi 10 brunva boloebzea damagrebuli. L1=20sm da L2=80sm. Zelaks sayrden wertilze horizontalurad aCereben da Semdeg xels uSveben. risi tolia a) pirveli nivTieri wertilis da b) meore nivTieri wertilis sawyisi aCqareba? ··55. erTi 10.44 suraTze blokis masaa avtomobilis muxla lilvs energia gadaaqvs, rodesac 1800br/wT siCqariT brunavs. ra brunvis moments iwvevs muxla lilvi? ·59. a) 10.18 suraTze R=12sm, M=400g da m=50g. gamoTvaleT blokis siCqare 50sm-iT dabla 5sm. gorgolaWi hori sur. 10.44. amocana 55. ufro masiuri bloki 5 wamis ganmavlobaSi 75smiT vardeba (Toki ar wydeba). a) ra aCqareba aqvs bloks? risi tolia b) ufro masiur blokze da g) naklebad masiur blokze mimagrebuli Tokis daWimuloba? d) ra kuTxur aCqarebas SeiZens gorgolaWi? e) ra brunviTi inercia aqvs gorgolaWs? SSM WWW 10.45a suraTze, uswormasworo formis, erTgvarovani sisqisa da simkvrivis plastmasis firfita RerZis garSe mo moZraobs, romelic firfitis zedapiris p e r p e n d i k u l a r u l i a da O wertilze gadis. firfitis brunviTi in ercia RerZis garSemo qvemoT aRwerili meTo paragrafi 10.10. muSaoba da brunvis kinetikuri energia Zravidan RerZze 100 cxZ (=76,4kvt) tempiT gorgolaWis radiusia ··56. rodesac t=3wm, ra a) kuTxuri aCqareba da b) kuTxuri siCqare aqvs mas? ·58. M=500g, meore blokis m=460g da masaa zontalur, xaxunisgan Tavisufal sayrdenzea damagrebuli. moZrao bis dawyebis Semdeg wamebSi. gorgolaWi Tavdapirvelad uZravia. dawevis Semdeg. amocana energiis Senaxvis principis daxmarebiT amoxseniT. b) a) SekiTxvas Tavidan gaeciT pasuxi, Tu R=5sm. ·60. 32 kg masis da 1,2 m radiusis borbali 280 br/wT siCqariT brunavs. 15 wamis Semdeg borbali unda gaCerdes. a) ra muSaoba unda Sesruldes borblis gaCerebaze? b) ra saSualo simZlavrea saWiro borblis gaCerebisTvis? ·61. 0,75 m sigrZis da 0,42 kg masis Zelaki erTi boloTia Camokidebuli. Semdeg mis meore bolos ubiZgeben da Zelaki qanqarasaviT moZraobs. moZraobis udables wertilze Zelakis kuTxuri siCqarea 4 radiani/wm. xa xuni da haeris winaRoba ugulebelyaviT da a) Zelakis kinetikuri siCqare udables wertilze gamoTvaleT. b) am wertilidan ra manZilze adis masis centri? ··62. 10.33 suraTze, 0,01 kg masis sami nivTieri wertili L = 6 sm sigrZis da umniSvnelo masis sur. 10.45. amocana 56. diT gamoiTvleba. 0,5kg masis da 2sm radiusis diskos firfitas awebeben da mis centrs Zelakzea damagrebuli. Zelaki mis erT boloze da O wertilze gamavali perpendikularuli RerZis garSemo brunavs. ra muSaobaa saWiro, raTa Zelakis brunvis tempi a) 0-dan 20 radia- O wertils amTxveven (sur. 10.45b). Semdeg diskos garedan Toks Semoaxvenen da 5 wamis ni/wm-mde, b) 20 radiani/wm-dan 40 radia- ganmavlobaSi am Toks qaCaven. Sedegad disko ni/wm-mde Seicvalos? d) ra simrude aqvs Zela kis kinetikuri energiis brunvis tempis kvad ratze damokidebulebis funqciis grafiks? da firfita 0,4n Zalis moqmedebis Sedegad brunavs. es Zala diskos kidis mxebia. kuTxuri siCqare 114radiani/wm-ia. ra brunviTi inercia aqvs firfitas RerZis garSemo? ···57. gorgolaWis brunviTi inercia RerZis garSemo aris 1×10-3kg·m2 da misi radiusia 10sm. masze fersos mxebi Zala moqmedebs da misi sidide droSi F=0,5t+0,3t2 formuliT mixedviT icvleba, sadac F mocemulia niutonebSi da t - ni/wm-mde da g) 40 radiani/wm-dan 60 radia- ··63. metrebSi dagraduirebuli saxazavi erT boloze vertikalurad gaaswores da Semdeg xeli gauSves. gamoTvaleT saxazavis meore bolos siCqare zedapirze dacemamde, Tu davuSvebT, rom saxazavis meore bolo zedapirze ar curdeba (miniSneba: saxazavi viwro Zelakad miiCnieT da energiis Senaxvis principi miusadageT). SSM ILW 359 ··64. 10sm radiusis da 20kg masis cilindri centraluri gaswvrivi RerZidan 5sm-iT daSorebuli paraleluri RerZis garSemo brunavs. a) ra brunviTi inercia aqvs cilindrs brunvis RerZis garSemo? b) Tu cilindri moZraobas maSin iwyebs, rodesac misi centraluri gaswvrivi RerZi brunvis RerZis simaRlezea, ra iqneba cilindris kuTxuri siCqare udables mdebareobaze? radiusis gorgolaWze gadis. Tokze m = 0,6 kg masis patara sagani Camokidebuli. gorgolaWis RerZze xaxunis Zala ar moqmedebs da Toki ar wydeba. ra siCqariT moZraobs sagani, rodesac 82 sm-iT dabla daiwevs? ganixileT energiebi. damatebiTi amocanebi da 68. borblis kuTxuri aCqarebaa α = 6t4 – 4t2, sadac α mocemulia radiani/wm2-ebSi, xolo t - wamebSi. rodesac t = 0, borblis kuTxuri siCqarea +2 radiani/wm da kuTxuri mdebareobaa +1 radiani. CawereT a) kuTxuri siCqare da R = 0,15 m radiusis) b) kuTxuri mdebareoba drois funqciis saxiT. da radialuri Ze 69. 10.48 suraTze, 6,2 kg masis ori bloki 2,4 sm radi usis da 7,4×10–4kg·m2 ··65. 10.46 suraTze naCvenebia sgan (m ferso masis lakisgan (m masis sur. 10.46. amocana 65. da L = 2R sigrZis) Semdgari sistema. sistema Zelakis da fersos sibrtyeSi mdebare da Zelakis qveda boloze gamavali brunvis RerZis garSemo brunavs. davuSvaT sistemisTvis biZgis miniWebis dros gada cemuli energia umniSvneloa. ra iqneba sistemis kuTxuri siCqare brunvis RerZis garSemo, rodesac is amotrialdeba? ···66. 55m sigrZis Zelaki maRlidan vardeba. rodesac Zelaki vertikalisadmi 35° kuTxes adgens, risi tolia misi a) kuTxuri siCqare, b) Zelakis wveris radialuri aCqareba da g) Zelakis wveris mxebi aCqareba (miniSneba: gamoiyeneT energia da ara mabruni momenti). brunviTi inerciis mqone gorgolaWze sur. 10.48. amocana 69. gadakiduli, masis armqone TokiTaa gadabmuli. Toki ar wydeba. magidasa da mcurav bloks Soris xaxuni an arsebobs an ara, xolo gorgolaWis RerZze xaxuni ar moqmedebs. rodesac sistema moZraobas iwyebs, gorgolaWi 91 mwm ganmavlobaSi 1,3 radianiT brundeba da blokebis aCqareba mudmivia. ra a) kuTxuri aCqareba aqvs gorgolaWs, b) ra aCqareba aqvs blokebs, risi tolia g) T1 Tokis daWimuloba da b) T2 Tokis daWimuloba? d) ra θ kuTxis dros aris mxebi aCqareba g-s toli? 70. rensleris politeqnikuri institutis ···67. M = 4,5 kg masis da R = 8,5 sm radiusis gton jeil feris umcrosma, 1893 wels, CikagoSi eSmakis borbali aago. im droisTvis sferuli garsi xaxunisgan Tavisufal sa yrdenze damagrebuli vertikaluri RerZis garSemo brunavs (sur. 10.47). masis armqone Toki garsis ekvatorzea Semoxveuli da I = 3×10–3 kg·m2 brunviTi inerciis da r = 5 sm kursdamTavrebulma, inJinerma jorj vaSin gasaocari masStabis atraqcioni 36 xis kabinisgan Sedgeboda, romelic 60 kacs itevda, xolo atraqcionis wris radiusi 76 m iyo. 36ve kabinis gavsebis Semdeg eSmakis borbali erT srul bruns 2 wuTis ganmavlobaSi mudmivi kuTxuri siCqariT asrulebda. gamoTvaleT mgzavrebiT savse kabinebis dros atraqcionis mier Sesrulebuli muSaoba. 71. 10.49 suraTze, r = 2 sm radiusis pa tara disko R = 4 sm radiusis sur. 10.47. amocana 67 mozr dil diskozea miwe bebuli. diskoebi sur. 10.49. amocana 71. 360 Tavi 10 brunva erT sibrtyeSia ganla-gebuli da mozrdili 74. WurWlis Tavsaxuris gasaxsnelebma sasme diskis centralur O wertilze gamavali per-pendikularuli RerZis garSemo brunavs. lis WurWlis dizaini bevrad gaaumjobesa. orive diskos 1,4×10 kg/m simkvrive da 5 mm centraluri sisqe aqvs. ra brunviTi inercia aqvs diskebs O wertilze gamavali brunvis RerZis garSemo? romelic WurWlis Tavsaxurzea damagrebuli. 72. meore bolo WurWlis Tavsaxurs dabla awveba. 3 3 10.50 suraTze naCvenebia saerTo centris mqone ori wriuli rgolisgan Semdgari konstruqcia. rgolebi erTmaneTze sur. 10.50. amocana 72. umniSvnelo masis sami ZelakiTaa gadabmuli. konstruqcia saerTo centris garSemo brunavs da am centrze umniSvnelo masis kidev erTi Zelakia ganlagebuli. qvemoT mocemul cxrilSi naCvenebia rgolebis masebi, Sida radiusebi da gasaxsneli WurWlis Tavsaxurze damagrebuli WanWikis garSemo brunavs, rodesac gasaxsnelis erT bolos maRla wevT, Tu maRla biZgi 10n-ia, daaxloebiT ra Zala moqmedebs WurWlis Tavsaxurze? 75. morbenali cxovelebis gaswvriv manqaniT moZravma damkvirveblebma daadgines, rom gepardebis maqsimaluri siCqarea 114km/sT (daaxloebiT 71mili/sT). warmoidgineT, rom gepards manqaniT gverdze misdevT da Tan spidometrs uyurebT, romelic 114km/sT siCqares aCvenebs. manqanasa da gepards Soris manZilia 8m. manqanis xmauri gepards aSinebs da gare radiusebi. gare rgolis kideze 12 n mxebi is manqanisgan moSorebiT 92m radiusis wriuli Zala 0,3 wamis ganmavlobaSi moqmedebs. rogor icvleba konstruqciis kuTxuri siCqare drois am SualedSi? traeqtoriiT moZraobs. maSasadame, Tqveni rgoli masa (kg) Sida radiusi (m) 1 2 0,12 0,24 0,016 0,09 gare radiusi (m) 0,045 0,14 73. 1908 wlis 30 ivniss, saRamos 7 saaTsa da 14 wuTze, satelefono kavSiris kvanZ siberiis (61° ganedsa da 102° grZedze) Tavze afeTqeba moxda. warmoqmnili cecxlovani burTi birTvuli iaraRis gamogonebamde yvelaze ufro mkveTr naTebas qmnida, rac ki kacobriobas enaxa. tunguskis movlenam `cis udidesi nawili dafara~ da savaraudod daaxloebiT 140m siganis qviani asteroidis afeTqeba iyo. a) mxolod dedamiwis brunva ganixileT da gansazRvreT, ra drois Semdeg unda gamoCeniliyo asteroidi, raTa afeTqeba helsinkis Tavze, 25° ganedze momxdariyo (amgvari asteroidis Camovardna qalaqs saerTod gaaqrobda). b) asteroidi dedamiwaze Camovardeboda, liTonis rom yofiliyo. ra drois Semdeg unda gamoCeniliyo asteroidi, raTa atlantikis okeaneSi, 20° grZedze manqana 100m radiusis wriul traeqtorias Semowers. a) ra kuTxuri siCqare aqvs manqanas da gepards wriuli traeqtoriebis garSemo? b) ra wrfivi siCqare aqvs gepards? (Tu wriul moZraobas mxedvelobaSi ar miiRebT, gepardis siCqare 114km/sT gegonebaT. swored amgvari Secdoma dauSves zemoT xsenebulma damkvirveblebma). 76. diskebis rbo la. 10.51 suraTze naCvenebia tris cen garSemo mrbunavi disko. t = 0, ori rodesac sur. 10.51. amocana 76. diskoebis aTvlis wrfeebis mimarTulebebi erTmaneTs emTxveva. am dros A disko 9,5 radiani/wm mudmivi kuTxuri siCqariT ukve brunavs, xolo B disko 2,2 radiani/wm2 mudmivi kuTxuri aCqarebiT moZraobas axla iwyebs. a) ra t dros eqnebaT diskoebis aTvlis wrfeebs Tanabari θ kuTxuri gadaadgileba? b) iqneba Tu ara es pirveli t dro (t = 0 drois Semdeg), rodesac aTvlis wrfeebis mimarTulebebi erTmaneTs daemTxveva? Camovardniliyo? (Sedegad miRebuli cunami 77. 10.52 suraTze gamosaxuli myari sxeuli atlantikis okeanis orive mxareze arsebul sami civilizacias gaaqrobda). Zelakisgan burTisa da Sedgeba. sami damakavSirebuli M = 1,6 kg, L = 0,6 m 361 da θ = 30°. burTebi nivTier wertilebad da d) SekiTxvebze pasuxis gacema damatebiTi SegviZlia ganvixiloT, xolo Zelakebis masebi informaciis gareSe? umniSvneloa. gamoTvaleT sxeulis brunvis 80. vertmfrenis rotoris frTa 7,8m sigrZis da 100kg masisaa da rotoris RerZTan erTi kinetikuri energia, Tu mas 1,2 radiani/wm kuTxuri siCqare a) P wertilze gamavali WanWikis da a) ra Zala moqmedebs RerZidan WanWikze, suraTis sibrtyis perpendikularuli RerZis garSemo aqvs da igive siCqare b) P wertilze gamavali, 2L sigrZis Zelakis saSualebiT aris damagrebuli. rodesac rotori 320br/wT kuTxuri siCqariT brunavs? (miniSneba: gamoTvlebis dros frTa perpendikularuli da suraTis sibrtyeSi wertilovan mdebare RerZis garSemo aqvs. b) gamoTvaleT rotorze modebuli Zalis masad SegviZlia miviCnioT) momenti, romelic mas srul siCqares 6,7 wamis ganmavlobaSi SesZens (amjerad frTas wertilovan masad ver miviCnevT. ratom?) g) ra muSaobas asrulebs Zalis momenti frTaze, raTa frTam 320br/wT kuTxuri siCqare SeiZinos? 81. 10.54 suraTze oTxi disko ori RvediTaa gadabmuli. A disko (15sm radiusis) amZravia da 10 radiani/wm kuTxuri siCqariT brunavs. B disko (10sm radiusis) pirveli Rvedis saSualebiT A diskTanaa dakavSirebuli. B’ disko (5sm radiusis) B-s koncentrulia da masze xistadaa mimagrebuli. C disko (25sm radiusis) meore Rvedis saSualebiT B’ sur. 10.52. amocana 77. 78. 10.53 suraTze, m1 = 400 g masis da m2 = 600 g masis ori bloki masis armqone TokiTaa erTmaneTze gadabmuli. M = 500 g masis da R = 12 sm radiusis Toki diskozea diskosTanaa dakavSirebuli. gamoTvaleT a) Semoxveuli sur. 10.53. amocana 78. horizontaluri pirveli Rvedis wertilis wrfivi siCqare, b) B diskos kuTxuri siCqare, g) B’-is kuTxuri da mis centrze gamavali RerZis garSemo xaxunis siCqare, d) meore Rvedis wertilis wrfivi gareSe brunavs. gamoTvaleT a) blokebis siCqare da e) C diskos kuTxuri siCqare. aCqareba, b) Tokis T1 daWimuloba marcxniv da g) Tokis T2 daWimuloba marjvniv. (miniSneba: Tu diskoebis damakavSirebeli Rvedi ar wydeba, maSin maTi fersoebis wrfivi 79. 1,2 m sigrZis da 6,4 kg masis foladis siCqareebi Tanabaria). Zelakis erT boloze 1,06 kg masis patara burTia damagrebuli. Zelaki mis Sua wer tilze gamavali vertikaluri RerZis garSemo, horizontalur sibrtyeSi brunavs. garkveul momentSi Zelakis kuTxuri siCqare 39 br/wm. xaxunis Sedegad Zelaki neldeba da 32 wamis Semdeg Cerdeba. davuSvaT xaxunis Sedegad Semanelebeli mudmivi Zalis momenti moqmedebs. gamoTvaleT a) kuTxuri aCqareba, b) Semanelebeli Zalis momenti, g) xaxunis Sedegad meqanikuri energiidan siTbur energiaSi gadasuli energia da d) 32 wamis gan sur. 10.54 amocana 81 mavlobaSi Sesrulebuli brunis raodenoba. 82. golfis burTi horizontalisadmi 20° kuTxiT, 60 m/wm siCqariT da 90 radiani/wm e) davuSvaT Semanelebeli Zalis momenti kuTxuri siCqariT isroles. haeris wevis Zala mudmivi ar aris. SesaZlebelia Tu ara a), b), g) ugulebelyaviT da burTis mier Sesrule 362 brunva Tavi 10 buli brunebis ricxvi maqsimaluri simaRlis ganivi koWebis saSualebiT miwaze gamaval miRwevamde gamoTvaleT. myar Zelakzea damagrebuli (sur. 10.57). ziani 83. 0,2m radiusis nacionaluri safexburTo ligis ukanaxazels borbali xaxunis miadga, rodesac is erT-erT skamze dajda. Tavisufal, moTamaSis TqmiT man fexebi da muceli magidasa horizontalur da skams Soris moaqcia da magidas daeyrdno. gan RerZzea damagre buli. borblis brunviTi sur. 10.55. amocana 83. Semdeg is skamze dajda da gasworda, ramac mTeli magidis amobruneba gamoiwvia. moTamaSe inercia RerZis garSemo aris 0,05kg·m . masis armqone Toki borbalzea Semoxveuli da masze 2kg masis blokia 2 miwaze daeca da Tavi daartya. suraTze gamosaxuli sigrZeebia d1 = 60 sm, mimagrebuli, romelic xaxunisgan Tavisufal, d2 = 13 sm, d3 = 5 sm da d4 = 18 sm. magidis masaa 90 kg. moTamaSis masaa 130 kg da rodesac is horizontalur zedapirze gadaadgildeba. skamze dajda, misi masis centri skamis gare Tu P=3n horizontaluri Zala blokze 10.55 kididan d5 = 20 sm manZiliT gareT iyo. davuS suraTis msgavsad moqmedebs, ra kuTxuri vaT mizidulobis Zala moTamaSis masis cent- aCqareba rze moqmedebda. a) SesaZlebelia Tu ara magidis aqvs borbals? davuSvaT, rom borbalze Semoxveuli Toki ar wydeba. amobruneba? b) daaxloebiT ra minimaluri 84. myari sxeuli L=0,6m sigrZis masaa saWiro brunvis gamowvevisTvis? sami identuri Ze lakisgan Sedgeba. H-is Zelakebs sur. 10.56. amocana 84. forma aqvT. sxeu li H-is erT-erTi fexis gaswvriv gamavali horizontaluri RerZis garSemo Tavisuflad brunavs. Semdeg sxeuli im adgilidan vardeba, sadac H-is sibrtye horizontaluria. ra sur. 10.57. amocana 86. kuTxuri siCqare eqneba sxeuls, rodesac H-is sibrtye vertikaluri iqneba? 87. or myar sferos toli, 1,65kg masa aqvs, xolo maTi radiusebia 0,226m da 0,854m. 85. a) daamtkiceT, rom M masis da R radiusis TiToeuli sfero centrze gamavali RerZis myari cilindris brunviTi inercia garSemo brunavs. a) ra τ Zalis momentia erTmaneTis saWiro, raTa patara sfero 15,5 wamis Semdeg 317 radiani/wm kuTxuri siCqariT moZraobdes? b) ra F Zala unda moqmedebdes sferos ekva tolia. b) daamtkiceT, rom M masis sxeulis I torze, raTa masze amgvarma mabrunma momentma centraluri RerZis garSemo da M masis da R / 2 radiusis fersos brunviTi inercia centraluri RerZis garSemo brunviTi inercia nebismieri RerZis garSemo eqvivalenturi fersos brunviTi inerciis tolia igive RerZis garSemo, Tu fersos igive M masa aqvs da misi k radiusia: k= imoqmedos? ra mniSvnelobebi aqvs g) τ-s da d) F-s ufro didi sferos SemTxvevaSi? 88. 10.58 suraTze, 0,2m radiusis bor bali xaxunisgan Tavisufal, hor eqvivalenturi fersos k radiuss mocemuli izontalur Rer sxeulis brunvis radiusi ewodeba. Zzea damagrebuli. 86. sapiknike magidam adamians ziani miayena borbalze gadaxveul, masis armqone Tokze da Tqven magidis mwarmoeblis winaaRmdeg 2 kg masis yuTia sur. 10.58. amocana 88. mimagrebuli, romelic eqspertad dagiqiraves. sapiknike magida Tavad horizontalisadmi θ = 20° kuTxiT daxril, magidisa da ori skamisgan Sedgeba. skamebi xaxunisgan Tavisufal zedapirze gadaad 363 gildeba. yuTi 2m/wm2 aCqarebiT qvemoT srial mdebareobs. a) ra sxvaobaa A wertilis a deba. ra brunviTi inercia aqvs borbals centriskenul aCqarebas da 0,15m radialur RerZis garSemo? manZilze mdebare wertilis centriskenul 89. 10.59 suraTze ga aCqarebas Soris? g) gamoTvaleT a aCqarebis mosaxuli erTgvarovani radialur manZilze damokidebulebis fun Zelakis sigrZea 2m. Ze qciis grafikis simrude. laki mis erT boloze 93. 1,3kg masis patara burTi 0,78m sigrZis gamavali, xaxunisgan Ta da umniSvnelo masis Zelakis erT boloze visufali RerZis gar Semo brunavs. sur. 10.59. amocana 89. daamagres. miRebuli sistema Zelakis meore bolos garSemo horizontalur wreze 5010 br/ Zelaki moZraobas horizontalidan maRla, θ = 40° wT kuTxuri siCqariT brunavs. a) gamoTvaleT kuTxiT iwyebs. gamoiyeneT energiis Senaxvis sistemis brunviTi inercia brunvis RerZis principi da horizontaluri mdebareobis garSemo. b) burTze 2,3×10–2 n haeris wevis dros Zelakis kuTxuri siCqare gamoTvaleT. Zala moqmedebs da moZraobas ewinaaRmdegeba. 90. 0,2kg masis oT ra sididis Zalis momenti unda moqmedebdes xi nivTieri wertili sistemaze, raTa man mudmivi siCqariT brunva kvadratis wveroebzea ganagrZos? ganlagebuli. kvadra 94. io-ios gareg tis gverdis sigrZea nuli formis mo 0,5 m. nivTieri wer wyobiloba tilebi erTmaneTTan nisgan Tavisufal, masis armqone Zela kebiT aris dakav sur. 10.60. amocana 90. xaxu h o r i z o n t a l u r RerZzea damagre Sirebuli. miRebuli sxeuli horizontaluri buli da 30kg masis RerZis yuTis garSemo vertikalur sibrtyeSi asawevad moZraobs. RerZi A wertilze gadis. sxeuli gamoiyeneba moZraobas iwyebs, rodesac AB Zelaki hori 10.62). zontaluradaa ganlagebuli (sur. (sur. mowyobi 10.60). lobis R gare ra sur. 10.62. amocana 94. a) ra brunviTi inercia aqvs sxeuls A RerZis diusia 0,5 m, xolo garSemo? b) ra kuTxuri siCqare aqvs sxeuls Tvlis morgvis r radiusia 0,2 m. rodesac A RerZis garSemo, rodesac AB mowyobilobaze Semoxveul Tokze 140 n Zala Zelaki vertikalur mdgomareobas iRebs? moqmedebs, Tvlis morgvze gadaxveul Tokze 91. Zravis mqnevara 25 radiani/wm kuTxuri Camokidebul yuTs 0,8m/wm2 maRla mimarTuli siCqariT brunavs. rodesac Zrava gamorTulia, aCqareba aqvs. ra brunviTi inercia aqvs mqnevara mudmivi tempiT neldeba da 20 wamis mowyobilobas brunvis RerZis garSemo? Semdeg Cerdeba. gamoTvaleT a) mqnevaras 95. t = 0 kuTxuri aCqareba, b) gaCerebis dros mqnevaras mudmiv kuTxur aCqarebas iZens. rodesac mier Semowerili kuTxe da g) mqnevaras mier Sesrulebuli brunebis ricxvi gaCerebamde. t = 2 wm, borblis kuTxuri siCqarea 5 radiani/wm. aCqareba grZeldeba t = 20 wm-mde da 92. 10.61 suraTze naCvenebia B wertilze gama Semdeg swrafad mcirdeba. ra kuTxes Semowers vali suraTis sibrtyis perpendikularuli RerZis garSemo 2000br/wT kuTxuri siCqariT mbrunavi propeleris frTa. A wertili frTis wvers emTxveva da 1,5m radialur manZilze sur. 10.61. amocana 92. droidan moyolebuli borbali borbali 0-dan 40 wm-mde drois SualedSi? 96. firsakravis kuTxuri siCqare gamorTvis momentSi aris 8 radiani/wm. sami wamis Semdeg firsakravis kuTxuri siCqarea 2,6 radiani/wm. ramden radians Semowers firsakravi gamorTvidan gaCerebamde? (α mudmivia). 364 brunva Tavi 10 97. t = 0 dros, gamorTvis momentSi, firsakravis kuTxuri siCqarea 3,5 radiani/wm. ro desac t = 1,6 wm, firsakravi Cerdeba. fir sakravis radiusia 15 sm. a) Tu kuTxuri aCqareba 101. 10.64 suraTze, 0,2m radiusis bor mudmivia, ramden radians Semowers firsakravi zontalur t = 0 drois Semdeg? b) ra wrfivi aCqareba aqvs firsakravis fersoze mdebare wertils t = 1 wm dros? (miniSneba: wrfiv aCqarebas mxebi da radialuri komponentebi aqvs). xaxunisgan Tavisufal, hori RerZzea damagrebuli. RerZis garSemo borblis brunviTi inerciaa 0,4kg·m2. 98. ori gadabmuli masis borbalze armqone sur. 10.64. amocana 101. To kia Semoxveuli da masze 6kg masis yuTia Zelaki (TiToeuli 0,2 kg masis) 10.63 suraTze bali Camokidebuli. sistema moZraobas iwyebs. gamosa rodesac yuTis kinetikuri energiaa 6j, a) risi xul myar sxeuls tolia borblis brunvis kinetikuri energia qmnis. da b) ra manZiliT Caiwia dabla yuTma? erT-erTi Zelakis sigrZea L1 = 0,4 m, meore 102. 10.65 suraTze, 2kg masis da 3m sigrZis xolo Zelaki Zelakis sigrZea horizontaluri RerZis garSemo Tavisuflad brunavs. horizontaluri RerZi L2 = 0,5 m. sur. 10.63. amocana 98. ra brunviTi ine rcia aqvs sxeuls a) saxelmZRvanelos gverdis Zelakis perpendikularulia da misi erTi bolodan d=1m manZiliTaa daSorebuli. Zelaki sibrtyisadmi perpendikularul da mokle moZraobas horizontaluri mdebareobidan Zelakis centrze gamavali RerZis garSemo iwyebs. a) ra iqneba misi maqsimaluri kuTxuri da b) saxelmZRvanelos gverdis sibrtyisadmi siCqare? b) Tu Zelakis masas gavzrdiT, perpendikularuli da grZeli Zelakis cen a)-s pasuxi gaizrdeba, Semcirdeba Tu ar trze gamavali RerZis garSemo? Seicvleba? 99. 0,5 kg masis sami nivTieri wertili 0,6 m sigrZis gverdis mqone tolgverda samkuTxeds qmnis. nivTieri wertilebi umniSvnelo masis ZelakebiTaa erTmaneTTan dakavSirebuli. ra brunviTi inercia aqvs miRebul sxeuls a) erT- sur. 10.65. amocana 102. erT nivTier wertilze gamavali da danarCeni 103. 35g masis nivTieri wertili firsakravis centridan 3,5sm manZilze moaTavses. statikuri xaxunis koeficientia 0,4. firsakravis siCqare nel-nela ω0-mde izrdeba da am dros nivTieri wertili srialdeba. a) gamoTvaleT ω0. b) ga ori nivTieri wertilis damakavSirebeli Zelakis paraleluri RerZis garSemo, b) erTerTi gverdis Sua wertilze gamavali da samkuTxedis sibrtyis perpendikularuli RerZis garSemo da g) samkuTxedis erTi gverdis paraleluri da danarCeni ori gverdis Sua wertilebze gamavali RerZis garSemo? 100. xuTi nivTieri wertili xy sibrtyeSia ganlagebuli, myar sxeuls qmnis da maTi monacemebi cxrilSia mocemulia. ra brunviTi inercia aqvs sxeuls a) x, b) y da g) z RerZis garSemo? d) ra aris sxeulis masis centri? sagani masa (g) x (sm) y (sm) 1 500 15 20 2 400 –13 13 3 300 17 –6 4 600 –4 –7 5 450 –5 9 mosaxeT srialis traeqtoria. 104. 10.66 suraTze gamosaxuli myari sxeuli masis ar mqone ZelakebiT d a k a v S i r e b u l i sami nivTieri we rtilisgan Sedge ba. sxeuli P wer tilze gamavali da sibrtyis misi sur. 10.66. amocana 104. perpendikularuli RerZis garSemo brunavs. 365 Tu M = 0,4 kg, a = 30 sm da b = 50 sm, ra muSao- gamocades. 5124 kg masis betonis bloki baa saWiro sxeulis kuTxuri siCqaris 5 ra muxis or Zelis saSualebiT maRla aitanes. diani/wm-mde gazrdisTvis? Zelebis zedapirebi sapoxiT iyo dafaruli 105. disko θ1 = 10 radiani kuTxuri mde bareobidan θ2 = 70 radiani kuTxur mdeba reobamde mudmivi kuTxuri aCqarebiT 6 wamSi brunavs. θ2 mdebareobaze diskos kuTxuri siCqarea 15 radiani/wm. a) ra iqneba diskos siCqare θ1 mdebareobaze? (sur. 10.67). 10 m sigrZis Zelebi miwidan or svetze iyo gadebuli, romlebzec bloki unda moeTavsebinaT. svetebis simaRle 3,9 m iyo, xolo bloksa da Zelebs Soris statikuri xaxunis koeficienti 0,22-s udrida. bloks masze da naZvis 4,5 m sigrZis or morze diskos? Semoxveuli Tokis saSualebiT ubiZgebdnen. g) ra kuTxur mdebareobazea disko uZravi? rodesac platformaze sakmarisi raodenobis d) aageT moZraobis dawyebis Semdeg diskos θ muSa dgeboda an jdeboda, naZvis mori svetis mdebareobis t droze damokidebulebis da ω wveris kuTxuri siCqaris t droze damokidebulebis Zelze gadaaadgildeba. blokis damWeri Toki funqciebis grafikebi. daaxloebiT morebis perpendikularuli iyo, 106. Cveni mze irmis naxtomis galaqtikis centridan 2,3×104 sinaTlis weliwadiTaa daSorebuli da am centris garSemo 250 km/wm xolo brunvis wertilsa da moris garSemo b) ra kuTxuri aCqareba aqvs siCqariT wrewirze brunavs. a) ra droSi Semouvlis mze galaqtikis centrs? b) ram deni bruni Seasrula mzem, mas Semdeg rac garSemo trialdeboda da bloks Tokis Semoxvevis wertils Soris manZili 0,7m-s aRwevda. davuSvaT, TiToeuli muSis masa 85 kg iyo, ra minimaluri raodenobis muSa iqneboda saWiro, raTa bloks Zelebze maRla asvla daewyo? daaxloebiT 4,5×109 wlis winaT warmoiSva? 107. firsakravi 33 13 br/wT siCqariT brunavs, neldeba da Zravis gamorTvis Semdeg 30 wamSi Cerdeba. a) gamoTvaleT kuTxuri aCqareba (br/wT2-ebSi). b) ramden bruns Seasrulebs bloki naZvis xis mori firsakravi 30 wamis ganmavlobaSi? 108. sferuli garsis radiusia 1,9 m. 960 n·m muSa Zalis momenti garss centrze gamavali RerZis platforma garSemo 6,2 radiani/wm2 kuTxur aCqarebas sveti aniWebs. a) ra brunviTi inercia aqvs garss am muxis Zeli RerZis garSemo da b) ra masa aqvs garss? sur. 10.67. amocana 111. 109. borbali α = 4at – 3bt kuTxuri aCqarebiT brunavs, sadac t droa, xolo a da b mudmivebi. Tu borblis sawyisi kuTxuri siCqarea ω0 , 3 2 a) misi kuTxuri siCqare da b) kuTxuri mde 112. molekulis brunviTi inerciaa 14000 u·pm2, sadac u masis atomuri erTeulia (1 u = 1,66×10–27 kg). Tu molekula 4,3×1023 radiani/wm kuTxuri siCqariT brunavs, ra iqneba misi bareoba drois funqciis saxiT CawereT. brunvis kinetikuri energia? 110. t drois SualedSi generatoris mqnevara θ = at + bt3 – ct4 kuTxiT brundeba, sadac a, b da c mudmivebia. dawereT borblis 113. 70 kg masis velosipedisti pedlebs mTeli masiT awveba da aRmarTze adis. davuSvaT a) kuTxuri siCqaris da b) kuTxuri aCqarebis ra maqsimalur Zalis moments warmoSobs gamosaxuleba. velosipedisti? 111. meTodi, romliTac stounhenjis say 114. Tavdapirvelad uZravi borbali mudmivi rden qvebze masiuri qvebi daalages, didi kuTxuri aCqarebis Sedegad brunvas iwyebs. xnis ΔT = 15 wm ganmavlobaSi SesaZlo meTodi iyo sadavo. CexeTis erT-erTi patara qalaqSi pedlebi 0,4 m radiusis drois wreze SualedSi brunavs, borbali 90 bruns asrulebs da misi kuTxuri siCqare 366 brunva Tavi 10 10 br/wm xdeba. a) ra kuTxuri siCqare hqonda borbals ΔT drois Sualedis dawyebisas? b) ra dro gavidan brunvis dawyebasa da ΔT drois 120. 10.69 suraTze Sualedis dawyebas Soris? gzavri, 115. 0,75m diametris salesi qvis fersoze naCvenebia sako munikacio Tanam romelic 1210kg masis, 1,21m ganlagebuli wertilis siCqare mudmivi tempiT diametris da 1,75m 12m/wm-dan 25m/wm-mde 6,2 wamis ganmavlobaSi sigrZis cilindria. icvleba. ra saSualo kuTxuri aCqareba aqvs kosmosuri borbals? didan gamoSvebamde 116. 10.68 suraTze, Tanamgzavri erTgvarovani 4m da 1,52br/wm kuTxuri siCqariT Ze sakuTari grZeli RerZis garSemo brunavs. ra masis a) brunviTi inercia aqvs Tanamgzavs brunvis sigrZis) RerZis garSemo da b) risi tolia Tanamgzavris (3kg laki sur. 10.69. amocana 120. xomal misi bolodan brunvis kinetikuri energia? d=1m manZiliT 121. vinilis firfita firsakravze 33 13 br/wT daSorebul tilze wer kuTxuri siCqariT brunavs. a) ra kuTxuri gamavali siCqare aqvs firfitas radiani/wm-ebSi? ra da Zelakis perpen A dikularuli h o r i z o n t a l u r i RerZis wrfivi sur. 10.68. amocana 116. garSemo siCqare aqvs firfitis wertils, romelic firsakravis RerZidan b) 15 sm da g) 7,4 sm manZilzea. mareobaSi Zelakis kinetikuri energiaa 20 j. 122. ra kuTxuri siCqare aqvs 110 m radiusis wrewirze 50 km/sT siCqariT moZrav manqanas? a) ra brunviTi inercia aqvs Zelaks A RerZis 123. O2 Jangbadis molekulis masaa 5,3×10–26 kg, garSemo? b) ra wrfivi siCqare aqvs Zelakis xolo atomebis SemaerTebeli wrfis centrze B vertikalur gamavali da am wrfis perpendikularuli mdgomareobaSia? g) ra θ kuTxeze gaCerdeba RerZis garSemo molekulis brunviTi inerciaa Zelaki wamierad maRla asvlis dros? 1,94×10–46 kg·m2. davuSvaT O2 molekulis masis centri airSi 500 m/wm gadataniTi siCqariT Tavisuflad bolos, brunavs. rodesac vertikalur Zelaki mdgo 117. manqana 30 m radiusis wriul trasaze moZraobs. misi siCqare 0,5 m/wm2 mudmivi moZraobs da molekulis brunvis kinetikuri tempiT izrdeba. a) ra jamuri wrfivi aCqareba energia masis centris gadataniTi kinetikuri aqvs manqanas 15 wamis Semdeg? b) ra kuTxes energiis 2/3-ia. ra kuTxuri siCqare aqvs adgens jamuri aCqarebis veqtori manqanis molekulas masis centris garSemo? siCqarisadmi am dros? 124. warmoidgineT, rom uZravi dedamiwa erTi 118. 8 sm diametris borbalze 5,6 m sigrZis Tokia Semoxveuli. borbali 1,5 radiani/wm2 dRis ganmavlobaSi RerZis garSemo moZraobas mudmivi kuTxuri aCqarebiT iwyebs moZraobas. iZens. ra sididis a) Zalis momenti, b) energia a) ra kuTxiT unda Semobrundes borbali, da g) saSualo simZlavrea saWiro dedamiwis raTa Toki mTlianad gaiSalos? b) ra dro dasWirdeba amas? 119. mZime mqnevara centraluri RerZis garSemo brunavs da sayrdeni wertilebze xaxunis Sedegad neldeba. Senelebis pirveli wuTis miwuruls, mqnevaras sawyisi 250br/wT kuTxuri siCqaris 0,9 nawili aqvs darCenili. davuSvaT kuTxuri aCqareba mudmivia, ra kuTxuri siCqare eqneba mqnevaras meore wuTis miwuruls? iwyebs da dRes arsebul kuTxur siCqares aCqarebisTvis? 125. davuSvaT, dedamiwa erTgvarovani simkvrivis mqone sferoa. gamoTvaleT misi a) brunviTi inercia da b) brunvis kinetikuri energia. g) davuSvaT, am energiis gamoyeneba SesaZlebelia. daaxloebiT ra drois gan mavlobaSi SeZlebs dedamiwa TiToeuli adamia nisTvis energiis 1 kvt tempiT miwodebas? 11 gorva, Zalis momenti da kuTxuri impulsi 1997 wels, nevadas Savi kldis udabnoSi manqana Thrush SSC-m dedamiwaze moZraobis siGqaris rekordi daamyara. manqanis siGqare erTi mimarTulebiT 1222km/sT iyo, xolo sawinaaRmdego mimarTulebiT - 1233km/sT. orive Sedegma bgeris siGqares (1207km/sT) gadaaWarba da manqana ultrabgeriTi gaxda, rasac manqanis Tavze gamoGenili mrudebi cxadyofs. mrudebis gaGena manqanis mier haerSi warmoqmnilma dartymiTma talRebma gamoiwvies. cxadia, amgvari rekordis damyareba did safrTxeebTan iyo dakavSirebuli. erT-erTi maTgani manqanis borblebs ukavSirdeba. ra safrTxe eqmneboda manqanis borblebs? pasuxs am TavSi SeityobT 11.1. ra aris fizika? me-10 TavSi ukve vTqviT, rom fizika brunvas Seiswavlis. SesaZlebelia fizikis es dargi yvelaze metad borblebis da borblis magvari sagnebis gorvis dros iyos gamosadegi. magaliTad, aRdgomis kunZulis preistoriuli xanis mcxovreblebi qvis gigantur qandakebebs mgorav Zelebze awyobdnen da ise gadahqondaT kunZulis teritoriaze. mogvianebiT, rodesac es xalxi CrdiloeT amerikaSi bargdeboda, TavianTi qoneba jer urmebiT, xolo Semdgom matarebliT gadahqondaT. dResdReobiT, moswons es vinmes Tu ara, samyaro manqanebiT, motociklebiT, velosipedebiT da sxva mgoravi transportiT savsea. gorvis Seswavla da gamoyeneba Zalian didi xania grZeldeba da, SesaZloa gagiCndeT azri, rom axali ideebi aRar arsebobs. sinamdvileSi, magaliTad, skeitbordebi arcTu ise didi xnis win gamoigones da Seqmnes, ramac didi finansuri warmatebac moitana. quCis toboganebma da TviTmmarTvadma segveebma (sur. 11.1) SesaZloa did qalaqebSi adamianebis gadaadgilebis stili sruliad Secvalos. gorvis maxasiaTeblebis ga moyenebas kvlav SeuZlia axal aRmoCenebamde da gan sacvifrebel Sedegebamde migviyvanos. gorvis Seswavlas misi sur. 11.1. adamianis gadamyvani TviT gamartivebiT viwyebT. marTvadi segvei 367 368 Tavi 11 gorva, Zalis momenti da kuTxuri impulsi 11.2. gorva, rogorc gadataniTi moZraobis da brunvis gaerTianeba am TavSi mxolod iseT sagnebs ganvixilavT, romlebic zedapiris gaswvriv Tanabrad goravs anu zedapirze mocurebis da asxletis gareSe gadaadgildeba. 11.2 suraTze naCvenebia rTuli gorva. wrfeze sagnis centri moZraobs, zedapiris xolo paralelur fersoze mdebare wertilis traeqtoria gansxvavebulia. gorvis Seswavla gamartivdeba, Tu miviCnevT, rom masis centri gadataniT moZraobas asrulebs, xolo sur. 11.2. mgoravi diskos fotografia. diskos centrze da kideze patara naTurebia damagrebuli. diskos kide cikloidad wodebul mruds Semowers. sagnis danarCeni nawili centris garSemo brunavs. warmoidgineT, rom trotuarze dgaxarT da quCis gaswvriv mgorav velosipedis borbals akvirdebiT (sur. 11.3). borblis O masis centri νcom siCqariT wrfivad moZraobs. quCisa da borblis Sexebis P wertili aseve νcom siCqariT wrfivad moZraobs da, maSasadame, P wertili yovelTvis O wertilis qvemoTaa. t drois SualedSi O da P wertilebi s manZils gadis. velosipedistis TvalTaxedviT, borbali centris garSemo θ kuTxiT brunavs, xolo borblis wertili, romelic t drois dasawyisSi quCas exeboda, s sigrZis rkals Semowers. 10.17 formula rkalis s sigrZes da brunvis θ kuTxes erTmaneTTan akavSirebs: sur. 11.3. mgoravi borblis O masis centri com siCqariT s manZils gadis, xolo borbali θ kuTxiTY brundeba. borblisa da zedapiris Sexebis P wertilic s manZilze gadaadgildeba. s = θR , (11.1) sadac R aris borblis radiusi. borblis centris (anu masis centris) νcom wrfivi siCqarea ds/dt. borblis ω kuTxuri siCqare centris garSemo aris dθ/dt. aqedan gamomdinare, 11.1 formulis drois diferencireba (rodesac R mudmivia) gvaZlevs: νcom = ωR (gorviT moZraoba) (11.2) 11.4 suraTze Cans, rom borblis gorva gadataniTi da brunviTi moZraobebis gaerTianebaa. 11.4a suraTze gamosaxulia brunviTi moZraoba (TiTqos centrze gamavali brunvis RerZi uZravia). borblis yvela wertili centris garSemo ω kuTxuri siCqariT brunavs (amgvari moZraoba me-10 TavSi ganvixileT). borblis kidis yvela wertilis wrfivi siCqarea νcom. 11.4b suraTze a) brunviTi moZraoba b) gadataniTi moZraoba g) gorviT moZraoba sur. 11.4. borblis gorva gadataniTi moZraobis da brunvis gaerTianebaa. a) brunviTi moZraoba. borblis yvela wertili Tanabari ω kuTxuri siCqariT moZraobs. borblis kideze mdebare wertilebi Tanabari ν=νcom wrfivi siCqariT moZraobs. suraTze naCvenebia borblis wverze (T) da fskerze (P) mdebare wertilebis wrfivi siCqareebi. b) gadataniTi moZraoba. borblis yvela wertili marjvnisken Tanabari com wrfivi siCqariT moZraobs. . g) a) da b) suraTebze gamosaxuli ori moZraobis gaerTianebaa. 11.3. gorvis kinetikuri energia 369 gamosaxulia gadataniTi moZraoba (TiTqos borbali saerTod ar brunavs). borblis yvela wertili νcom siCqariT marjvnisken gadaadgildeba. 11.4a da 11.4b suraTebis gaerTianeba 11.4g suraTze borblis realur gorvas gvaZlevs. yuradReba miaqcieT, rom borblis qveda nawili (P wertilze) uZravia da borblis zeda nawili (T wertilze) 2νcom siCqariT anu borblis danarCen nawilze swrafad moZraobs. 11.5 suraTze miRebuli Sedegebia naCvenebi. borblis zeda nawili qveda nawilze swrafad moZraobs, vinaidan am nawilis manebi ufro bundovnad moCans. zedapirze mgoravi nebismieri mrgvali sxeulis moZraoba gadataniT moZraobad da brunvad SegviZlia davyoT, rogorc es aris 11.4a da 11.4b suraTebze. sur. 11.5. velosipedis mgoravi borblis fotografia. borblis zeda nawilTan axlos mdebare manebi ufro bundovnad Cans, vinaidan zeda nawili ufro swrafad moZraobs, rogorc es 11.4g suraTzea naCvenebi. gorva, rogorc Tanabari brunva 11.6 suraTze borblis gorvis axali gaazrebaa SemoTavazebuli, TiTqos borbali quCisa da borblis Sexebis wertilze gamavali RerZis garSemo brunavs. 11.4g suraTze, gorvas P wertilze gamavali da suraTis sibrtyis perpendikularuli RerZis garSemo brunvad ganvixilavT. 11.6 suraTze gamosaxuli veqtorebi mgoravi borblis wertilebis myisi siCqareebia. SekiTxva: am axali RerZis garSemo ra kuTxur siCqares mianiWebs uZravi damkvirvebeli velosipedis mgorav borbals? pasuxi: igive ω siCqares, rasac velosipedisti masis centrze gamavali RerZis garSemo mbrunav borbals. pasuxis SemowmebisTvis mgoravi borblis wveris wrfivi siCqare uZravi damkvirveblis TvalTaxedvidan gamovTvaloT. Tu borblis radiusia R, borblis wveri P wertilze gamavali RerZidan (sur. 11.6) 2R manZilzea da borblis wveris wrfivi siCqare unda iyos: νwveri = (ω)(2R) = 2(ωR) = 2νcom es formula 11.4g suraTTan srul TanxvedraSia. aseve SegiZliaT SemoawmoT borblis O wertilis da P wertilis wrfivi siCqareebi 11.4g suraTze. sur. 11.6. gorva Tanabar brunvad SegviZlia ganvixiloT. borbali P wer tilze gamavali RerZis garSemo ω kuT xuri siCqariT moZraobs. veqtorebi borbalze mdebare wertilebis myis wrfiv siCqareebs asaxaven. veqtorebi gadataniTi moZraobis da brunvis gaerTianebis Sedegad miiReba, rogorc es aris 11.4 suraTze. sakontrolo wertili 1. klounis velosipedis ukana borblis radiusi wina borblis radiusze orjer metia. a) moZraobis dros ukana borblis wveris wrfivi siCqare wina borblis wveris siCqareze metia, naklebia Tu ar gansxvavdeba? b) ukana borblis kuTxuri siCqare wina borblis kuTxur siCqareze metia, naklebia Tu ar gansxvavdeba? 370 gorva, Zalis momenti da kuTxuri impulsi Tavi 11 11.3. gorvis kinetikuri energia modiT, mgoravi borblis kinetikuri energia uZravi damkvirveblis TvalTaxedvidan gamov TvaloT. Tu gorvas 11.6 suraTis P wertilze gamavali RerZis garSemo brunvad miviCnevT, maSin 10.34 formula mogvcems: 1 2 K= IPω2 (11.3) sadac ω aris borblis kuTxuri siCqare, xolo IP - borblis brunviTi inercia P wertilze gamavali RerZis garSemo. 10.36 formuliT mocemuli (I=Icom+Mh2) paraleluri RerZebis Teoremidan gamomdinare: IP = Icom + MR2 , (11.4) sadac M aris borblis masa, Icom - borblis brunviTi inercia masis centrze gamavali RerZis garSemo, xolo R (borblis radiusi) h perpendikularuli manZilia. 11.4 formulis 11.3 formulaSi Casma gvaZlevs: 1 2 1 2 K = Icomω2 + MR2ω2 gamoviyenoT νcom=ωR (11.2 formula) da miviRebT: 1 2 1 2 K = Icomω2 + M (11.5) 1 I ω2 wevri masis centrze gamavali RerZis garSemo borblis brunvasTan (sur. 11.4a) 2 com 1 wevri borblis masis centris gadataniT dakavSirebuli kinetikuri energiaa, xolo M 2 moZraobasTan (sur. 11.4b) dakavSirebul kinetikur energias asaxavs. maSasadame: 1 2 mgorav sagans ori saxis kinetikuri energia gaaCnia. brunvis kinetikuri energia ( Icomω2) masis centris garSemo brunviT aris gamowveuli, xolo gadataniTi kinetikuri energia 1 2 ( M ) masis centris gadataniTi moZraobiT miiReba. amocanis amoxsnis nimuSi davuSvaT, Thrust SSC manqanis TiToeuli borbali M=170kg masis da erTgvarovani sisqis diskoa. borblebi goravs. rodesac manqanis siCqarea 1233km/sT, ra kinetikuri energia aqvs TiToeul borbals? amoxsna: 11.5 formula mgoravi sagnis kinetikur energias gvaZlevs, magram misi gamoyenebisTvis sami ram unda gavixsenoT: 1. rodesac mgoravi vsaubrobT, siCqares sagnis yovelTvis vgulisxmobT. siCqareze masis am centris SemTxvevaSi, νcom=1233km/sT = 342,5m/wm. 2. 11.5 formula mgoravi sagnis ω kuTxur siCqares saWiroebs. kuTxur siCqaresa da νcom siCqares 11.2 formula akavSirebs νcom , sadac R borblis radiusia). R 3. 11.5 formula sagnis Icom brunviT inerciasac saWiroebs masis centris garSemo. 10.2g (ω = cxrilidan gamomdinare, erTgvarovani 11.1 1 2 diskos SemTxvevaSi Icom = MR2. yovelive zemoTqmulidan gamomdinare, 11.5 formula gvaZlevs: 1 1 K = 2 Icomω2 + 2 M ν 2 1 1 1 =( 2 )( 2 MR2)( com ) + 2 M R 3 3 = 4M = 4 (170 kg)(342,5 m/wm)2 = 1,5 × 107j (pasuxi) (yuradReba miaqcieT, rom gamoTvlebis dros borblis R radiusi gabaTilda). miRebuli pasuxi Thrust SSC-s mier rekordis damyarebasTan asaxavs. dakavSirebul TiToeuli borblis safrTxes kinetikuri energia uzarmazaria da TiTqmis 10.7 amocanis amoxsnis nimuSSi moyvanili mbrunavi foladis diskos afeTqebis kinetikuri energiis (2,1×107j) tolia. nebismier dabrkolebasTan Sejaxebis SemTxvevaSi borbali diskos msgavsad afeTqdeba. foladis 11.4. gorvis Zalebi 371 11.4. gorvis Zalebi xaxuni da gorva Tu borbali 11.3 suraTis msgavsad mudmivi siCqariT goravs, is kontaqtis P wertilze ar srialebs da, maSasadame, xaxunis Zalac ar moqmedebs. Tu mgorav borbalze jamuri Zala moqmedebs da mis aCqarebas an Senelebas cdilobs, maSin jamuri Zala moZraobis mimarTulebis gaswvriv masis centris  acom aCqarebas iwvevs. jamuri Zala borbals ufro swrafad an ufro nela abrunebs anu α kuTxur aCqarebasac iwvevs. miRebuli aCqarebebi borbals P wertilze asrialeben. maSasadame, P wertilze xaxunis Zala unda moqmedebdes, romelic borblis srials SeewinaaRmdegeba. Tu borbali ar srialebs, maSin masze  f s statikuri xaxunis Zala moqmedebs da gorva Tanabaria. am SemTxvevaSi, 11.2 formulis droiT diferencirebis Sedegad (rodesac  R mudmivia) acom wrfiv aCqarebasa da α kuTxur aCqarebas Soris kavSirs miviRebT. formulis marcxena mxares acom=dνcom/dt, xolo marjvena mxares α=dω/dt. amgvarad, Tanabari gorsvisTvis gveqneba: acom=αR (Tanabari gorva) (11.6) Tu borbali srialebs da masze jamuri Zala moqmedebs, maSin 11.3 suraTis P wertilze moqmedi Zala  f k kinetikuri sur. 11.7. borbali horizontalur zedapirze srialis gareSe goravs da  acom wrfivi aCqareba aqvs. P wertilze f s xaxunis Zala moqmedebs da srials ewinaaRmdegeba. xaxunis Zalaa. am SemTxvevaSi gorva Tanabari ar aris da moZraobas 11.6 formula ar miesadageba. mocemul TavSi mxolod Tanabar gorvas ganvixilavT. 11.7 suraTze naCvenebi borbali brtyel zedapirze marjvnisken goravs da brunvis siswrafe izrdeba. swrafi brunva P wertilze borblis fskeris marcxnisken gasrialebas iwvevs. P wertilze moqmedi xaxunis Zala marjvniskenaa mimarTuli da gacurebas ewinaaRmdegeba. Tu borbali ar srialebs, xaxunis Zala statikuria, gorva Tanabaria da 11.5 formula moZraobas miesadageba. Tu 11.7 suraTze gamosaxuli borblis moZraoba neldeba, maSin masis centris  wertilze moqmedi f s xaxunis Zala marcxnisken iqneba mimarTuli.  acom aCqareba da P gorva daxril sibrtyeze 11.8 suraTze naCvenebia M masis da R radiusis mrgvali, erTgvarovani sxeuli, romelic θ kuTxiT daxril sibrtyeze x RerZis gaswvriv gordeba. Cven sxeulis acom,x aCqarebis gamosaxuleba unda miviRoT. amisTvis niutonis meore kanonis wrfivi (Ft= Ma) da kuTxuri (τt= Iα) versiebi gamoviyenoT. Tavdapirvelad, 11.8 suraTis msgavsad sxeulze moqmedi Zalebi gamovsaxoT:  1. sxeulze moqmedi Fg mizidulobis Zala qvemoTaa mimarTuli. veqtoris saTave sxeulis masis centrs emTxveva. sibrtyis gaswvrivi komponentia Fg sin θ = Mg sin θ.  2. FN drekadobis Zala sibrtyisadmi perpendikularulia. es Zala P kontaqtis wertilze moqmedebs, magram Zalis veqtoris saTave sxeulis masis centris mimarTulebas davamTxvieT. 3.  f s statikuri xaxunis Zala P kontaqtis wertilze moqmedebs da sibrtyis gaswvriv, aRmarTiskenaa mimarTuli (Tu sxeuli P wertilze mocurdeba, is sibrtyeze Camosrialdeba. maSasadame, mocurebis sawinaaRmdego xaxunis Zala aRmarTisken unda iyos mimarTuli) 372 Tavi 11 gorva, Zalis momenti da kuTxuri impulsi niutonis meore kanoni x RerZis gaswvrivi komponen tebisTvis ase Caiwereba: fs – Mg sin θ = Macom,x (11.7) formula fs da acom,x ucnobebs Seicavs (Cven ar vuSvebT, rom fs Zalas fs,max maqsimaluri mniSvneloba aqvs. fs Zala sakmarisia, raTa sxeuli daxril sibrtyeze Tanabrad, mocurebis gareSe Camogordes) amjerad niutonis meore kanonis kuTxuri forma masis centris garSemo sxeulis brunvisTvis unda daiweros. Tavdapirvelad, 10.41 formulis (τ=r┴F) gamoyenebiT masis centris garSemo sxeulze moqmedi Zalis momentebi CavweroT.  f s xaxunis Zalis momentis mxaria R da es Zala Rfs moments warmoqmnis. Zalis momenti dadebiTia, vinaidan xaxunis Zala sxeuls saaTis isris moZraobis sawinaaRmdego mimarTulebiT Semobrunebas cdilobs.  Fg  FN Zalebs masis centris garSemo  nulovani momentis mxrebi aqvT da, maSasadame, Fg da FN Zalis momentebi da iqneba nuli. aqedan gamomdinare, sxeulis masis centrze sur. 11.8. R radiusis mrgvali, erTgvarovani sxeuli daxril sibrtyeze  gordeba. masze Fg mizidulobis Zala,  Zala da aRmarTisken FN drekadobis  mimarTuli, f s xaxunis Zala moqmedebs  (sicxadisTvis FN veqtoris saTave sxeulis centrs davamTxvieT) gamavali RerZis garSemo niutonis meore kanonis kuTxuri forma iqneba: Rfs = Icomα (11.8) miRebuli formula fs da α ucnobebs Seicavs. sxeuli Tanabrad goravs, amitom 11.6 formuliT (acom=αR) acom,x da α ucnobebi erTmaneTs SegviZlia davukavSiroT. am SemTxvevaSi acom,x uaryofiTia (x RerZis uaryofiTi mimarTulebisaa), xolo α dadebiTia (saaTis isris moZraobis sawinaaRmdego mimarTulebisaa). 11.8 formulaSi α-s nacvlad acom,x CavsvaT da fs-sTvis amovxsnaT: R acom,x fs = –Icom 2 (11.9) R 11.7 formulaSi fs-is nacvlad 11.9 formulis marjvena mxare CavsvaT da miviRebT: g sin θ acom,x = – (11.10) 1 + Icom /MR2 am formulis gamoyenebiT horizontalisadmi θ kuTxiT daxril zedapirze mgoravi nebismieri sxeulis acom,x wrfivi aCqareba SegviZlia gamovTvaloT. – sakontrolo wertili 2. A da B identuri disko Tanabari siCqariT migoravs. Semdeg A disko aRmarTze gordeba da h maqsimalur simaRles aRwevs, xolo B disko identur, oRond xaxunisgan Tavisufal aRmarTze adis. B diskos mier miRweuli maqsimaluri simaRle h-ze metia, naklebia Tu igivea? amocanis amoxsnis nimuSi 11.2 M = 6 kg masis da R radiusis erTgvarovani burTi θ = 30° kuTxiT daxril sibrtyeze Ta ruls Kf kinetikur energiasTan SegviZlia nabrad gordeba. davakavSiroT. garda amisa, burTi-dedamiwis a) burTi h = 1,2 m vertikaluri simaRlidan sistemis meqanikuri energia burTis gorvis Camodis. ra iqneba misi siCqare daRmarTis dros miwuruls? mizidulobis amoxsna: burTis siCqare daRmarTis miwu inaxeba, vinaidan Zala burTze asrulebs mxolod muSaobas, 11.5. io-io 373 romelic Semnaxvelia. daxrili sibrtyidan burTze moqmedi drekadobis Zala muSaobas ar asrulebs, vinaidan is burTis traeqtoriis perpendikularulia. daxrili sibrtyidan burTze moqmed xaxunis Zalas energia siTbur energiaSi ar gadaaqvs, vinaidan burTi ar νcom = = 4,1 m/wm (pasuxi) yuradReba miaqcieT, rom pasuxi burTis M srialebs (Tanabrad goravs). masaze an R radiusze damokidebuli ar aris. maSasadame, meqanikuri energiis Senaxvis kanoni b) ra sididis da mimarTulebis xaxunis Zala aseTi saxiT SegviZlia davweroT: moqmedebs burTze daxril sibrtyeze gorvis Kf + Uf = Ki + Ui , (11.11) sadac f da i indeqsebi saboloo da sawyis mniSvnelobebs asaxavs. Tavdapirvelad mizi dulobis potenciuri energiaa Ui=Mgh (sa dac M burTis masaa), xolo sabolood Uf=0 xdeba. kinetikuri energia Tavdapirvelad aris Ki=0. Kf saboloo kinetikuri energiis gamoTvlisTvis gavixsenoT, rom kinetikuri energia gadataniT moZraobas da brunvas moicavs. amitom 11.5 formulis marjvena mxaris gamoyenebiT orive moZraobas viTvaliswinebT. yvela es gamosaxuleba 11.1 formulaSi CavsvaT da miviRebT: 1 1 (11.12) 2 2 sadac Icom aris burTis brunviTi inercia ( Icomω2 + Mνcom2) + 0 = 0 + Mgh, dros? amoxsna: vinaidan burTi Tanabrad goravs, masze moqmedi xaxunis Zala 11.9 formuliT gamoiTvleba. Tavdapirvelad ki 11.10 for mulidan burTis acom,x aCqareba unda gamov TvaloT: acom,x = – g sin θ g sin θ 2 = – 1 + Icom/MR 1 + 2 MR2/MR2 5 =– (9,8 m/wm2) sin 30° 1+ 2 = – 3,5 m/wm2. 5 yuradReba miaqcieT, rom burTis aCqarebis gamoTvlisTvis M dagvWirvebia. masa an R radiusi ar maSasadame, 30°-iT daxril sibrtyeze nebismieri zomis da erTgvarovani masis burTs amgvari aCqareba eqneba, Tu burTi masis centrze gamavali RerZis garSemo, Tanabrad goravs. νcom - CvenTvis saintereso siCqare daxrili sibrtyis bolos, xolo ω - kuTxuri siCqare 11.9 formulis amoxsnis Sedegad miviRebT: aseve daxrili sibrtyis bolos. 11.2 formulis gamoyenebiT ω-s nacvlad νcom/R CavsvaT da 11.12 formulaSi ucnobebis ricxvi SevamciroT. amis Semdeg 10.2 cxrilidan Icom-is nacvlad 2 MR2 CavsvaT da νcom-sTvis vinaidan amovxsnaT: burTi 5 Tanabrad goravs, fs = –Icom acom,x a 2 2 2 com,x = – 5 Macom,x 2 = – 5 MR R R2 2 (pasuxi) = – 5 (6 kg) (–3,5 m/wm2) = 8,4 n am SemTxvevaSi dagvWirda M masa, magram R radiusi ar dagvWirda. maSasadame, 30°-iT daxril sibrtyeze mgorav, 6kg masis burTze moqmedi xaxunis Zala, miuxedavad burTis radiusisa, 8,4 n-ia. 11.5. io-io io-io fizikis patara laboratoriaa. Tu io-io Tokis gaswvriv h manZilze gordeba, misi 1 potenciuri energia mgh-iT mcirdeba, xolo kinetikur energias gadataniTi ( mνcom2) da brunviTi 2 1 ( Icomω2) formiT iRebs. ukan dabrunebis dros io-io kinetikur energias kargavs da potenciur 2 energias iZens. io-ios Tanamedrove versiebSi Toki RerZze damagrebuli ki ar aris, aramed mis garSemoa Semoxveuli. rodesac io-io Tokis bolos Seexeba, Tokidan RerZze moqmedi maRla mimarTuli Zala qvemoT Casvlas aCerebs. amis Semdeg io-io brunavs da mxolod brunvis kinetikuri energia aqvs. 374 Tavi 11 gorva, Zalis momenti da kuTxuri impulsi brunva manamde grZeldeba (io-ios `sZinavs~), vidre Tqven Toks ubiZgebT (io-ios `gaaRviZebT~), Toki RerZs Semoexveva da io-io ukan asvlas iwyebs. Tokis boloze io-ios brunvis kinetikuri energia (da, maSasadame, Zilis xangrZlivoba) mniSvnelovnad gaizrdeba, Tu Cagorebis nacvlad io-ios isvriT da νcom da ω sawyis siCqareebs mianiWebT. Tokze Cagorebuli io-ios acom wrfivi aCqarebis gamosa xulebis misaRebad daxril sibrtyeze mgoravi sxeulis msgavsad niutonis meore kanoni SegviZlia gamoviyenoT. orive SemTxveva TiTqmis identuria da gansxvaveba mxolod SemdegSia: 1. horizontalisadmi θ kuTxiT daxril sibrtyeze gorvis nacvlad io-io horizontalisadmi θ=90° kuTxis mdgenel Tokze gordeba. 2. gare zedapirze gorvis nacvlad, io-io R0 radiusis RerZze gordeba (sur. 11.9a).  3. f s Zalis moqmedebis Sedegad Senelebis nacvlad, io-ios Tokidan moqmedi T Zala anelebs (sur. 11.9b). analizi am SemTxvevaSic 11.10 formulamde migviyvans. aqedan gamomdinare, 11.10 formulaSi ubralod aRniSvnebi SevcvaloT da θ=90° CavsvaT: acom = – g 1 + Icom/MR02 , (a) (b) sur. 11.9 a) io-ios ganivi kveTi. umniSvnelo sisqis Toki R0 radiusis RerZzea Semoxveuli. b) Tavisufali sxeulis grafiki vardnili io-iosTvis. naCvenebia mxolod RerZi. (11.13) sadac Icom aris io-ios brunviTi inercia centris garSemo, xolo M - misi masaa. Tokze ukan asvlis dros io-ios igive qvemoT mimarTuli aCqareba aqvs, vinaidan masze igive Zalebi moqmedebs, romlebic 11.9b suraTze. 11.6. Zalis momentis xelaxla ganxilva me-10 TavSi fiqsirebuli RerZis garSemo mbrunavi myari sxeulis τ Zalis momenti ganvsazRvreT, rodesac sxeulis TiToeuli nivTieri wertili am RerZze centrirebuli wriuli traeqtoriiT moZraobda. amjerad Zalis momentis gansazRvrebas ganvavrcobT da fiqsirebuli wertilisadmi (da ara fiqsirebu

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