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First published In 2(114 by
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2017
2016
2(115 2014
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Prlmed ln lta]y
A catalogue n>eord for thls title ls av.ll.ble from the Brltl.hUbrary
ISBN 978 1471809217
Contents
Acknowledgements ..............................................................................................................................................................vii
Introduction ............................................................................................................................................................. . ....... viii
Syllabus structure relating to book topics ............................................................................................................................ i)(
AS Level
Topic 1
1.1
1.2
1.1
Topic 2
2.1
2.2
Topic 1
1.1
Topic 4
4.1
4.2
4.3
Topic 5
5.1
5.2
5.1
5.4
Topic 6
6.2
6.1
6.3
6.4
Physical quantities and units ..
........ . ....... ........................... . . . . 1
Physical quantities
.2
Siquantitiesand baseunits
.2
Scalars and vec\ors
Measurement techniques .. .
....................... 15
Measurements
15
Erronand uncertainties
31
Kinematics ..
. ................... 40
Speed, displacement, velocilyand acceleralion
40
Dynamics
"
Relationships involving force and mass
"
Weight
58
The principle of conservalion of momenlum
61
Forces, density and pressure ...... ........ . . . . . . ..... ... .............................. .. . . ............... ................ .. 71
Types of force
71
Momenl of a force
71
Equilibrium of forces
73
Density and pressure
Work, energy, power.
76
. . ........ ................. . . ............... 80
Work
80
Energy
83
Kineticenergy
85
Power
Topic 9 Deformation of solids ..
9.1 & 9.2 Force and deformalion
Topic 14 Waves
14.1 Wave molion
14.2 Graphical represenlalion of waves
14.1 The determination of lhe frequency of sound using a calibraled c.r.o.
14.4 Doppler effecl
14.5 The eleclromagnelic speclrum
89
.................... 94
94
101
101
102
107
108
109
Contents
Topic 15 Superposition .... ............................... ................
15.3 Interference
15.1 Stationary wavH
14.3 Measuring the speed of sound using stationary waves
15.2 & 15.4 Diffraction
Topic 17
17.1
17.2
... ............. ... . .... ........ ... .. .................. 114
114
11.
12'
126
Electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... ............ ... . . . . . . . . ............... . . . . . . . . . . . . . . . . . . . . . . 136
Electric forces and fields
136
Electric field strength
13.
Topic 19 Current of electricity
19.1 Charge and current
19.2 Potential difference
19.3 Resistance
.. ......... 145
145
148
14.
Topic 20 D.C . circuits
20.1 Electrical circuits
20.2 KirchhoWs first and second laws
20.3 PotentiaJ dividers and potentiometers
Topic 26 Particle physics
26.1 Atomic structure and radioactivity
26.2 Fundamental particles
156
156
IS.
162
.................168
168
178
A Level
Topic 7
7.1
7.2
Motion in a circle...
.....................184
Radian measure and angular displacement
184
Centripetal acceleration and centripetal force
ISS
Topic 8 Gravitational fields . ...................... ...................................................................
8.1 Gravitational field
8.2 & 8.3 Gravitational field strength
8.4 Gravitational potential and gravitational potential energy
... ...........191
191
192
198
Topic 10 Ideal gases
10.1 & 10.3 Equation of state of an ideal gas ..
10.2 A microscopic model of a gas
205
Topic 11 Temperature
11.1 Temperature
11.2 Temperature scales
11.3 Thermometers
m
m
Topic 12
12.1
12.2
Thermal properties of materials.. .
Solid�, liquids and gases, and thermal (heat) energy
Internal energy
Topic 11 Oscillati ons ... ......................................................
11.1 Oscillations
13.2 Energy changes in simple harmonic motion
13.3 Free and damped oscillations
'"
'"
211
211
.................. 217
217
m
.. ......... ... . ................ ... .................2]0
230
m
m
Contents
Topic 14 Ultrasound
14.6 The generation and use of ultrasound
247
247
Topic 16 Communication
16.1 Communication channels
16.2 Modulation
16.3 Analogue and digital signals
16.4 Relative merits of cl'Iannels of communication
16.5 Signal attenuation
m
m
m
Topic 17
17.3
17.4
17.5
Electric fields ..
Point cl'larges
Electric field strenglh due l o a point charge
Electric potential energy and electric potential ..
Topic 18 Capacitance ..
18.1 Capacitors and capacitance.
18.2 Energy stored in a capacitor
Topics 19 & 20 Electronic sensors
19.4 Sensing devices
20.1 The use of potential dividers
Topic 21 Electronics
21.1 The ideal operational amplifier (op-amp)
21.2 Operational amplifier circuits
21.3 Output devices
262
265
26.
. . ........... 274
27'
216
..276
... ...................281
28'
285
'"
'"
295
298
298
'"
30S
Topic 22 Magnetic fields ..... . . . . . . . . . . . . . . .................................... ........................... . . . . . . . . . .................. 311
22.1 Concept of a magnetic field
311
22.2 Force on a current·carrying conductor
31'
22.3 Force on a moving charged particle in a magnetic field
318
22.4 Magnetic fields due to currents
32'
22.5 The use of (nuclear) magnetic resonance imaging ..
326
Topic 23
23.1
Electromagnetic induction ..
Magnetie fluKan d e leclromagnetic induction
Topic 24 Alternating currents
24.1 Characteristiesof alternaling currents
24.2 Transformers
24.3 Transmission of electrical energy
24.4 Rectification
Topic 25 Quantum physics
25.1 & 25.2 Photoelectric emission of electrons and energy of a photon
25.1 Wave-partideduality
25.4 Energy levels in aloms and line spectra
25.5 Band theory
25.6 The production and use of X-rays
............................... 332
m
341
..341
..343
34'
345
350
350
3SS
356
36.
m
Contents
26 Nuclear physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . . . . ............ . . . . . . . . . . . . . . . . 376
26.3 Mass defect and nuclear binding energy
'"
26.4 The spontaneous and random nature of radioactive decay
381
Topic
AS Level Answers to Now it's your turn and Examination style questions
A Level Answers to Now it's your turn and hamination style questions
Index
Student's CD contents
Interactive tests
Topic summaries
& Definitions and formulae
Revision checklists
Examination structure
Planning revision
Examination technique
Glossary of command words
Catculation of uncertainty in a result
Significant figures and decimal places
Proportionality and linearity
Straight·l ine graphs
388
'"
399
Acknowledgements
The publtshe� wooklilire to thank the following for pennlsslon to use copyright m:llerlal
Pholocredil.5:
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Acknowledgemenu,
Every effon has been nude 10 {nice �II copyright holders, but If any hl"e been inadwnently O\"('r\ooked, {he PublIShers wlU be
pleased 10 make the roecessary urnngemellis Jtthe fi1"5l opponunlty.
vii
Introduction
This book [s a 1It'\\' edlla'l cJ: j"NnlflIIOMIAS{mdA Lerrel Pbysfcslhal has been Il"Vlsed and amended 10 be COfIlp;ttlJle wtlh till'
N
Physics Syl13bus 9702 a C1mbodge InternatIOnal ExamlnaUons, published In 2014 for ftn;I examination In 2016
The book has Ix'en fully endorsed by Cambridge illlemalional Examinations and Is lisle(! :l.S an endorsed textbook for liIudems
smdylng thls syllabus.
so thal lhi' boclk: PlQllkI es comprehensl\'e rover ci the
so thal .ltudent s studying AS physics wUl fiod that pan cilhel>-yllabus
theCOOlenl ci the book goes sllgtnly beyond tile
ew m:lIerial has been locluded, where TJe«'SS3ry,
subject content The COIlleRi d lhe 000k has been re-Ql"dered
as being separate flQI)I the second part of the A Level course. In a few places,
syllabus requirements, either 10 prov1<k> some background Information or to arrive at a satlsf:lctOl')' termination cI a topIC.
All the assessment objectives !Ill! :if'(' lck>nUfled In the syllabus are CO\'ered In the bool..:. The learning OUICOlTlCS, as specified In
the s}1I3bus. ar e 115100 In each topic cJ the book using the same wordIng as In the syllabus so Ihalliludcnls may identify easily the
sectlon cllhe syllabus that Is beln8 COVI.'roo. The contem of each topic Is IdenUfied by learning outcome. 001 n<.>cess.1flly presemed
In sequenUal numerical ortk>r. Ill't accordln8 to the most sensible older for leamln8. I'or exampk>, tn AS l.evel Toplc 6. learning
Oluc ome 6.2 (Work) comes lx-fore 6.1 (Enef8Y). Th{' chan opposite 5ho,,""5 how topics are arranged In til{, book and how Ihls relates
C":t
to syllabusOO\'{'rage.
A new feature of Ih e syllabus Is Key conceptS, These are the essential Ideas, theories. principles or men tal tools that help leam{,1ll
to develop a deep 11l1d{,lllIandlng of !h�r subject, and make links between dlff{'f{'n\ topics. An k:0I1 Indl
ronce p t l sCQVered:
� ���:;� �� �
!es where each KIry
d r: �
hat
t
and ener8Y Imerle! IS thrOu8h forres and fields. The behaviour of the Unll't'rse Is 8ovemc<l IlY fun<l:lmental
forces that ad oyer dlffercrot leng!ll scales and ma8nltudes. These Indude the gr:J.v ltatlonal force and the electroml8neuc force.
KIry points, deflnilions and equations aft' hIghlighted In coloured panels. There Is a summary of IIX' Importam feltures thaI have
Ix'en covered after each section or topiC. Throu8houl each topic. "'"OI"ked examples are provided so that Sludents may familiarIse
c
lhemselves I>.ith tht' SUbject maHer. 111e worked examples ue followed by questions of simIlar difficulty. IISled under the !'{QW II�
yorrr film headings. In addillo n. there are questions which h:ive a brooder context and arc of IIX' examln:ulon style as regards
Sl
wording and Ie\'el of difficulty. Answers to bah types of queslon are provided at the ba k of the book.
..eT:l� of �"UbIecl mailer and style of questions
This book h:is been wrillt'n specifically for the Cambrld� syllabus. HO\\'l'\-er, 11.5 cO'
make-It suitable fQ'"stlldents who are
udy lng t0W3rdS physics quallflc:r.tlons of Olher aW:lrdlng bodies.
Geoff Goodwin
julyZOl4
viii
Syllabus structure relating
to book topics
I. Physic�1 qu.mt�� and un�s
In_national 11.111.5 L.wI
InWr.w1onal A/A5 Lwei Phy$k.
Pl'lysIUASl.eftITopits
A LewITopia
1. PIlysical qu.;Jolilil'5 and units
2.M��SI..Irement techniques
2. Me.asureJTll!nt tedmiques
4.0ynamks
4.0ynarnics
5. fOfces,density and plessure
5. Forces, densilyand press.ure
6. WOr\;, eI1llfgy. pclWE!I
6. wort, energy, JXlwer
7.Motion in a ( ��
7.Motion in a dr de
B GrOlYitabonal fields
9. Deformation of solids
8Gr.witationalf�1ds
9. Deformation of :;o�ds
12. Therm� PlQPertil!l of �lel1i1s
12. Thllfmal propertiE's of maU!rIoIls
13. Qscillations
13.O!idlbtions
COIoeI'S/4.I,14.l,14.3,14.4
;,ndl4.S
IS. Superposition
15.Superpo�ion
16.Communiatlon
16.Communi;atlon
17.E�tri(fields
COK'rS17.3,17.4�11_5
lB.Cap...:it;mce
19. (urreol of ele<triclty
18.CopOKiiance
19.CuJrent oll'll'Ctrkity
19& 20Ek>ctronic S!1l1sors
COOllfSI9.I,/9.2;J(j(j/9.3
t <J& 10E IN tronic sensors
COK'rS20.J.20.2;md20.3a.b
co�20.3C.d
21.Ek>ctronics
21.Electronk:s
22.M<lglleticl�ds
l2.Maglll'tKli{lk;Is
23. fll!<:trom.ll¥letlc llldoucUon
23 . Ell'c:trom.l9f1�tic induction
24. AIt�rn.lting currents
24. Altem�ting current�
25. Qu�ntum ph)'5ic$
25. Qu�llum physics
26. Pxtide �nd nude�r physics
26. P�tide;)oo nuclear physics
26. P�II' �oo nucle�r physics
COKn26.3�26.4
This page intentionally left blank
AS Level
1
Physical quantities and units
By the end of this topic, you will be able to:
(a) understand that alJ physlcalqualllltJesronsislofa
numerical maQnUudeand a unll
(b) lTIakereasonable estlm.ates ofphyslcalquamitJes
Included In the syllabus
(a) recali lhefoilowlOjlSl base quantUlesand theIr
untts:massCka),lcnith(nl),tlmeCs),cufrent(A),
temperature 00. amount of substance(mol)
(b) express derived unlts as products orquotJents of
theS! base unHs and use the named unIts listed
In Ihe sylboos as appropriate
(e) useSI base units to check the homogeneity of
physICal equations
(d) use the followIng prefx
l es
IndICate declmal submulUpies or mUltiples ofboI:h
base:,md derh'ed units:'
o
•
pIco(p)
nano(n)
•
mlcroCp)
mJ11J(m)
•
centl(e)
ded(d)
•
•
• kUo(k)
• mega
•
(�O
glga(G)
• tera(T)
(e) understand and use the colwenUons for labelling
gr:lph axes arx1 lable columns
(a) distlngulsh scalar ancl\'ectorquantilies andglve
examples ofeach
(b) add and subtract copIanar\'ectors
(c) represent a \'ector as two perpendlctJlar
components
Note: amount ofsubstance (mol) is only used In the A level
course but Is included here for completeness.
Starting points
.
• AcclM'ate measurement i5 very important in the development of physics
•
PhY5icists begin Ut observing, measuring and coIlecti og data,
• Tht'Se data are aoaly5ed to discover whether they fit i nto a pattem,
• If there i5 a pattem and this pattem un be used to expl ain other events, i t becomes
a theory
• The process is known a5the scientific method (see Figure
Flgur.1.18Ioc:kdl�r.IIT1toilintratetheSCIentdlCmethod
1,1),
II Physical quantities and unilS
1.1
Physical quantities
Figure 1.2 Brahe(lS46-1601) mealoUred
theelevatl omofst�rs;lhesedaysamodem
thl.'OdoliteiluledfOf meaSUring angular
el�vation
A physical quantity Is a fea ture of somclhlng whiCh can IX' ,neasured. for example.
Iength, ....'elght,orUme off.lll. Every physlCllI quantity has a nu merlcal value and a unk.
If someone says they have a wal.lt measurement of 50. they could be very .slI m or very
fat depending on whethe r the measurellll"f'i Is In centimetresor I nchesl nke care _ It Is
vital to gJve tile unit of measuremerv: whenever � quam.y Is measured or wrltten down
large and small quariltlesare usually expressed In sclenllOcOOatla1. i.e. asa slmple
number mUltiplied by a power often. FOr example. 0.00034 would be written as
3.4 x ]()o-_ and 154000000 as 1.54 x 10-. There \s far less chance of (lllIklng a m lSlake
wilhlhe number of zeros!
Figure 1.3 The eJephant l s larlJj! In comparisorrw lth the boy
but sma� compared with theJumbo Jet
1.2 SI quantities and base units
In very much the same way that lansuag<!S
world, ma�' different systems of mea5l.Jremeflt have e\\jll'ed. JuS( as languages can be
tnnsiated from one to aTlOlher. units of measurement can also be corll'erted between
sy.ltems. Although 5OIlll' COIIVE'IliIon factor.; are easy to remember. some are very
....
dlfJlcu•. It Is much bener 10 ha'
juSl one s}'S(em of unitS. For this reason. scientists
around the world use the Systeme Internato
i nal
s y.ltem of measurement
1.2 SI quan1i1ies and base units
If a quanUty Is (0 be measured accuratcly.
11K> unl! in w!lich It Is me:lSUrW
mUSI be
deflned as predsely as possible.
SlisfoondedOfl�fundamentalorbaseunits.
The base quanllUes and the units wUh wflldl they are measured are IISied In
'Thbie
1.1. Forcornpleteness, thecandeb tusbren
lndllded. but this unit \\111 oot be
used In the- AlAS course.l1ll' IIlOk> "111 only be used In the A Lewl course
Table 1.1 Thl'b.-quantll.leS�d urnb
qwontny
symbol
��tricQJffent
amp�re(.lmp)
A
thermodynamic temperatllre
Flgurel.4Tnem�ssoflnlsjewelcould
be mealllrl.'dlnl<lla.gr�ml,pollndl,c�rats.
grains,etc
lumi nous Intensity
EachqualUity h.asJuSl one unit
and this
unit can hal'e multiples and sub.multiples
to cater for larger or smaller value5. The unit ISSlvcn a Ilrefix to denote the muklpie
(see Table 1.2). For example. one tllooJsandth of a metre Is knawn as a
mi llimetre (mm) and 1.0 millimetre equals 1.0" 10-) metfe5 (m).
or sub·multlple
Table 1.2 � more commontt usK!
Pfeillll!'S
,,-,
prefix
multlplylngfxtor
1011
""
,<>,
,,,.
,,,.
'ed
10"
Cl'nti
10-
''''
1010·
,,,-.
10-'1
Beware when COIlwnlng unlls for areas and volumes!
Squarlng bc:Xh sides
Imm'=(10-1}lm'=lo-6ml
lcm)= (1o-zymJ= lo-'m1
llle box In �lgure
Figur. 1,5 Thisbalc h�sa
WoiufTll!ofl0xlOlcm)
1.5 has a
volume a 1.0"
l()lcm1or 1.0" to'mm) or 1.0" l�mJ.
A dlSlaTlCt'ofthlny melres shoukl be wrlttcn as 30m and 001 3Qms or3Qm s.l1ll'
le!ter s lsnen'rloc!uded In a unll forlhe plural. If a space Is lef! belwe e n lwo i ellers.
the lroers denoK> different unlls. So.30m 5 would mean Ihlny mc(f(' seccnds and
3Omsme:ans30mlUlsecaJds .
I
II Physical quantities and unilS
Example
Calculate the number of micrograms in 1.0 mWligram.
1.0g .. I.0" 100mg
and 1.0g .. 1.0" l06rnicrograms(llg)
�,1.0,,'Qlmg .. '.0"'0'Ilg
and 1.0mg .. (1.0" 10'lJ1:1.0" 1()l).1.0w 10lllg
Now it's your turn
1
CaIaJiatetheare.a,inw,ofthetop ofa tablewithsides ofl,2mandO.gm.
)
Write down, u-;ing 5dentifk notation, the vailies of the following QlJaf1tities'
« rM! the number of,ubk metres in one cubic kilometre.
2
De1 mi
(a16.8pF,
(b) 32IlC,
(e)
4
6OGW.
How many electric fires, each rated at 2.SkW, can be POWefed from a genefator
providing 20
. MW
r-- o.26 nm �
5
1.6,
An atom of gold, Figure
is 5.6 w
Flgure 1.6 Atomofgokl
10-lpm.
has a diameter ofa.26nm and the dia meter ofits nucleus
Ca kulate the ratio ofthe diametef
m les
01 the atom to that ofthe nucleus
Derived units
AU qu: Ut
, apan from tlK' OOSl" quanUtIes, on
h
Derived units consist ofsorT\2 rombirliltion of the
�t
mul tiplied t
See Table
be expressed In terms cI tleri\'etl
bas.e units. The bas.e units may be
er Of dimed byone anothef, but newr oldded Of sublracted.
1.j for examples of de!1\"OO
unUs. some quantities have a named unU. For
example, t h e unit
In terms of base units. Qualllllk's whldJ do I'lOl have a named unit are expressed In
terms of other units . For example, specific latent heat (lbpIc 12) Is measured In joules
perkllogTamOkg-').
Table 1,) Some examples of deriwd umtswhlch maybe �td IntheAlASCOl.l�e
.....frequency
velocity
')(cell:>ration
force
energy
��tri{char9'!
hertz
m.w
(HZ)
,-,
)
ton {N
kgws-l
wall{W)
kgffils-1
(Qulomb(C)
llm ( )
potential ditfereoce
volt
e�tri(iil resiltance
o
(V)
kgffils-1A-1
O
kgffils-1A-l
�ificlle<lt(�p<Kity Hg-'I('
Example
kgms-.2
joull:>(J)
d��
mls-lj(-'
'Mlal aretlw base units of speed?
speed is defined as
and � the unit is
7-
Division bya unit is shown us.ing a negidiwindeK, that Is. S-I.
The bas.e units of speed are ms-I.
1.2 SI quan1i1ies and base units
Now it's your turn
Uw th!1 information in Tables 1.1 and 1.310 determine the base units of the fonowing
quantities .
• Oro"
(. ';:;;;�,I
, "�+:';;;I
Checking equations
It IsposslJ!elOworl!:OlJ.theKXalrlUrrberaoransesIn t\\ooogsIforoeoogronulns
four and!hI> other lin" (tbt- anS\\\"r Is nine!). This exercise would, c{ 00lI1'Se. be nonsense
IfCl'lf' OOgCOfWalfll'd threeOOlnges and the othef foor mangoes, In theS:lmew a y, for any
equatlon to make 5e1l5e, each term Invo..
..
ed in the equalloo mu;t luve the same base
unls
l .
(or groups) Is added 10, or subtracted from. OIher terms.
Ihetenns are v, II,andat.
In anyequation where each term 1m the same baseunits,the eq uation is saki to be
homogeneous or b
' alanced',
In tile example above, each term h a s the base unllsm S-I, If the equation Is ncr
tlomogeneous, then It Is lncorre<:t and IS OCII valid. When an equation Is known to be
tlomogeneous,the f\tiK'balaTldngabase unuprcw1desarneansaflndlng lhe unlts
aan unknown qu anUty.
Example
Use base units to ww that the following equation is homogenrous.
1mB: done =gain in kiletic energy + gain in
gra'lifatiooal potential energy
Thl! tl!rms i n t heequation arl!'M)rk ,
potl!ntiall!nergy.
'MlI'kdonl! =foKl!xdi!il.iln<:1!1TKMld
anciso ttw>baseunitsarl!kg mpxm"kgmZ,-l.
I
ki!'M!tkl!nergy= "lTld5s,,{5p4!!!d)l
I
Sintl!ooy�rell!.lm ber surnilS has nounit,thl!baSl!unjtsarekg"(ms-I)l,,kgmJ,-J
potrotial energy =m.1SS"gravitatioflal field strength g" distance
ThebaSi!unitsarl!kg"ms-1"m",kgmZs-z.
Conclusion: All terms have the same base units and the equnlon 15 homogeneous
Now it's your turn
Usebase units to check whethl!f thl! following equations ale balanced:
(a) pressure =depth "density" gravitatiooalfieldsffengrh,
(b) energy=mass"lspeoooflg
i ht)l.
II Thl!thl!fIDal energyQneedl.'d to mi'lt a solid of mass m without anychange of
tem�ratureis givenbythl!equatioo
8
Q,.mL
whi!fe Lis a constanl. find the base unilSol L.
10 Determine ttw> base units of the following quantities
(a)e-nergyl=force"d'st,mce),
(b)spedflCheiltGipacity.
(th('rma/('nf'fgychilnge=rna.u" 5pf!dfic hEat upadry " lemperaftEechange)
11 Show that the left-hand side of the equation
pressure
+�" deruity" (spl'i1d)2. constant
is homogeneous and find the base units of theconst3l1t on the right·hand side.
I
II Physical quantities and unilS
Conventions for symbols and units
You may have fl()(iced thatwhen symbols and unlS are prfnted. the)' appear In
dlffl'!"enI slyles d type. The syn:bol for a phySICalq uanlly Is pOnled In
.
type,whereas Is unlt Is In roman (upright) type. For exa mple
IluJIC�ng)
l·eIoctty IJ Is italic. but
Its unit m s-' Is roman. Of COU!'lle, youwll1l'101 be able t o make this dl$l:lnctlon In
handwriting.
At
AjAS lewl andbeyond, tnere Is J specbl COfII't'ntlott for IaIX'H1ng columns of
data In t:1blesand graph axes. l1Ie symbol IS prlntedflrsl On Italic), separ:lIed by3
forward slash (!:he printing term Is a solidus) from the un i t (In roman). l1len the data
Is presellled in a column, or along an axIS, as pure numbers, his Is Ulu:;trnted In
Figure 1.7,which shows a table or data and the resuiling grnph for Ille velocity IJ of a
T
panicle at wrlous times t.
.r
--"....... ,,.?'
_._./
�::"
�
=--.,.
�
-... ,,­
<�
-..,;
-.-
'. L:
,.. 4.0
-�-:��
2.0
,
2.0
lis
Figura 1.7 Th�convenlo
i n!of
If you remember that a physical quantity COfIialns a pure number and a Unll. the
reason for h s
I l SlyIed presentallon bemme:s clear. By dMd lng a phySICalqll3ntlly such
as tlffiI'" � number and a untO by the approprtlte unit you are left with 3 pure number.
It Istben algetmtlcaUycurect fcr the data In tables. and along graph axes. to appear as
.
pure numbe!s.
Figure 1.8 TlM!r�boofthe fTUSSo f the
humpboiCk wtYle to thl massofthe mouse
� �ut 10" T�t is nIIflutt comp�red
the r ;rtlOoftlM!m�ssof �g..Jl.1)('fl0thtfTUSs
of�nude\r.i(H)'·11
You may also see examp!es tn which the symbol for the physlclllqll3ntly Isfolm·ed
by the slash, and then bya poI\·erd 10, and then the unit. for example (/10ls. This
means that the column of data has been dMded by 100, to 5;l1'e repe;itlng lots of
zelO5 In the table. If you set'a table o rg ph Iabelledrjl02s and the OgUI\'S 1,2,31n
the table column or along the graph
rn
nls, thiS ffil'"ans that the expertmcnaal dat3was
obialnedatl'llluesdrofIOOs,MOs,300s.
Try to get out of the habit of he:tdlng table columns and grnphs In Y,"Jys such as
'r In 5', 'I(s)' or e"en of recordIng each fl.>Jdlng In the l�l>Ie as 1.0 oS, 2.0 s, 3.0 s.
Order of magnitude of quantities
It Is often useful to be able to estlm�te til(> size, or order of" m a2nitude. ofa quantity.
Strl<:tlyspeaklng, the order of magnRude IS the P'J"'·er of ten lowhlch the numlx'rts
T':llsed. The ability to
Table 1.4 SOme�alul!5 01 d lS\�nc
dil
Eollth to edge
f � g�laxy
di
r thtotheSun
di
length 01 �c�r
,, "
=
�
"-
; :
a
::�r;;: �:!:�st�:c: �� ��:� ��:�:
h
distancewhich may be met In the AjAS Physlcs course.
1.2 .. 101'
out experiments orwhen suggesting theorIeS. HavIng an Idea of lhe eXpeaed result
15 ,, 10"
.S .. 10!
. -j
,, ..0'
.=
---to', -0
S" 10-4
f'
�� ::
e
���-� � �
t
1.4" 1(1211
olob��bleUnM!rse
d�
i 1T'li!1i!I
estimate Is (Xlnlcularly Imponant In a ,SI.IJ>te<=t ilkI.' physicswhere
quantities hal"l"such widely different wlues. A sbondlstance For an astrophystcist
�===========:;:�..�
, �g'm=1 �:
3" 10- '0
tallce lrom
The abtllty 10 estimate orders of magnlrudE.' Is valuable when planning and carrying
o
l311
pfOllldes a useful check lhat a silly error has not been mxle.Thls ls also troewher!
l311celIs
using a calculator. For example, the acrelf'nltlOfl of rree Fall al lhe f.arth� surface
abouI 10m s-'. If a value of 9800m rz to!; calculated. then this Is otMously wrong and
a simple erta" In the JX)'IH'r of tm Ls likely to be the cause. Similarly, a calculation In
whlch theco;lofboUinga k enJeorwa\{'r lsFoundlObesel't'f;JldoIb�rntherlhan a
;
d;=
f
may Indicate that the eneflO' has been measured In wau-hours rnther than
ew cen ts,
1.J Scalars and vectors
Example
It is worthwhile to remember the liiZH of some common objects so that comparisons can
be made. For example, Oil jar 01 peanut butter has a mass 01 about 5009 and a canon 01
orange juke has a yolume 01 1000cm1 (1 litre).
Now it's your turn
12 Estimate the lollowing quantities:
(a) the mas.s 01 an orange,
(b) the mass 01 an adult human,
(e) the height ol a room in a houst',
(d) thediameterol a pmdl.
(e) theYDlume ola 5lTlal beao,
(f) theYDIume ola humanhl1ad,
(g) the SJX!ed ola jumbo jet,
(h) thetemperatureolt:he r.uman body.
1.3 Scalars and vectors
AU physical quantlUes haV{' a magnl!ude and a unIt. I'or some quantl!les. magnl!ude
and unIts do not gl"e us enough Information to describe fully the quantity. I'or
example. If w\" are gtven the time for whICh a car trnl'eLs at a certain speed. then we
em calculate the distance travelled. However. we cannct find out how 13r the car Is
from Its .ltartl ng (X>lnt unJess we are rold the dlrecllon d tral'e!. In thLsC2se, the speed
and direction must be spe<:1fled.
A quantity which can be de;uibed fully by giving rts magnnude is tc.nov.n as a Kalar
quantity. A vector quantity hu magnnude and direction.
Some examples d SGllar and .l'CtOr quantilies are gll"eO In l:tbie 1.5.
TOIblel.5 Somesc.1lars andvec�
quantity
weight
"...,
Flgur. ,.9 Altllough the .1thlete nJns IOkm
In therro!, hll flnaidist�efrom tllestGlrtirog
pointmay well be zerot
v@locity
temperature
Note: ltntayseemthat electrlc currentshould be trealcd asa vector quantky. We give
currem a direction when w\" deal wtth. for example. the motor effect (see 10plc 22)
and when we predict the direction of the magnetIC field due to currem-carrylng coils
and wtres. Hov
..ever. electric cum'nt does not follow the \lWS of IOOor addition and
slKJuki be treated asa scalar quantlty.
Example
A 'big wheel' at a theme park has a diametllf ol 14m and people on the ride complern one
revolution in 24s. Calrulate:
(a) the distanc:e a rider fTlOYe!i in J.O minutes,
(b) the distanc:e 01 the rider from the starting poliition
I
II Physical quantities and unilS
�60 " 7.5 revolutions.
(a)
In 3.0 minlltes, thl! r'der complete� 3.0
f wh
distance travelloo '" 7.5 >< cirrumfereflce o
eel
", 7.5 >< 21 1 >< 7.0
", 130m
(b) 7.5 �utions are completoo. Rider is Y, revolution from thl! starting point. Thf.
rider i'iat the opposite end
diameter of the big wheel. So, the dist�l1(e from
ofa
starting position '" 14m
Now i1's your 1urn
13 State wheth
following
er the
quantities are scaiars Ofwectors:
(al timeofdepartureofatrain,
(b) gr vitat ona
(c) ernityofa
14 Sta whether the
quantities are sca ars Of wect s
I_j mowmentof the
ofa dock,
(b) frequency of vibration,
Ie) flow of water in a pipe.
Spero and velocity have tl".eSdme units. h�ain why is sca ar ql.lantitywhereas
velocity is a vector ql.lantity.
student slates that a bag of sugar has a weght of
that this wei9hl isa
qlJantity. DisClJSS whether the stlJdent is correct when stating Ihal wei9hl is a
a
i
d
te
l field strength,
liquid.
following
l
harnls
or
a
speed
15
l
I O N and
16 A
VeclOf
Vector representation
When you hi a tennls ball,)"OO have
we!I as how hard
to
to
judge the direction you wall/ it to move In. as
hi I. llle forr:e you exert IS thcrefore a
\'eCtOfquantity
andGinnol
be represemed by a number alone. One way to represent a vector Is by mean of an
s
arrow.
The dlrectklfl of the arrow IS the directIOn of the\'0001" quall/lty. The length of
the alTU'\\'. dra"'"!lto SClIle. represefU IS magnitude. This Is UJustr:ued In Figure 1.10.
Scale: I unil represents 5 ms-1
�)velocily l5 ms-l, dueeast
Figure 1.10
Representation
I
b)V8loclty lOms-l, du� south
of � vl'C Of q �ntl
t u ly
Addition of vectors
The addlUOIl of t"l','O scalar quantnles which have the S:IITK' unit Is no prohlem. The
quantltles are added usingthe normal rulesaf addttlon, l'orex:tmple, a beakerof
volume 250cml and a bucket of volulIll' 9.0 Ilres have a tOl:al vo/UIIll'
of 9250cm1.
Adding together two vectors Is more difficult because they hal'(' dlnx:tlon as V<"l'll as
magnitude. U the {V<U H'Ctors are In the same dIrection.
then they Cln simply be added
together. 1\\"0 objects ofwe\gt( 50N and 40N ha,'e a combined ....
'eight of90N because
both weighls act n
I the same dlrecrlon (\enk."311y do.vnwards).
Figure LII
shows
the
effect of adding "'"0 fon:es of magnlude:s JON and 20N which act along the salIll'
line in the salIll' dlrecrklfl or In oppo!ilte dlrecrlOflS. TIle angle between the forces Is
0" when they act n
I the same dlrecrlon and 180" when they are in opposite directions.
For all OI.her angles between tlie
dlrectlOn5 of the forces, the combined elToo. or
re5ullanl, is some \'3.Jue between ION arKI50N.
1.J Scalars and vectors
r1
"V"
-
-
.1\
Figure 1.11 Veaoraddi/iof\
FtgUTII 1.12 Ve<:tor trlang1es
l"('Suttam Is found by means of a vector triangle. l!:lcll one of the two vectors VI and
1 20N
---+-
V.
---+-
� ION
pposke directions, the
In cases where the twoveaors do no! act In the salOO Of o
VI Is represented In magnitude and direction by the side of a triangle. Nelle thai boIh
vectcrs must be In either a cbckwlsl' or an anuclockWlse direction (see Figure 1.12)
The combined effea. or resull3!U R. Is given In magnitude aod dlrectloo by the third
side of the triangle. It Is Important to wmember that. If VI and VI are dr:lIm cbckwlse.
then R Is amIcJock"1se; If v, and v, are antlClcd;wlse, R Is clockwise.
1lIe rerullanl may be found by means of a SClIe dbgrnm. AltemaU\'t'Iy, having
drawn a sketch ofthe vectortrlangle, the problcm nlay besol.,.ed uslng trigonomel:ry
(5eelhe Maths Noreon page 14).
4.0kmh-1
A V1ipis traYl'"ingdUl! north with il � of 12km h-l relatjye tothe water. fhE.re isa
currnnl inthe water flowing at 4.0km h-1 inan easterly direction. Determine the velocity of
theVlipby:
(a) 'IGIle drawing,
{b) cakulation.
(al BySGlle drawing {Figure l.13):
Scale: l cm represents 2km h-1
re5llitant R
Thevelodtyis
6.3 " 2 ,, I2.6km h-i in a dired:km iBo east of north.
{bl By calculalion
Referring to the diagram (Figure 1.14) and using Pythagoras' theorem.
R' .. 121 + 42 .. 160
R. ,ff(jj .. 12.6
lana ..
';;' .. 0.33
a . 18.4·
The�locilyofthe Vlip is l2.6km h-i i n a djrectlon 18.4·ustof nor1h.
Now it's your turn
17 Explain how an arnm may be uloed to represent a YK1CH" quantity.
18 Two forces are of rnagnitude450N ard 240N respectively. Determine:
(A) the maximum magnitude of the reloU1tant force,
(b) the minimum rnagnitudeof the resultant force,
(e) there!.Ukantfort:e whentheforces actatrightangles toeat:h othef.
Use a YK1CH" diagram and then checX your reloU1t by t:alcUation.
Flgur. '.14
I
II Physical quantities and unilS
19 A boat can be rowed irt a speed of 7.0km h-I in stili water. A river flows at a constant
speed of l.Skm h-I. Use a 'l£ale diagram to determine the angle to the bank at which
theboalmustbe rowed io order thatthe boattravels directly across tl\eriver.
a point P ilSshown in Figure 1.15. Draw a vector diagram. 10 'l£ale, 10
20 Two forcesiKI
OIl
determinelhe resultantforce. Chedyourv.orkby caiculation
21 A swimmer who can wmn in st�1 water at a speed of 4km h-I is swimming in a river.
The river flows at a speed of 3km h-I. CalaJlatethe speed of the swimmer relaliveto
th'i'river bank when 'iheswiITl5:
(.) downstream.
(b) upstream.
n Draw to Kille a vector triangle to determine the resultant ofthe two forces shown in
Figure 1.16. Ched: )'OUr answer bycak:ulating the resultant.
TIle use d a vector triangle for nndlng the resultant of tWO \'t'CIOrS an be
demonstrated by ITlt"aflS of a simple laboratory experiment. A welglu Is anaclled to
each end of a flexible thread and the thrt'ad IS t»(>n
suspended Cfo,'ef two pulleys, as
shown 10 Figure 1.17. A third weight Is anached to a point l' nearthe centre of the
Flgur. l.16
thread. The string moves on'f the pu1Jeys and then comes 10 rest. The po6ltlons of the
threads are marked on a ple<:e of paper held on a boord lX'hlnd the threads. 11115 Is
easy to do If light from a small lamp Is shone at the boon!. Having noted the sizes WI
a vector triangle can then be drawn
the paper, as shown 10 Figure 1.111. ThI: resultant of W, and wl Is Found to be equal
and W, of the I'o'l'lghts OIl the ends of the thread.
on
In magnitude but oppolilll' In direction to the weight w�.
If IhlS were not $0, there
would be a resultant force at P and till-' thread and weights would mewe. The use of
a vector triangle Is tustlfJed. The three
forces WI. Wl and WJ are In eqUilibrium. The
rondlUOIl for the vector diagram of these fofces to represent the equilibrium slu:llJon Is
that the three vectOG should form
a closed triangle
Flgurli! 1.17 Apparalusto dledlhe usfol<lvector trlingie
10
1.J Scalars and vectors
4J
Flgur. l.18The�tof trl�ngle
We have considered onJy tht> addllion 0(1\\,10 ,'eclOf5. When three or more Vedors
need lobe added, the same prtndples apply. pl'Olllded the \'t'C\Orsare coplanar
(;1,11
In the same plane). The vector Ulangil' then becomes II "ector polygon: the resultalll
forms the missing side 10 clo5e the polygon.
To subtract two ,"t'Cta"s. r�"t.'rSl' the direCUOfl Oha! IS, change the sign) of the ,'eCtOr
10 be sub!racted, aoo then add.
Resolution of vectors
On pages 8--10 we saw 111:1.1 1,,"0 vectors may be added togecher to produce a
single resultant. ThIs resultant beha\"t�s In IIII' same way as the 1",,10 Individual
veelors. II follows that :l. single vector may be split up, or resoh'e<!, InlO IWO
vectors, or components. The combined effect of the components Is the SUlle as
Ihe original vector. In laler chapters. we will sec Ihal resolution of a VOODl" IntlO
."''0 perpendicular components Is :l. very useful means of solving certain types
of
problem.
COnsider a folU" of magnllude F acllng 3t an angle of 6t>e1ow the hOrlZOfllal (see
Figure 1.l9} A vector ulangle can be drawn wl1h 3 component Fu tn the hort7.0ntal
dtre<:!lon and a component F" acting vertICally. Rememt>erlng thatI-: F" and F"form a
rlght.angledulangJe, then
F,, =FCOS6
and F,, = Fsln8
The force F
lias been Il.'sotved IniO t\"O pl'rpefldlcular components. F" and F". The
example chosen is concerned wIll forces. but tile method applies to all types of vector
Flgur. '.'9 ResoMng�
quantlly.
'>«torintocomponents
Example
A glider is lalJl"1dtl!d by an airuaft with a cabl@, asshown Wl Figure l.20. AI one particular
morTII!r1t.theten� inthe cable is620N and thecablemakes anangleof2S·withthe
horilOrltalbee Figurt' 1.211 Calrulate:
(al the forcppuiting the g6derhorilOfllally,
(bl the vertical forc:e exerted by the cable on the ooseof the glider.
(al horizontal rompooent F" " 620 (OS 25 • S&O N
(b) vertical {omponent F" ", 620sin 2 S _ 260N
Now it's your turn
23 An airc:raft is travelling
of the aircraft in·
Rgure l.20
�620:'
3S0 ea51. of north al a speed of 310km h-l. Cak:ulate the speed
(a) the r.ortherly ctirection,
(b) the &lSterly ctirection.
of 9.2m S-I. The hillsicle makes an ang11' 01
6.3° with the horizootal. Calculale, for the cyclist:
24 A cydist i5 1ravelling down a hill at a :;peOO
(al the verticaI 5pero,
(bl the horizootal spero.
!
Rgur. '.Z1
11
I
II Physical quantities and unilS
. AII �l quantities hiM!a magnitudl! (s.i.ze) and a unit.
• The SI base units 01 mass, length, tinll', electric rurren!, thermodynamic temperature
and amount 01 substance are the kilogram, metre, se<:ond, ampere. kelvin and mole
�spectively.
• Units 01 all medlankal electrical, magnetic and thermal quant�les may be deriYl!!! in
termsolthese baSl!unil5.
• Ph�al equations must be homogeneous (balanced). Each term in an equation must
ha-..e the..amebaseunits.
• Ttw CDn-..ention for p"inting OOadings in tables of d.lta, and IOf label�ng graph <W!S. is
the symbol IOf the physical quantity (in italic), follolWd by a forward slash. follmwd
bythe abbreviation for the unit (in roman). !n hanawritlng, one cannot d istinguish
between italic and
• Ttw order of magnitude of a numbe!" is the power of ten to wNct1 the number is
rafsed. Th.e Ofder of magnitude can be used to make a check on whether a cakulation
roman type.
givesa sensible amWl'r.
• A 5Calar quantity has magnitude only.
• A Yectorquant�yhas magnitude anddirection.
• A Ye<:tor quant�y may be representi!d by an arrow, wfth the length
01 the arrow
drawn to 5Cale to give the magnitude.
• The rombined effectoltwo(or more)vectors is calledth.e resultant
• Coplanar vectors may be added (or subtracted) using a wctOf diagram.
• The resultant may be lound u5ing a scale drawing
01 the Ye<:tordiagram. or by
cakulation
• A single VectOf may be divided into two separate components.
01 the wctOf.
a Yector is resolved into two componems at right angies to each other.
• The dividing 01 a Yedor into compooents is known as the resolutkln
• tn generai,
Examination style questions
1
i EKPI�n what is meant by a ba� urir.
ii Give four examples of base units.
b State what is meant by a derivedunit
(; i for any equation to be valid. it must be
homogeneous. Explain what is meant by a
homogeneous equation.
ii The pres:surepofan ideal gasofdensity p is given by
the equation
p_jp(c2)
ooere <cI> is the mean-square-speed (i.e. itisa
quantity measured as [speedJ2)
Use base units to show that the equation is
homogeneous.
2
b
a
The period
Tol a pendulum of mass AI is given by the
4
Explainwt.yyourans� to a mean that cautionis
required when the homogeneity of an equation is
being tested.
a
Di!>linguish between a scaiilrand a vector quantity.
b A mass of weight 120N is hung from t'NO strings as
shOYlfl in fig. 1.22
Y
"
<0': :0'
120N
Fig. 1.22
Deterrnine, by scale drawing or by calculation, the
tension in:
Detefrnine the base units ofthe quantityl.
]
"
a
Deterrnine the base units of:
i v.<)f1<.done,
ii the moment of a foroe.
i
RA.
ii RB.
EJ<amination styl, questions
<:
Use )'OtX answers '" b to detennine the horizontal
The 5peed of the wind is 36m s-' and the speed of the
airuaft is2SOm s-1.
i RA.
i
component of the tension in:
i RB.
S
Aflelder ina cricket miltch throwsthe ball tothewicket­
ke�r. At one moment of time. the ball has a horizontal
Yelocityof 16m s-' and a velocity in the vMicaly upward
direction of 8.9m s·'.
a Determine, fOf the ball:
Cambridge International AS and A Level Physics,
9702123 Oct/NcN2012 0
7
is trave!ling re!ative to the
,= sa
Determine the base units of c.
ball just as il is caught:
(3]
Cambridge International AS and A Level Physics,
9702/21 Mayl)une2012 0 1
i itsvertical speed,
ii the angle that the path of the
8
ball makes with the horirontal.
Suggest with a reason v.ohethef
theball,atthe moment itis
caught, is rising Of hilling.
a The spacing between two atoms
ina aystal is3.8x IO-10m,
State this distan(e in pm.
Ms.
(2/
9
Flg. 1.2l
a T'M) of the SI base quantities are mass and time. Slate
three otherSl base quantities.
/3J
b A sphere of radius r is IllO'.'ing at speed v ttYough air of
demiity p. The resistive force I' acting on the sphere is
given by the expression
where B and II are constants without units
momentum
(1]
e The velocity vector diagram for an air<:raft heading due
north is shown to scale in Fig, 1.23. There is a wind
blowing from the north-west
IIJ
F= lJrptl'
d ldentify all thevector quantities in thelist beiow
'M)rk
m
9702102 MayfJune 2008 0 ,
/1/
rrom the Sun to the Earth,
m
m
c the mass of 11 plastic 30cm ruler
Cambridge Internaional
t
AS and A level Physics,
c The distance from the Earth to
theSunis O.1STm. Caiculatethe
time in minutes for light to travel
Make reasonable estimates of the following quantities.
a the frequency of an ilUdibie sound w_
b the w_length, in MI, of ultr<Niolet radiation
d the density of air at atmospheric pressuAL'
{II
b Calculate the time of one day in
weight
m
where pis the pressure diffeAL'nce between the ends
01 the pipe of radius rand length I, The constant C
depends on the frictional effects of the liquid
velocity remains unchanged. The
speed of the ball ilt the moment
when the wicket-keeper catches
itis 1 9 m S-' . Calculate, for the
energy
I
m
v '"'"
horizontal.
b During Ihe flight of the ball to
the wicket·keeper, the horizontal
distance
11 i State the SI base units of volume.
ii Sllowttwt theSl base units ofpressure aAL'
kg m-1 S-l.
b The volume vof liquid that flO'NS throogh a pipe in
time lis given by the equation
ii the direction in wt-ich it
6
[2J
theailUaft.
i itsresultantspeed.
<:
Makeacopy o/ Fig. l.23. Oraw an a m:lW loshowthe
direction of the reSlAtant velocity ofthe aircraft
m
ii Determine the magnitude 01 the rmtant velocity 0/
Comment on your answer.
i State the SI base units of ,.; pand v,
ii Use base units to determine the value of II
/3J
12J
Cambridge Internationctl AS and A Level Physics,
9702/21 Oct/Nov 2010 Q 1
13
I
II Physical quantities and unilS
Sine rule
Flgur. 1.24
For �ny triangle (Figure 1.24).
�
�
�C
Si:A = SI' IJ = Sl'
Cosine rule
I'or �ny triangle,
tr= tr .. & - 2Ix: COI> A
tr=a' . . & - UlCcosIJ
c' = tr .. tr-UWcosc
14
Flgur. 1.25
For a rlghHngk'd trlangle (l'lgure 1.2S).
.. u·
AJso for a rlghl·�nglcd triangle
sin 8 = *
1:058= *
tan 8= ;
h' = rr
AS Level
2 Measurement techniques
By the end of this topic, you will be able to:
• use a c::lthode-ray oscilloscope (c.r.o,)
(a) use techniques for the meaS\lremem of length,
• use a caUbrnted Hall probe
volume, angle, mass, !lme, temperature and
(b) use both analogue scales and d1iltaJ dIsplays
electrical quantHles, all to appropriate rnnges of
magnitude. In panlculaf, YOtl should be able to:
(c) use callbratlon curves
(a) undemand andexplaln Uu.· effectsof systematlc
• measure lenj/ths tlslng a ruler, calipers and
errors (Including zero errors) and random error.;
ln measurement
• measure welilu and hence mass using balances
-ren preclslon and
(b) understand the distinction bet ....
• measure an anale uslna a prolractor
• measure Ume Intervals using docks,
accuracy
(c) aSlieSS the uncertainty ln a derived quantlty
StOP"''atches aod the calibrated lime-base of a
by simple addillon or absolute, fr:loCtlonal or
cathode-ray oscilloscope (c.r.o.)
percemageuncertalnlles
• measure temperature using a thermometer
• use ammeters and \'OItmelers with appropriate
Note: U!ie of a callbraled liall
probe Is required only n
I the
A len�l course bUI IS InCluded here for completeness.
• use a g:&lvanomeler n
I null methods
Starting points
• n-x-etical idea'> in phfsics <lfe generally tested by e.periment before being fully
accepted.
• Experimental work is an important part
of a physics course.
• Make a sernible choK:e of the in5trument to use to measure a particular physical
quantity.
• Thefeafl"sotln::esof errorilfld uocertainty in e;q>erimerrtal worl:.
2.1
Measurements
All l'X]X'"rlmems that 3R'" designed to
d:>taln a qu�ntI1all�'e reSl.Il1 for a ptwsk;al quantl1y
Involve rneaSUR'"ments. These measurements ntUS1 I)C of some oomblnallon of the base
quamllies length, mass, Ume. temperature and current
Inclu{\(> quamlty
In experimental
of subslance and luminous
(1b complete the list. we shoukl
Intenslly. 001 llH.ose are ne( encountered
\\-urk In AlAS PhysICs,) In Ihe following sections we will look at the
rncthcx!s avallabie formeasurlng thebaS(' quantttles lna schoo] Of collegelaboratOl"}".
By understanding the
principles of the available nlC'lhods, we will lX' able to make an
InlOrmed dedsloo about the choice of a panlCular technique. With respect to ntlklng
the
experln1lt'U as precise and reproduc1b1e as possible. and avoiding 1lltln::es of
systematIC error. For all of the quanlltles. the effective choice wtU lX' IImled by what
Instruments are available In )'our laboratory. Howl'''er. In one type of eX:lmln:ation
questlon.. 011 planning and design. you may be asked to deYlS(' an expet1n1lt'U and draw
OII)uur theoreUcal. ralhertltanpr.K"l.k::al. kOOYl1edge of\'aOOUstypesofapparatus.
At AlAS Ie,"t'!. students wnerally 35.9.1me that the callbl'"atlon of the InSironl('1IlS they
use Is rorrect.. Jic/I\'cver, It Is worth thinking about hcf,t" to' compare the calibration of
one Inslroment against anol:fier, ewn lr thiS IS 3 check )'0.1 will \"t'ry seldom make.
15
II Measurement techniques
Figure 2.1 StudenISdoin !l � eJ<.l)enmenl ln�
phy5kslaoorJtory
In a planning/design que.ltIOfl. you might be asked to suggest a m('lhod of calibration.
Gener:J11y I1 lsea�yto compare thl- callbr:JIlOn cldlfferffit lnstrumffits. but nct so easy
to determine whtch cl tv..-o lnslfumffitsglvlng dlfferffit readings Is corre<1.
On the pages that follow. we will spl'nd most time on m('lhods for measuring length,
be<:ause lensth-measurlng Instruments of severnl different tylX's will be available In
yourlabor:ltay.
Methods of measuring length
The metre rule
The simples!: Iffigth-measurlng Instrument to be found In )'our Iabor.Itory is a me'lre
(or half-metre) rule. U has the great advantages clbelng d\e;j.p. COO\'Cfllent and simple
to usc. A rHltln�ly unskilled student should have no dlffku_y In taking a reading \\1th
an uOCCltalruy aO.5mm. HOI\·C\w. )"OU should be aware clthree possible sources of
Flgure 2.2 ZeroeITorwllh � m etreruit
ermrln uslng a metre rule.
"The firsl may aT"\se If the end of the rule IS worn, gMog rise loa rero error
(PIgure 22). For this reaSOCl, 1t Is ood pr:lcUce to pla.ce tlte zt'fOend of the rule asalTl5l
one end of the object to be measured and to take the reading at the otlte1" end. You
should place- the objoct asalTl5l the rule so that a readIng Is made at each endclthe
object. The lengtb of the objecr is then obtained by subtractton of the two readIngs
A �ero error like this Is a systematic error. b«"<lusc It Is Invol.,.oo every time a reading
Is taken from the rero end of the rule. (j.. more detailed explJnatlon of systematic
errors will be found on page 35.) In gen(>fal, the zero reading or any Instroment may
be subject to an error. We shall rneet thiS type of error again In the micrometer screw
gauge and In the ammeter.
Th(' calibration of the metre ruk> may give rlsc to an�lter syswmaUc error because
the markings are locorre<."t. Try comparing th(' 30cm gt:l(\u�ted leng1h cI one plastic
rule with the same nominal length 011 another. You are qUIte likely 10 find a discrepancy
of one or two mUllmetres. One of till' reasons why wooden Of pla5t1c metre rules are
cheap Is that the manufacturerdol's no! claim any gre:lt accuracy for the scale markings.
The calibration may be che<:kOO by laying the ruk> alongside a lnore aocurate rule. .such
as a Slee! JOan or metre rule, aTKl noting any discrepancy. If you compare an engineer's
steel rule wlh a plasttc rule, )'QU wUl see at 0I'lCC that Ute engr:JI'E!"d marks Ofltill' .9.eeI
rule are much finer than the Imp!l.'5lit'd marks on the pbSlIc. Of COUTSt'. the extra care
whtch hasbt'en taken In engraving thesteeJ rule has t o bc paid fOf. A OIl('-metre steel
rule Is many times lllCJe expens/\'e than a pb5t1c metre rule.
Another soolU'" of error with the me'lre role lS parallax error. If the object to be
ntea.surOO Is TlOI. on the S3me IC\-el as the graduated .surface d the role. the angle at
which the scale Is v\e\\� wUI affect the resul (I:lgure 2.3). This IS 3. r.mdcm error U;ee
Figure 2.] P�r�n� errorWlIh � metre rul.
16
paW }5), bec:ru5e the angle dview may be different for dIfferent
readIngs. It may
2.1 Measurements
be reduced by arranging the rult' soth:lt tlll're Is no gap between the SCI!e aoo the
oIJtect. Parallax error Is also Imporrant In reading any lnsuunleflt ln which a needle
mo\:es 0'IeT a SClIe. A ratm sophlsUcated way d eliminating IJ'Irllllax elTO'" Is to place
a mJlTO'" alongside tbe SClIe. When the needle and SClIe are viewed directly, the needle
and Its Image In the mirror coincide. ThIS ensures that the SGlIe reading Is alw:i.ys
taken at the sante vll'wlng anglt'.
The smallesl: dtvll;\on on tbe metre rult' Is I mOl. If you take precautions to a\'OkI
parllilax error, rOll should be ablt' 1Oe51lmate a readIng to about O.!jmm. If)'OlI are
measuring the length of an object by taking a reading at each end. the uncertatntles
add logl\l'a total uocertalny of Imm. 1be range dtlte rnecre rule lsflOlll lmm 10
1000mm. lb measure a length of ITIOCl.' than I m with a metre rule wUl ITII.roduce a
further uncertainty, of perhapos I mm or 2mm. bec:luse d the difficulty d maklng :l
reference mark at the I m end ofthe rule and rIl(Wlng therulesothl1the zeroexactly
mrresponds wlh this reference. It Is usually better to usc a ;teet tape to measure
lengths of more th:in 1m.
The micrometer screw gauge
The type of micrometer screw gauge available In a SCIIOOI or college laboratory may
be used to measure the dimensions ofobtects up to 3 maximum of about 50mm
Measuremeflts can easily be mad(' with an uncertainty d 10).101 or I('ss. 111e principle
d the Instrument Is the magniflcallon of linear motion using tlte circular motion of a
SCIl'W. The Instrument consists ofa U..shapcd piece d steel wUh a fixed. plane, rod·
plffe A (see I'Igure vi). Opposite this Is a screw Wlh a corresponding end·pIe<:e B.
llle posllon athe screw can be adjusted USing the ratcllCt C which Is con1l('(:ted to
the thimble D. There are graduations along the barrel dtlte Instruntent (the bearing In
which the screw turns), and round the drcumference of llie thimble. The purpose of
the ratchet Isto ensure that the same torque (that IS. till' �mount oftwl<;() Is applied 10
the Ihlmble for each reading. If this torque Is exceeded. lhe r.ltchet slips. The object 10
be measured Is placed In the jaws of thE' gauge between end-pieces It. and B. �nd B Is
screwed dol\'Tl ontothectJtect, U5lng thE' ratchel C,unUl thE' r.ltchel sltps.
1llI" SCft.'W advances exactly I mm for twO r(.>'.'O/o1l0n5. That ts, tile pitch dthe
screw IsO.5mmor 500�m. [f)'OlIlookat the gradU:ttlons O!l thebarTei ofthe SCll'W
bearing you will see th:lt there are dtvlslonS every OSmm. The reading on thE' barrei
corresponds to the pootioo a the edge dthe thimblE' (see figure 2.5). When I�klng
Flgur. 2.5 SCrewg.1ugesc�eswlth.1
rudmg ol986mm
� reading lis Impat�nt 10 check which jQlfdthe mUlimetTe the edge ofthE' bJrreI
Is In. The graduations round the dlUlmference d the tillmblc run from 0 10 50. Each
dlvllilon COffi'Sponds to o;ne..hundredth d J mm, or 1O�lm. llIe re;idlng al the thimble
17
II Measurement techniques
Is added 10 the reading on the IxII1l'L Thus. l'Igure 2.S show.'l � reading 0I9.Smm (II
the barrel plus O.j6mm on the t
h
im
.... 9.86mm In lObI. You can easll)' read 10 the
neaR'S( dlvtsklfJ on the thimble; thai is. to the neaR'S( om mOl (lO�n\).
The mk:rorllt.1er screw gauge Is 'l'ry likely 10 ha\'e a systematic zero error. ,,"very
time you use the gauge, you shoukl check the zero error by movIng f e B so that II
nukes rontaCl wllh face A. The screw nlUS( be lightened WIth the ratchel C. so that a
reprodudble zero Is obIalned. 11len take the reading on the oorrel and the thlmble
This gI\'es the zero error. which mU51 be aJlov.oo for In �Il subsequent readings.
Figure 2.00 shoIol.-sa screw gauge "1tha zero error 01 +O.12mm. lf thts"'ere the error
which applied when the reading 0I9.86mm was oblalned In Figure 2.5. the true
IengIh of the obfect ""OlIld be 9.86mm - O.12mm = 9.74mm. Figure 2.61> shows a zero
error ol--O.08mm. In this case, the true length of t e ob!ect In Figure 2.5 ""OlIkI be
9.86mm ... O.08mm = 9.94mm.
In the case oIa ,,"OOden or plastic me!re rule, • IS good pr.1CI1ce to check the
calibrntlorJ agalnS! an engineer's :lleel rule, If one IS avall�l>le In your LalXJr.ltory. II
would be unusual to do the same wnh a micrometer screw gauge, but If there Is
doubl: about the callbratlon ofa panlcular g.luge. It can be checked by mea.s\Jling the
dlmenslOflS ofa serJesol gauge blocks. A g.luge l)lock ISa rect�ngular.'jleel blockwllh
faces which are accumwly plane and pamllel. The length of the gauge block Is known
to an uncenalruy 01 less than I)lm. Howt.'Ver, not many school laborJtories possess
gauge blocks.
:llC
Flgur.Z.6")Zl!fO�rroro1.0 '2mm b)ztfO
I'rfOro1-0.08mm
h
The vernier caliper
A �'ernler caliper Is a versatile instrument fOf measuring the dimensions of an olllfft,
the diameter of a hole. Of the depth ci a hole. Its range IS up to about loomm, and
It can be read 10 O.lmm Of 0.050101 depending on tile type 01 vernier with which II
Is filled. It consists of a Sleel mm .'lGlle A with t\\"O reference postS at the zero mark
(:;ee Figure 2.7).
Flgure l.7Vemier(olliper
Sliding
IYernlefl
111II111II1""IIII""ijJ"
o
10
20
30mm
Flgunt Z.8 VerOler sCOlleWlth .. rtidingo1
2S.4mm
18
A slkllng pan B lTIO\"es along the .'lGI1e. The slider has the vernier scale engr:lI'OO on II.
The zero oIlhe vernier corresponds with reference posI.'l on the sliding pari. One set
of reference politS, lhose wlhthe straig.hl pans onthe lnside, l.'l usect l!ketheJa,,-s ofa
screw gau8if": the obfect to be measured Is placed between the jaws or reference poI'ilS,
and the slidlng pan B Is lDO\"ed along unlii the object Is gripped tightly. A reading to
the neall'S! mm Is taken on the fixed .'lGI1e. aI the zero end 0( the I'emler .'lGI1e. TIle
reading to:l. temh of:l. mOl Is obIalned by finding where a graduation 01 the ll'mJer
scale coincides \\1th a gradualloo oIthe fixed SClIe. l'Igure 2.8 shcFo\-s the SClIe ofa
\'emJer caUper glvtng a reading of25.4mm,
2.1
Measurements
11Ie serond set ot jaws has 11K> straight pans on lhe OUtside. These can be used 10
measure the diameter d a hole. Thl> j::Iws are placed Inside lhe hole :md are 1llO'I-l'd
apan unUI ltiey are I n CUlI:ICI "'11h Ihe ed9t'S oIlhe lloie. Tlle salie3nd wmler can
lhen be read.
A pin al lhe end otlhe sliding pan dlhe caUper can be used to measure lhe depth
ol a blind hole: for example. a hole which hils been drtlJed In. bul I10l rlg:IJ: lhrough. a
hole. and
the pin lTlO\'ed IntOlhe hole untU It reaches the Ixxlom (see Rgure 2.9). The reading of
wooden board. The end of the fixed salle IS ptaced on the board. aCfOli;S the
FlgurIt 2.9 MNsurementofthedepthof;J
bhndhole
Ihe salleaoo \-emler gfve!S lhe de-pthofthe hole.
AS with the mlcromele!" s;:rewgaugt', lhe\-emler callper should 00 che<:ked fora
eory
syslematic zero erlU before taking a re:ldlf18.
The k-amlng ctJjeclln's for thts course do not require students to !law knowledR!" d
the vernier scale for the th
papers. Howewr, some bbor.llorles lTLly be eqUipped
wllh\-emler callpers forpracllrnl worll.
Examples
1
Figure 2.1Oa show; the scale
of a micrometer screw gauge when the zefO error is �ing
checked and Figure 2.IOb :;hows the scale wI1en the gauge is t9htened on an object
What is the length of the objecl7
From Figure 2.10a. the zero emJr is +O.12mm. The leading in Figure 2.1Ob is lS.62mm
TN> !ength olthe object isthus(I S.62 - 0.12)mm_ 1S.50mm.
I""�jl
5
6
7 cm
Flgurlt 2.11
Figure 2.10 a);IDd b)
2
�jjjjljjjmjj\�
Figure 2.11 :;how; the scale 01 a wmier ca6per. What Is the reading?
The zro> 01 the vemier scale is �tween the 5.5crn and 5.6cm divisions of thefiXf>d
sea"'. T�ls coinddence bi>twet>n the third graduation of the �nier scale and one 01
the gradudtions 01 the fixed scale. The reading is thus S.Slon or SS.lmm.
Now it's your turn
1
Figu� 2.12a and 2.12b 'ihow thescdlesof a miaometer wew gauge when the zero
Is being cheded. and iI9iIin when medsuring the diameter 01 an object, WlJat ls the
diameter?
Figurlt 2.12
i) Mld b)
Choice of method
A summary of the range and reading uncertainty of length-meaSUring Instruments Is
glVl'fl ln Table 2.1.
Table 2.1 lengln-mealurimj imtrum�rrt5
untltrtalnty
In length
ctleck zefO, c�lbration errors
micrometer screw g�uge
ctleckzefO erfQf
vers��1e: Inside �nd outside diam�ten.
d".
In deciding which Instrumenl 10 use In a jJQrtlcular expertmt'm. you should ronskler
grea
flrg the nature of the length measuremt'n! )'OU hal'\! to ITLlke. for example. If you
nee<! IO ftnd the dlameter d a steel 5phefe. the screwgau8E.' and caliper te<:hnlques are
obvious candidates. You shouki thl'nronslderwhetheryou nero the
ter precl,;lon
"
I
II Measurement techniques
of the micrometer. In :.1 parllc\llar experlnK"m. lhe uncertainly tn the dlarnt'ler d the
sphere may be lhe domillani uocerlatnty
<- section 2.2 Errors aod unrenalnt\eS page
�l), aodln such a C35eltle f3CI tlut tht' preclSlOIl aI':lUablewtththe screwgauge lslell
limes that for tht' \-emJer caliper wllJ decide the argument. In an experlmenl which
llIay lag some time. you should also think about the resources of)'our I:!bor.uory.
Is II 5ens!ble 10 use what may be Ofl{' doo/y a .srtU1l number d aV;llbble screw
gauges, when theymay also be In demand by other gudenl:s For Cllner expertments?
Somellmes, In de9gn qUl.'5lIons, you are asked to thInk about till.' COSC of se«lng up an
expe!1menl. "We ha'-e menlloned tne difference In cosc of a Slee! mecre mit' compared
"'1th wooden 01' pla5tlc rules. It would be foolish economics to supply a gee! rule for
each of:.1 number of gudenl5, when I would be perfectly adequate to provide each
of them wlh a wooden rule and haw ooe gee! rule available In the laboratory for
callbr:itJon purpoo;es.
Application: measurement of prl!ssure differl!r'Kl!
A difference In gas pressure may be measured by comparing the helghl5 of liquId In
the two anns of a U-lUbe. FIgure 2.t} shows :.1 U-Iube coonected to a COIllalner of gas.
The pressure abovl"the llquld In tube A Is atll"106pheric pressureP..... The pressure
!JIr
al:.o.'e the liquid In tube B. and hence the pressure <:ig.u In the container. Isp. The
relallon between the pressures Is
p = p-.. + /!,hpg
where
/!,h Is the difference In vertical hetgJu between Ute levels of the liquid In the two
anns ofttle tube, p Is the denSIty of the liquid. and S Is the acceleration d free fall. Tb
find the pll'S!ilJre of the gas. all we need 10 do IS 10 measure tlte difference /!,h between
the Itqukl levels. assumlng thatp_, p ()lnd s) are known.
In some laboratories, a U-tube mounted on :.1 board to which a ''('''leal mllllm(1re
Flgur. 2.1l
SC'31e Is anached may be avallable_ This devIoce IS calle<l a m.:morneler. It Is then :.1
sImple malle!' to measure M. If
the manometer COIltalns all or water. the liquid In
the lube wlU ha"e a cunro 5urfilce which Is conca,'e oownwards ("lgure 2.14a). This
surfilce Is GlUed the meniscus. Use a 5eI-squal\' to nnd the reading on the ,"('nlcal
SC'31e COIll'5pondtng to the
boItom of the menIscus In each 5kk! d the U-tube. The
surface of the liquid In a manomeler nile<! Wlh mercury will be COfI\"CJ: (PlgurP 2.14b),
and In IhlsClSe YOll sboukl read 10 the top a the menlscus. Agaln. use a :set-square
10 3void parallax error. Ifa rnanometerts llOl avalilble,YOll \\1Il h:l\"C to arrange YOllr
own �-ysl:em of U-tube and SC'31e; for example. a hatf-metrP rule. Make sure that lhe
scale tsdamped ,·ertlcally.
topot
menIscus
Flgur. l_14 a)andb)
20
2.1
Measurements
Methods of measuring mass
l1le method of measurtng mass Is "1th a balance. In fact. balances compa.re the
u'f!l8bl d the unknol\'n mass "1th the weight d a standard mass. Bul because weigh!
Is proportlorlll.1 10 mass, equality bet,,"l.'ffi the unknov.'n weight aoo the weigh! d the
standard mass means that the unknown ma5li IS equal 10 the standard nl:lSli
[n your labor:llory. roo may h:a,l' acce5li to a number of dlffcrt'ft types d balance,
tncltidtng the top-pan balance.the le\l'rba13nce aoothe sprtng baUnce. l t l s lmport3nt
th:at you lJIoukl famillartse youOOfwlth the
use of a ll type5 tlut are available to you,
type. NOll.' also thai. some
so tlut you do 001 resu1ct your choice 10 one particular
types ofsprtng balance may be callbrnll.'d In force unlls (tlut Is. In OC\\1on) rathertlun
In nuss unls (kilogram).
The top-pan balance
l1le top-pan balance (Flgure 2.15) l s a dlrecl-readlng Instrumenl. oosed on a pressure
sensor, or sometImes a sprtng. The unknown mass Is placed 011 the pan. and Its weight
appllc-s a force to the sensor. The mass corresponding to tltlS
f orce Is displayed on a
dIgital read-out.
When ustng the balance, ensure that the Inl1lal (unloaded) reading Is 7.ero. There Is
a control for adjusting the zero reading. The balance may 11al'(' a tare facility, for use In
backing off the mass of an empty COntainer so that the mass of matertal added to the
container Is obtained directly. This workS In the same way as adjusting the balance for
zero ermr.
The uncertainty In the reading of a particular top-pan balance will be quoted In the
manufacturer's manual AS "1th other dlgl1al lnstrumcnt.s. U Is likely to be expressed
as a perreuage uncertainty of the reading shown on the scale. together with the
Figure Z.15 TOP-Pin b<lIanCf
unr;:ertalnly In the flnal flgure ofthe dlsplay.
The spring balance and the lever balance
9): thl;> exten�1on of
meast.lred directly. by a
Spring balances (figure 2.16) are based on Hooke'S law (see TOpIc
a looded sprtng Is proportional 10 the load. l1le eJttenskln Is
ma!Xer IllOYlngaJong a Slraight !il1Iie, or by a polnter moving OI'er a clR;:uJar scaie. As
�1th a ny lnstrument uslng a SGIieand polnter. )"O.Ishould take care notlO lntrodoce
a parallax error when you take readIngS.. PosItton your.;elf so tlut your line ofstgh: Is
p!'fllt'"OdJcubr 10 the- scale. Before placing the obfeCt of unlmcw.ll mass on the pan.
check fO!" zero error. 1bere Is Ukely to be a rero-error adjUstment §crew on the balance.
lever balances are based on the prtnclp!e cI rnorJ"Ients. l n ont' common type
(Figure 2.17), the unknown mass Is placed on a pan, and balance Is aclileved by sliding
a mass along a bar. calibrated In mass units, umll the bar Is Ilortzontal. This represents
the condition In which the momem of the load IS equal and opposite
of the slkllng mass and the bar. A reading IS taken
\0 the momem
f rom tlte edse of lhe sHdlng mass
on the dlvlstons malked on the bar. In thts case. par:tUax error ts less likely to be
serious. Again, check for zero error before taking a reading.
Figure Z.1.. Spring balillce
Flgure Z.17 lewrbabnce
FIgure 2.18 t.ever balanc�w,tll
CrculMSCal�
11
II Measurement techniques
Anolher type of Ie\"er balance has a pointer mC)IIlng along a circular scale (Figu re 2.UO.
A weight Ofl t\le pointer arm Is placed In one of t\\'O pa>.ions In order 10 change thl>
range (for examp!e, fru n O---lOOg IO O-tkg).
Both oftile5e typesofbalance are used more for the convenience ofobtalnlng a
rapld, approxlmate reading. rathf'r lhan for an accuratetletefll1lnaUoo. An Indlc:UlorI
of thl> uncertainly In\w-ed In readings with a p:1ItICUJar balance C;ln be obtained from
thesma1lesl dMsIon OIl Ihf' SClIIe.
Example
The ma�s of a quantity of chemical is determined using a Iewr balance. Owr the range of
masSol!s invol'o'l'd. the ....paration betwoon mass graduatlons on the bar is2g. The reading
for the mass of the empty contaifM'r is 56g, and the reading for the mass of the oontaifM'r
p/us the chemkal is 100g. Rnd the mall of the chemical, and the unceltainty in this valoo
By subtrdctioo. Ihe ma�� of Ihe chemical il 104 - 56. 48g
°7
.0 '° °r
o ",
The uncertainly in each reading il li�e� to be half of the smallest division of the mass
graduation � o n the beamthat il±lg. Eachofthe two readingshasan ur.c:ertainty of±lg'
theun<ertainly inthe ma�50fthe chemical ilthus±2g.
Now it's your turn
2 TIo!! mass of a chemical used 10 make up a solution is determined as follows. A dish
containing the chemical is plac:ed 00 the pan of a spring balance. The pojnter reading
on the filiI' is shown in Figure 2.19a. The chemical Is then tipped into a known I'Olu�
ofwater. and the empty dish replaced 00 the pan. giviflg the pointer reading shown in
Figure 2.19b. What is the mall of chemical. and what is the uncert�nty in thisv��1
b)
•
30
'"
'"
"
masslg
50
massIiI
50
Figure 2.19
Choice of method
AS Slated aba\'e, thf' top-pan and the spnng bal�nce are direcl-re:Jdlng In.'ilnJments
This means that readings can be obtained quICkly and conven\('ntly. The lever Ixllance
requln's adlustmern of the sliding mass, but tillS takes only a very short tlme. There may
be some rules In your laboratory about wh ICh type c1 lxllance should be used for whICh
task. In general, the pan of the balance 5hou1d IX' kept clean and dry. Do not weigh
out loose chf'mlCals OIl thf' pan, always use a container. tile mass of which you have
determined beforehand. or for whICh you h�ve mlde allowance using the tare (OflIro!
In general. choose a balance c1senstttvly approprllte to ttle experlmenr you are
C;lrrylng oul.
Application: current balance
A U-shaped magnells plaU'd on a lop�n balance (l'Igure 2.20). A wire Is tbmped so
that I runs along the channel cI the maglK'l. The WIre ls COflnected In a dn:ult wtth a
d.c. supply, a rheo5tat (var1:lbll' resistor). an ammeter �nd a switch. When the supply
Is swlched Ofl, the balance reading Is !iI.'ffi to change. because a ron:e Is exerted on
the wtre In the magnet:1C field. By NewtOri'S thIrd llw, a ron:e ls also eJef1ed on the
"
2.1 Measurements
magnet. This Is derected by 11K> cllange lim In the OllSS reading. This change must
be oonll'ned lnIo forr:e F by mullip/ylng llm by g Thevarillioll \\1{h current / �
the magnellc face F may be determined.
Tlle eqwtlon
(:;ee lbplc22)
where lis the length or the wi!\' In the magnetic 1lekI. may be verified. The direction
ofthoi' force, as predicted IheoM.lc:.Jlly by �1emJng'S left-hand rule, may al:so be I't'rtfled
by cheddng whetlter the mass reading Increases or decreases for � 8"'....n (\lITen!
direction.
-+ -1 f----{::::;;Z::J----{
Figure 2.20 Currl'lll bal�n(l� eXpeOmllfl1
Measuring an angle
Angles are measured using an Il1SIrumem called a proU1lCtor. This looks like :l seml�
drrulJr. or somt'ltmes drrular, ruk>r. "1th lis scale marked 00\ In angular measu(�"
lnl':ulably degrees mlher Illan radians. The refUm 0( the clrde Is dearly marlced
To measure the angle between twO lines. the CE11Ite of the circle of tile plQt13ctor
Is placed exactly O'"l"f the pain 0( lnIell'ieCtlOn of the Unes and one line Is aligned
with thl'OO dlrectlon oC the proll1lClOr(F1gure 2.21). The angJe becwet'll Ihl> Unes ls
then gWen by the reading on IhP scale at WhICh the second line passes through the
circumference a the circle.
If the direction ofa stngIe line Ileeds lobe de.flncd, lhls is almysreferred to the
direction of lhe Oxaxis, lhe hortzofllal axiS poInllng IOl'o':lrdslhe rlghl, :l.SZ('ro.
Figure 2,21 USIIlg a protrxlDr
13
I
II Measurement techniques
MOSl plUlractorS used In schools and colleges have a dlam«er d about 10em. The size
of the scale atthe drrumfen'llU' dthe drcie Is Ihen such that the hltC1"Val oo\\'l;'ell
SC'lledlvlslOfls ls 1�. lIlseaSYIOm.lkea readlng lolhe ncan'Sl degree. and !i()lneUmes
IOhalfa degree, lflhe llne being measured LSnoe eoough. PlUlractors d larger
dlallll.'U."r may he matted In half degrees.
Methods of measuring time
The experlmens you \\111 me« In )UJr prncUc.ll physks course dealwilh the
mea:;uremeN: of time Inll'rv.r.ls, nlthet" than "'1lh absolute lime. The basic mt1hod
of mea:;urlng a lime 1rv.erv.r.1 Is with a stopdock or SlOPW·:IIch. In each case, the
In,;trumen Is Slarled and stopped by pressing a Ieverora buUOI1. and re-set by
pressing anothl'r COlItroI. You should brnlllarl5e yourself willi lhe way of operallng lhe
In,;truJllerll before you start a UrnlngexlX'rlmenl In earnest. Remernherlhal lhe reaction
time d Ihe experimenter (a few lernM of a second) Is Ukely 10 he much gn'aler than
lhe ullU'rlalnly of thi' Instrument II.>elf. If you do not reduce lhe l.'!Teets of reaction
lime. an unaa:epubie sySiemaUc error may he buill In 10 Illl.' experiment. .A5 explained
in the Sl.'CIIon on Errors and uncertaintIeS (page }i). one ....'3yof reduclnglhe effect
ofn'action time Is to time ffiOUgh events (for ex.1ntple, till.' swtngs of a pendulum) 10
m3kelhl' Interval being measured very much 13rscr lh3n IllC l.'xperitnenter's reaction
lime. A gcxxl tl.'\:hnlque Is to munt the ewnts (the swings). mmmendng by muntlng
down to zero. and starting the timer at thl.' zero munt Wherever pos.5Ibll.'. work with
3t least 20 SI.'COf'Ids' wonh of events (06dllaUons). and Il'pl.'lt l.'leh set of timings thret'
limes. (Sometimes, when carrying OlIt experlnlCms on damped O6dllltlons. )'0lI will
have to be satl.lfte<i \\1th fewer swings. but try not to go below IntCfVJls of 10 seroods.)
Thestopclod:.
A mechanlcll.l. sprlng-p<mered stopdock will ha,,, an analogue dlspla)'; 11I3t Is. a hand
(or hands) which ffi()\l' round a dbl (FIgure 2.22). SUCh �n Instrument Is likely 10 read
Figure 2.22 An.lbgue stopdock
10 the III.'3rest one-flfth ofa
second.
The stopwatch or digital tiner
This IlISIrull"lCN: has a dlglal display (Figun' 2.23). II Is based on the oscillations of a
qua"z cry!ital. l1Ie read.<Jlll will probably be to the TlC3reS1 ooe-huooredth of 3 second.
[n add.iofI lo the start. stop and n'-set controls, dig.al stop\\':ltches often h:J'"e a 'lap'
'3tch ls stlll running.
fadlll),. whlch allo\l.·s one reading lO be held In the dlspby whIle the ....
Because of this complexll)" Il ls ,...al lhat you knew.' the fuoctlons of all the controls.
Figure 2.2l Oiglt�1 timer
electromagnel
You r own wrl.ltwatch may wel l h:Jve a built-In Slopwlltch. which may be ju,;t as
precise as the w:nches available In the Iabornlory. HO\\'e\·er. the start �nd :ilop COlltrois
on wriSi .ltop\\'3tches are sometllllCS mther small. and il Is Imponanl Ihal you �houkl
nOl fumble a :ilanor .ltopsignal.
Choice of method
Qftt'll. SludelliS an' attmcted to a digital siopw:Uch Ix'C\use II rt'ads to one_hundredth
of a second. Howeve r. In all eXpl.'rlmenlS In which tilc &a" and stop signals are
applied manuall)', such plKlsioflls Unnl.'Cess,1ry and Inappropriate. Thl.' r('3C1lon lime
of the experi mente r. which Is likely to be a fl.'w tl.'nllis of J second. will cance l out the
x ligtlt iate
precision of the watch. It woukl be misleadIng. and bad practice. to entl.'r times such as
'2t.)12s' Ina table of ll'Sutts If the sySiematlC error duc to rt'3ctlon time had not been
fully acrounted for. Thus, If you are doing an experlnlCnt 011 the timing of oscillations
and then' 31l' no d!gltal tlmen; available. you will be at 00 dlSldVJntage If)UJ have to
use an analogue Slopdock.
""..
Application: determination of the acceleration of free fall
x lli:fll aate
Figure 2,24 Determlt1atK>n of
theKcele�lOI1 offreefal
14
A SleeI sphere is released from an electromagnet and falls under graYlt)·. AS It btls, II
p::1S5eS Ihrough Ught gates which 5"'1«:h an t'Il.'c1ronlc t imer on and then off O'lgun'
2.24). lbe aa:eler:itlon of free fall Gin be detemllne<i from the valUt'S dthe time
Inll'lV3lsaoo dl:il:allCt'S.
2.1 Measurements
This Is an experlmem In which eie<::uoolC �V.'llChlng IS essentbl ln order to reduct'
the poIenllaJly very large error caU5l'd by the reactlon Umeolthe e:tper!mt'llier. Here,
timing to one-hundredth ofa second � �lal
Application: measurement of frequency using a cathode-ray oscilloscope
A calhode-rny o:scillosoopl' (er.oJ has a callJr.ned tlme-oosc, so that measuremenlS [tool
the screen ofthe c.r.o. canbeu!ed(()gI""<l1ue:sclllme l�';)Is. Oneappl\t'atloo \s1O
!Jl!-"asu�the frequency ot a perlodlc slgIl3L foreJ(;lmple lhe slne-W;l\'e OOlpul ol 3 s1gnaJ
generntoc.l1lf' slgnai lsronnected 10 the Y-Inpu oI"the c.r.o.. ancl the Y-alq:)l\fler and
tlrne..m.>e ronlrolsaR' ad!usted untll a Ir.IU' olat IeaSiOfle, but fewer lhan about thl'",
('OIl\/lIeI:e cydes athesignal ls otxalned on the scrren. l1lf' dlsUnce Lon the gr.n1ru1e
(j:he- saUl' on the screen) <.UTe5pOfidlng to 011(> complele cycle Is measured (J'lgUfe 2.25).
It 1s g!Xxl practice to measure the length ci, say. IOorC)'clcs, and \hen divide by foor so
as to oblaln an average value ofL 11x> gr.lIICuIe "1I1 proiJ:lllly lX' dlvided lruo c('1lllmelre
and perllaps millimetre or two-millimetre dMSIOnS. If the time-base setUng Is x (which
will beln unlts of SOOJOds.mllJiseronds ormlCl'06CCOOds pef cenlhnefJe). thellme Tfor
OIX"cyc\(> Is given by T= Ix llle fR'quency/oflhe Slgnli is then obtained from/= 1(1:
Theuncenalruyofthe delerrnln:nlonwlli depend on llOW wcll rOU ClneSlimatethe
measuremem of the length of the cycle from the gr:!Ucule. Rememlx>ring that the trace has
a Hnil:e Width. you can probably measure this length to In unrertaintr of about *2mm.
Flgu,..
2.25 The USf' of � U.o.to rm><Ilure frequency
AS wil:h rra;l irul:rumenlS ),ou will U5l'. )'OUr IabornlOfY Ume wtll be so Umlled that
you ,,111 probably ha\"C to take the Ume-ba5e SClilngs on trusl. 1·!cIY.t'\'t'r. It is worth
thinking about � methods of checking the altbr.ltlon. You could If)' checking
agalnsl a GlUbr:lted signal RffiCr:nor. but who IS to say which of the signal generator or
CI.O. has the rorrect GlUbr.lUon? Another method would be to ronne<1 a microphone
to the Y-inpul. arxl sound a tuning fork ci kflOWn frequency near the mlc:ropllonf'.
Example
The output ofa signal gerlCfatoris connoctoo 10 the Y·input of a c.r.o. lNhen the timl!-- baSf'
control is set al O.SO millisecond� per centimetre, the trace stlown in Figure 2.26 is
obtain�. W'hat illhe ffequcncy oflhe signall
Two compjele cydes of the triKe oceupy 6.0em on the graticule. The length of onl! cycle is
thelefore 3.0cm. Thetime·b.Jse Sf'lting is O.50ms em-I , W 3.0cm is equivalef1t to
3.0x O.50= 1.5 ms. The freqUef1CY is thllS 111.5 x 10-l . 670H:
Figure 2.26
15
I
II Measurement techniques
Now it's your turn
) T� same signal is applied to t� V-input of t� c.r.o. as In t� example on page 25.
but the time-base control is changed to 2.0 miHiseconds per centimetre. How mafIY
complete cydes ofthe trace wiJ appear on the screen, which is 8.0an wide?
Methods of measuring temperature
Tl1.e SI unlt or temperaflJre, the keh1n (K). IS bJsed on the Ideal g:tS (or
thermodynamic) scale d temperature. The sale may be an1wd at using an Instrument
called a CO!lSlam-n>luffieg:ls thermomeI.er0;ee pages 212-3>. The equation re!aUng the
celslus temperature scale to the thermodynamIC scale is
9 : T - 273.15
where 91s In degrees Ceislus aoo TIS In kclvln.
Forrunately, In your praaJcal course you "1U CQIlle across nOlhlng more CQIllpllcated
than a l!quld-In-glass (probably a mercury-In-glass) Ihermomeler, You may, hcJI>,'ever,
do experIments OIl thennocouple thernlC>lfl('tef'li and reslstanre thermometers, and
assess their suitabIlity for use as a thermometer.
Themercury-in-glass thermometer
LIquId-In_glass thennometers are bJsed on the t/)(>nn31 exp�nslon of a liquId. A
quamlty of liquId IS COflt:llne<i In a bulb at tile end of a tilin capillary lUbe. The space
abafe the liquId contains an lnen gas at k:M' pressure. [f!lle bulb Is placed In a
beaker or water which IS gradually heated. the liquid expands and the thTe:Jd dllquld
OITupies fIlOfl" and more at the capiLlary rube. The caplillry tube Is graduated: the
posllon or the end or the thread gives the temperature.
Most thermal physic<; a:perlml'llls which you "111 auT)' CU Wlll lnvol\'e the
rneasuremen at temperntures betwa>n OOC (the temper-l1uTe at nlt'll'1llce) and IOO"C
(l:hetemper:llur e at .'leam :aixJo-e bolHng water ata pressu.re of l atmosphere). Themosl
use(ul themtomerer COll'flngth!s f1lTl8l' l S a metcur)'-In-gw.sthermorncWr (Flgure 2.zn
wlh g:radu.atkn'i flUIl -IO"C to 1I0OC, In I"C 1I1Ierv:ds. You Will nod I easy 10 t:a�
readings to the nearest half degree, and perNp5 10 O.2"C. There are a nulriJer at
precautkn'i )U.J shoukltake ,,'henusIng the thetmomett'!'. Always alb.\' time ferthe
thermomelt'!' 10 reach thermal equWbrtum wlh IS surTOlLndlngs. If you are rnea;;urlng the
temper:ll.ure of a beaker at liquid which l'l belng healed, the liquid IllU5t be thon:lughJy
stlrred beb'elaldngthereadlng. (Becauseof oon\'ectlon currenlS, there lsa l�re
�"'Y
dlffffi'OCe or several degrees bet"a'Il the top and the bouom of the liquid;) TIle
thermometer ls callb!:J.ted focuse at 3 .ltancbrd depth cJlm� thl'l nuybe sta1ed on
Figure 2,27 Mercury-in-gills thermometer
the stem. If l is nol, try 10 eru;ure that the 1hennOineter IS alwars Immen;cd In 1he liquid
to the same depl:h. TIle length at the bulb plus abou1 20mm Is a reallOllable guide
There are some points to be made abou1 safesy. Thennomesers are rel:Jtlvely
fragile Instruments. Because or their stL1PC. tlley h,we a tendency to roll along the
bench.top. Make sure that your thermometer docs not roll oil and faU to the ground
If a thermometer does break and the mercury In It corflt's Ollt. do nOl be tempted
to play with II. Mercury Is a polson. Tb reduce tile risk cJ breakage. do not use
the thennometer as a stIrrer. unless lt ts ofa robust typc deslgnated as a 'stIrrIng
thermometer'. If you haw to fit a thermometer tllrougll a rubber bung. make sure that
the hole In the bung IS large enough. and lubrICate the rubber thoroughly wlth soap.
Wear glc"es and grlp the thermomerer so that. if it breaks. your wrist wtlJ IlOI be cut.
Example
Thet(!{T\pcrature ofa rnixtur�ofice. saltand water ismeasured using a mercury-in-9lass
the�ter. When tllermalequilibriumhasbe-en reached. the mercuryttvead inthe
t��ter is as Y\O'NTI in Figure 2.28. What is t� temperature ofthe mllrture? What is
the uncmaintyin this value?
Ft'llur. 2.28
16
By int�rpolation between the scale divisions, the temperature reading is _2.S�. The
unc«taintyis probablyabout :l:.O.S°c.
2.1 Measurements
I:
Now it's your turn
T� templ!filtu re of iI solidifying �quid is measured using a liquid-in.glass thermometer.
When thMnal l!quili b rium has been reached, the �quid thread in the the rmomet\!f is as
5hown in Figure 2.29. What is the so�dification temperature? What is the uncertainty in
this value?
•
"
�
"
FlgureZ.Z9
Figure Z.30 Therroocoupie lhermometer
The thermocouple thermometer
A thermocouple thel1llO!l'leteT consists a two wires macle a different metals or
alloys. joined at one end. lbe Olhl- r ends of the Wires are connected to the tennJnals
cl a mllll\·okmeter. 11tls m:ly be a digital Insnumem, which Is allbr.lted In "C �
Figure 2.30). The thermocouple may also be connected to a datalogger. The lunctlon Is
placed In themul contact with the object, the temper.uure ofwhich Is �ulred.
The thermocouple thermometer actually measures the dlffereOC'e In temper:Hure
be!:ween the lunctlon ofthe ""u metals(lhe hot lunctloo) and a cold junction. In some
applications, the cokl lunction Is pL1ced In an ice-v,';1ler mlxture. so as 10 achl{",'e a
known Il'ference (see Figure 119. page 216).
The thermocouple may be conneded to a mllll\'Olt1IlCler which has not: been
calibrated In temperature units. In this case. you Will hal'e to make use of the known
variation of thermoeledrlc e.mJ. for that particular pair cl metals wkh tempemure.
You will need to draw a calibration cun:e, a graptl cl e.mJ.. agalnSl temperature so
that you can read off the temperature corresponding to an e.ml. reading. NOlI.' that
this graph Is often a curve rather than a !llralght line, as shown In I'tgure 231. In some
casesthe curV:lIure Is so much that tile iIoame e.mJ. an be obcalned fa- t....u dlfferent
temper:uures. Clearly, this resl:rtcts the tempernture range ..
c.cr wMeh tile thermocouple
Choice of method
lbe heat Glpactly of the bub of a llquld-In-glass thermornet:cr IS !ll.h
.IC greater Ihan that
cl tite hoi: Junction of a thermocouple. for Ihl.S reason, lhe thermocouple Is panlrnbrly
u!j('{ul when a rapklly varying temperature I.S 10 be measured, or when lhe objecI, tite
temperature of whtdr. Is required, has a smlll heat capacity.
Merrury-in-gLass thermometers are available to CO'o'er the temperature range from
about -40"C to 350"C. Thermocouples U5lng dlffcrertl pairs cl tTII.'!al or alloy v.1res can
much Larger range.
The choice of a panlcuiar thermometer in a given application will depend on the
CQ\'eI"3
range of temperatures to be CQ\"{'red. the heal Glpadty cl tite obiea, and wllelhe£ the
tempwatlll'eFC
Flgure Z.31 Cahbratlon curw for a
thmoocoupie
temperature Is varying rapidly.
Methods of measuring current
and potential difference
Your physics Laboratory will probably hJ,'e :I selection of InSlnl1ncnts for measuring
curreru and potential difference (voltage). The two main types are analogue meters.
In which 3 poIruer moves over 3 scale (I'lgure 2.32:>. and digital. In which the value Is
displayed on 3 read-DUt conslstlng ol' a senes cllrtlegcrs (l'lgure 2.33)
Analogue meters
llle normal analogue meter Is resl:rlcted to the me3surement of the relev:Jnt quantity
O\'l'ra slngJe range. Forexample. a O-IAd.c. ammeterwlll mt':lsure dlrect currents
In the range from zero lo lA. A O-30V d.c. voltmeter w1ll nleasure steady potential
differences In lhe range from zero to 3Ov, some analOfluc meters haw 3 dlCll-range
factUly, with a common negalh'e terminal and t....
'O positll'\! terminals, each cl which
1$ assoctlted with 3 separate scale on tile InSlfumenl. ThUs, one .scale might be 0-3A,
and the OIiter O--IOA. Each of the pc:Ntl\'e temllnals Is marked WIth the SC31e IOwhkh
II refers. Be Glreful to lake lhe reading on the SC31e corresponding to the pair of
termtnaJsrOll hal'l' seIected.
17
II Measurement techniques
Figure 2.11 Oignal meter
Figure 2.12 Anala.gue mete-r
Analogue mt'lers are subject to zero error. Before switching on the drcult, chKk
whether the neffile ls exacrly atthe zeromark. l f U ls nol, relurn the neffile to zero by
adjusUng the.\Cl\'w at the needleplvOl.. Thel\' ls also tnepos5lhllltyof parallaxerror.
The needle should be read from 3 posttlOO directly above k and the SC'dle. and not
from one side. Sometimes a strip a mirror 15 pr<l"1ded close to the scale so that the
experimenter can align the neffile "'Ih IS Image In tOO mirror. t'fl5l.lrlng that
b)
galvanomlter
�
Figure 2.14.1) G.w�nof1'leleJWlth shunt. for
cuJrl'nt �suJementsbl ��nomtterWlth
mutllpher. forvon�e �surements
viewing
Is H'nlGll. llle uncertainty assocbted "1th 3 currt'fll or voIta� reading from an
analogue meter Is usu.aHytakt'fl 10 bt' ("J;) the srnallest satle re:ldlng
A galvanomt'ler Is a serullh-e curl\'nI·measurlng analogue lTM.'Ier. It rna}' be
('Ofl\'efled Into an allUllt.1er by the ('Oflne<.11On of a SUitable resistor In parnllel wlh the
meter (FIgUI\'
2.,}4a). Such a resl5lor Is called a $bum. The meter may be OOIlll'rted
Into a I'Obneler by the connection a a sulable resistor In series wlh the meter (FIgUI\'
2.j4b). SUdl a resistor IsGllleda /rmltlplter.
The manufacturers PIOl'!de 5hunl:S and multtpllCrs wltlch are clearty Labelled with the
('Ofl\"t.'fNoo. fuOClJon and fuU-sc:r.1e deIlectJon. for a!lachmenl to the oo.slc gal\"anometer
(FIgure
2.35). All you need do Is to select the shunt Of multiplier required for your
experiment aoo make 5l.Ire that you apply the correct fadorwht'fl re;ldlng the satle.
A galvanometer wah a cenln'-zero satle sho*;>.'S nega!!\'e currents when the Ilf'edle Is
to the left-hand skJt> a the zero mark aoo posklll! currents when It Is to the rlghl. This
type of meter Is aten used as a null IndlGltor; thai Is. to detect when Ihe current In a
pan ofa circuIt Is zero.
Flgure 2.15 Shunts�ndmultrplll!fSforuse
wllh � g.w;inomettr
18
2,1 Measurements
Digital meters
Dlgl:al melers may have a zero error. Bel"ore 5Wllclllng on tile circuit. check whether
tile reading Is zero. If II Is f)()( zero. make a llOI:e alile relding and take . InlO acrounl
wilen Il."adlng tile curren! or Iu/!:.IgIl'. The use 0( a dlghl meier may S31'e you tile
trouble d selecting an IllSlrument will a liUllabie (;lflgC for )'OOf appllC:llton. OS{ hal'e
....
an au!o-ranglng filnc1lon: Utat Is, the InSlrument sclec!stlle mosI senslUl'(' r:mge forthe
pantcutar value 0( current or lul!age being meuured. AU the experimenter llas !O 00 Is
check wtll'lher 1hf>R' Is a zero error and make an adju,.ment If rteCessal)', noIe wheiller
tile dlspby 1OO1ca!eS 'A' or 'rnA', and d�..
r e Ille position 0( the decimal point!
The uncer!alnty In the R'adlng o(a dlgl1al meter is expressed In tenru; 01 the o',erall
uncenalnly and the uncertainty In the last digit. When In use, you ",111 note that tile
ta,. dlgtt 0( the display nUClUJtes from OIle figure 10 anolher, You can try to e:.tlmate
tile mean 01 the fluCluatlons,but !flhts fluCiua!ton occurs, !heR' Is ctearly uncertatnty
!n tlle last dtgll 01 the value.
Multimeters
Ftgure 2,]6Mlllllmeler
Muttlmeters. or multtfunClIon IllSlrunll'n!s, 3re 3vall�ble ln holh analogue and digital
forms (FlguR' 2.36). SUch meters may loclude SWltctted opllons for the llll'asurement
01 dIrect and altemallng currents and voltages. ancl
of resls!an«,. wtlh several ranl!"s
for each quantity beIng measured. If you use � mul!l!TK'ter. make sure that you are
easure the quanUty
f.Jmlllar with the COIltrols, so that you can set the Instrument to m
you require.
Choice of method
Much "'ill depend on the selection of meters available In your bbor;ItOfY. Before you
SCI up your clreul, make a rough calculation 10 delermlne !Ile ranges 01 currenlS and
\�ages that)UlI "'111 have to measuR'. ThiS Is a IIlIal p:lrt olthe planning process,
and will help to make sure that }UlI sek.'a the approprlale InSlrumenl from thor;e
[n some
tabomta'ies, mu�meters are provided for use prlmartty �s test [nslruments,
10 be a\'3lbble 10 anyone who wishes 10 make � rnpld check on currents, potential
dltferences orre51stances l n a clTcult. l f lhLsts the ruie [nyour laboratory, Ltlsbad
praCiIce 10 use a muJUmeler In a long experlmenl, when � sl�un('lIon 3nd single­
range IllSlrumeru woukl do Ihe job equally succesully.
sf
Remember that, to measuR' a current In � componcnl ln a clrt'ULt, an allllrllt'«'
shoukl be connected In ser1es wlh the component, To meaSUre !he potenllal
difference across tile componenl, a l'O!tmeter shoukl be con!lCCled In p3rnLIeI wllh
the component. The arrangemenl IS shown In Bgure 2.37,
:� �
component
�..
,
YOItmetec
in pa.raI�1
Figure 2.]7 An amme� is connedi'd WI series
wtth !hl> component, <I "«>itmetI'T III par�11e-I
"
II Measurement techniques
Application: measurement of voltage using a cathode-ray osdlloscope
NQle, kno\\1edge dthislJl!"3suremenl tecllnlque Isonly required In tile A 1,eo.'e! (.'OU1liI"
bu: Is Included here for cr:mpIeteness. The C3ti1ode-r.ly O5dUoscope, ....h.. lIS caIlbr.ued
y.,amplfler. may be used to nll'aSUrt' !he ampllude cl an allCmlUrlg YOIuge slg:n.1L
('«e hal'e already J;effi haw !he Ilrne-basil' cl lhe c.r,o. m:ly be used to measul{' time.)
The sIgn:IJ IS connected to tlll'" Y-Inpll. and !he Y.,ampllflcr and Ume-ba5e settlngs :Jll;'
a$1SU'd u f"([/ a J,lllabie lr.lCl' lS oOOilned (flgure 2.38). The ampllrude A cl the lr.lCl' l s
measured. lfthe Y-ampliller Sl'lUng Is Q (lnunllS cll'OIsper cenllmetre), tlll'" peak:lI:lrue Vo
01 the slgml lsgl\'t'1I br V
O =AQ. The peak:-tO-pe;lk V2rue Is 2VOo and the r.flU.
(IQOI-
1JI!"31NllU3rt')I'ObgeIS voI.Ji . (Rf>rnl>ntx>r lh:ltlhe readingOOlalJled onanaru.1ogue
ordlglal miUDele£ lSihe r.m.s.\"3lUl'.)
Figure 1.38 M�asufl'lTI!'ntof altern.otingYOlt�ge
Example
The output from a sigrlill generator is wnr.ected to t� Y-Input 01 a C.r.O. � the
Y_amplili�r wntrol is set to 5.0 milliYolts per centimetre, the trace shown in FiJufl' 2.39 is
obtained. find thepeak'oOltage ofthe !iignal, andthe r.m.s.voltage.
Mea\.Urethe amplitud� of the trace on the graticule: this is 1,4cm. Tile Y-amplfier 'l('ning is
5.0mVcm-'. 1.4cm is thuo;equivalenl 10 1.4 " 5.0. 7.0m"'. The peak voltage of the signal
is 7.0mV. Th£. r.m.s.voltage is given 'at7.0/.J2 • 4JlmV r.m.'
Figure 1.19
Now it's your turn
5
The outputfrom a signal generaloris connected totheY-inp;.it ofa c.r.o. Wh£.nthe
Y-amplifier control is set 10 20 milliyollS per centimetre. Ih£. trace shown in Figure 2.40
is obtained. Find
(II)
the peak-to-peak voltage 01 tile !iignal,
(b) the r.m.s. yoitage.
Figure 1.40
30
2,2 Errors and uncertainties
Measuring magnetic flux density
l1leflux densly ofa magrlC'llc f\eId may be measurcd tlslng a Hall probe, a device
which makes use ofthe Hall erreel (see page 322). The Hall prcbe apparatus used
ln sdiool or college Iaborntor1e:sCOfls!stS ofa thlnsitce a a semlcooductor lIUIerlal
which Is pl:tced wlh Is plalle at rlgill angie5 10 the dlroctlon of the magnctlc fleki.
lbe mnr.rol unlt Is alT3nged to pass a renaln current through lhe semk:'or'duClor slice;
the Hall ,mage, which Is proponknll lO the magnetic flux de!lstly, Is read off on an
aMIogUl" or dlgll:al meter, which Is alrwdy callbrnted In unllS of magnetic nux dt-ll5ly
(j:esl.1). The amlngemenlls Illustrated In "tgure 2.41.
lbe use of the Hall probe to mea,;ure m;agnet1C nux density Is only required for the
A ]e,'l"l syllabus but l s lnduded here for coolple1:eocss.
2.2 Errors and uncertainties
Accuracy and precision
Accuracy i'ithe degr@('to v.tlkh a mea5lJfementapproacheslhe 'truevalue'.
AcrUrncy dt-pends on lhe equlpmenl u5ed, the skill oflhe experimenter and
the techniques used. Reducing systematic (>ITQI" or uncenalnly (see pJge 3�) In a
measuremenl ImplUI'es Its accuracy.
Prec:islon illhe determinedby the 5ize ofthe randomerror{s@('page3S)in the
Precision Is that pan of accuracy which Is within the control of Ihe experinlC'nter. The
experiment('r may choose different measliring InstrumentS and may use Ihem with
dlfTerent levels ofsklll. thus affecllng the preClslon of measurement.
If we W3nt 10 measure the diameter of a steel Spt10ere or a marble. we could use 3
metre rule, or a vernier Glilper. or a mlcron1oeter screw !puge. The choice of measuring
Instrument would depend on the precision wah which we wan/the mC"J.5Uremeni to be
madt-. For example, the metre rule could be used 10 n10easure to tile ne:J� millimetre,
the vernier caliper to the nearest tenth of a millimetre, and Ihe micronw;'ler screw gauge
to the nearest one-bmdredth G a mUllmetle. \'Ii'e OOOJId show the readings as follows:
\'emlercaUper:
mlcrunel:e!' SCJl"\\' gauge:
31
II Measurement techniques
ILL
I�
III
IL
T
rNeling
�)pr8Cise and accUfale
r
reaalng
b)impre<:ise t>ut accurale
Figure 2.42
r
ruaiog
� pr8Cise[)ut notaccUfate
T
reaIlloe
mprecise aoo oot accur.te
b) i
Figure 2.43
The degree
a predsiofl to wlllcll tlie IlK':lsur£'mem IS made locreases as we merl'e from
tile metre rule to the vernler caliper aoo f1natly 10 Ille mlcrorJleIC1 .screw gauge. NoI:e
Ilial tile number a sIgIllflrnm figures quelled for tile meaSUrement Increases as tile
precision locre:Jses. [n facI, tile number a stgOlflrnm figures 10 a measurement gin's
an Indlcalloo a llie precision a llie nlt'3suremeol.
When a measuremem Is repealed many limes wllh a precise Instrumeot. tile
reading<; are all cb;e together, as mcrwn In Figure 2Ala. UsIng a measuring InsI:rumen:
with less precision means tllat there would be a gtt'<lter spread a readIng<;. as shown
In I'lgure Vi2b, resulting In greater uncenalmy.
PrecIsion Is panly 10 do with the accuracy of ao ooserv;ltion Of measurement. A
readlog may be very prectse but I need ["10( be accurate. ACCUracy Is concerned will
hoow close a re:Jdlng Is to Is tnJe \"alue. FOreX<lmple. a mtcrometerscrew gauge may
be precise 10 ±O.OOl cm but, If there Is a large zero error. thCfl tlie re�dlogs from the
SOlie for the diameter oI'a sphere or marolewould 001 be accur,lIe. The distinction
. 3. On each of the graphs the
between precision and accuracy Is Illustrated 10 Figure 2 4
value Tis the true value of the quarulty.
Uncertainty
In the Ilst OIl page 31, each ofthe measunmwms IS shown With II.\i precision. I'orCX:Jrnple.
uslog the metre rule. the measun>ment ofthedlarnelerls 1.2cm WIth a precLslon ofO.lcm.
In reaUty. predslonls OOIthe onlyfacror a(fcctlng theaccurJCY ci the measurement.
Thelolal range of values wilhin wtlich the measurement Is likely to lie is koowo as its
uocartainty.
For example. a measurement of 46.0 ±
OScm Implies that the most likely value Is
46.0cm, buIll could be as low as 45.5cm or as high as 46Scm. The uncertainty In the
measurC1llefll ls ±O.5cm or %(0.5/46) x 100'M. = ±I"'Il ls important to understand thaI. when wrlllng OONn rne:asuremeots, the number
of .'llgnlficltO figures 0( the measurement IndiCates IS uncertainty. SOme examples fA
uOCC113ln1y are gn-en In llIbie 22
TOIble2.2 E;l.:imp1esofuncertalnty
stopw�u:h with 0.1 s divkioll5
16.2s
,lmll'M'tl'r wiih 0.1 A diYisions
NOlI' that while a particular tempernture
Is shown as a number w.1t the uo. "C. a
temperature flJlem'll ls correctly shown as a number "'111t the Uok deg C. However, IllOSl
pI.'Ople usethe unll "Cfor bolh a partlcutu temperature and a temperature llllC1V:l.I.
� ng Is 001 wholly coonned to
II should be remembered that the uncertainty In a re di
the reading of Its scale Of to the skill a the ('xpcrlmeot(.'f. Any measuring lostrumem
ha� a buill·In uocenalnty. I'or example. a lJl('(al metre rule expands a� Its temperature
rises. At only ooe temperature will readings of the scale be precise. At all OIher
temperatures. there will be an uncertainty due to the expaosion of the scale. Knowing
by how much Ihe rule expands would enabl(' thiS uocertalnty to be removed and
hence lmplU\'t' predskln.
:ICh meter. For example. a
Manufacrurers of digital mete� QUOI(' the uncertainty for e
digital voltmeter may be QUOIed as ±l% ±2 digits. Tile ±I% applies 10 the tOlal readIng
sliown on the scale aoo the ±2 dlgls Is the uncertainty In the nnal displa)' figure.
This means that the uncertainly In a reading of 4.00V \\"OU1d be (±4.00 x 1/100) ±
0.02 = ±O.06v. Tills uncertainty would be added 10 any further uncertalnl)' due 10 a
flUCIuatlng readl.ng.
The uncenalnty 10 a nlt'3surement Is .sometImes referred 10 as being lis
error. This Is
not strictly true. Error \\"OU1d Imply that a mistake has been made. 1here Is no mistake
In taking the measurC1l.lefll but there is alW:iys some doubc Of" some uncertainty as to
31
2.2 Errors ,nd uncertainties
Example
A studrot lal:i!os il larg� numb@rol imprecise readingsfor thecurrent in a wire. He uses an
a�erv.ith ill(>ro error of -M, meaning that al scale readings are too small by!J.I. Th£.
Irueva� oftheanrent js/. Sketch a d istributionC\Jrveofthenumberof readingspioned
against therneasured value of the current Label any relevaotva!ues
This ist�c.ase ilustrated inFigure2.43b.The peak oftheclJf'o'eis centred ona value
of/_ Ai.
Now it 's your turn
,
A lar!J(' number of precise reading5 for the di<Jmeter D of a wire is made using a
miClOfTW!ler <,new gauge. The gauge has a zero erl'Of +1::, which means that all fNmngs
are too large. Sketch a mstrib.ltion curve of the nUlTlbef of reamngs plotted against the
� vall.'eofthe diameter.
7
The manufacturer of a digital ammeter quotes fa uocertainty as :1:1.5% :1:2 digits.
{aj Determine the uncertainty in a <:Oflstant reading of 2.64A.
(b) The meter is uSl'd to m&I:iUfe the CUnei'll from a d.c. power supply. The current is
found to fluctuate randomly betwoon 1.9SA 31"1d 2.04A. Determine the most likely
�allJe of the current, with fts lJm:ertainty.
Choice of instruments
The pl"fflsion ofan lnSlTument requlR'd for a panlcular nteasurement Is related to the
measurement being made. Obvlously. 1ftill.' diameter d a hair Is being measuR'd. a
hlgh_predslon mlcromel:erscrew g3uge IS required, rntller than a mC'lre rule. Simllarty.
a jplvanomC'ler should be used to measure currents d tlle order or a few milliamperes,
r.Jther than an ammeter. Choice Is often fairly otwlOUs where single me\lsurements are
being made, but care has to be taken where tWO readings are subtracted. Consider the
following example.
The distance of a IeIls from a Ilxed point IS measuR'd using a mC'lre rule. TIle
The lens IS now 11"10\'00 closer to the flxed poInI
rTlOI'ed? The answer Is obvklus:
dlSlance Is 95.2an (see FIgure 2.44)..
and the oew dlSlance Is 9}.7cm. HoI\' rar has the lens
<95.2 - 93.7) "" 1.5cm. But haw precise IS the meotSUremet"l?
t
Q
Figure 1.44
We have seen th.:itthe uncertainty In each ml.'3surement using a metre rule Is,
optimistically, :1:1 mOl.
cimm at the zero end ci tile rule plus i-mm when finding the
1eIl$ This means that each seplr.Jte measurement 01"
0 /940 )( IOO:m. le. about 0.1". That appears to
rTlOI'ed 1s:l:2 nun. because both
distances ltave an uncenalnly. and these uncertalmles add up. so the uncelt:llllly
position of the centre 01" the
length has an uncertainty of about
be good! Hc,..,.ever. the uncertainty In the dIStance
Is ",(2/15 x 1(0)% = "'13910. This uncertainty IS. qulte dearly. unacceptable. AIK)(!ler
means by which the distance rTlOI-ed could be measured must be devised 10 reduce the
uncertainly.
During your AlAS course. )'ou will meet wlth many dlITerenI measuring Instruments.
You must learn to recognise which Instrument IS nt05I appropfiate for a pal1lcubr
33
I
II Measurement techniques
mearuremeu. A Slopw:!tch lIl3y be sulabie for me3rurlng tile period of osdllatlon of
a pendulum but )"00 would han'
dlffk:uhy using I to nnd the tin", taken for a stone 10
faU vertic'al!y from res( through a dlslafKl." of I m. Choice of appropriate InSlrumen!S Is
Ukely to be examlned when lOU are plannIng experiments.
Example
Suggl!Slappropriate in�ruments for the mNl.Urement of the dimensions of a single page
ofthisbook.
The obvious instrument to meal.Ure the height andwidthof a I><Ige Is a 30cm ruler. whim
tan bereac! to :o:l rrvn. The width. the!imill\er dimension. is about 210mm. so the actual
un�ntyis210rrvn ± l mmand the percentageuncer1aintyis about±0.S% . tt h not
�nsible to try to measure the thidneis of a single page. ewn with a micrcmeter Sl:TeW
gauge. asthe percentage uncm:ainty wm be ve.ryhigh. lnsteac!. usetheWf!Wga�to
measure the thidneis of a large number of l><lges (but darn ooude the covers!). Four
hUMred pages are about 18mm
The unc.ertainty iTl lhis measurement. using a screw
Ihkt..
gauge. is ±O.01 mm. givirog a perc.entage uncertaITltyofabout :tO.OS% in the thidness 0f
all 400 pages. This is also the percentage ur.certaiTlty in the thidness ofa single page. If an
uncerlainlyof ±O.S% is<Kceptable. a vefTIiercaliper should be used insteac! ofthe
Now it's your turn
8 Suggest appropriate instruments for the measurement of:
(a) the discharge current ofa capacitor (of the order of 10-6 /V.
(b) the time fora feather to fall in air through a distaTlce of about 40cm.
(d
the time for a ball to fall vertical� through a distaTlce of about 40cm.
(d) thelength ofa penduum having a period of about Is,
(a) the temperature of some water iI!i il cools to room temperature.
(f) the tl!fTlperatureofa roaring 8unsenflame.
9
(g) the weighl of 20sman glas.5beads.
(h) the weight of a house bride
The diameter of a ball is measured U5ing a metre rule and a two set-SQUares. as
ilklstrated in Figure 2.45. The readings on the rule are 16.8cm and 20.4cm. Each
reac!ing has an lJfKerlaintyof±l rrvn. (alculate. for the diameter ol the billI·
(.) itsactual unc.ertainty.
(b) its percentage unc.ertainty.
n
readi g
reading
Figure 1.45
Suggl!St an altemative. but IOOre precise. meth-od by ....nich the rj;ametercould be
measured.
Systematic and random uncertainty (error)
with
acreptabk.> predskln. but also the te<ilnlques a' measurement must optImise acrur:l.C)'.
Not (Illy Is the cholce a' instrume"/ll Important. so tllal an)' measurement Is made
That Is, your expertmental technique must reduce as far as possible ::my UfKl."rtalnUes
In readIngs. These uncertaInties may be da.ssed as eUIE-r systematic or nndom.
34
2.2 Errors ,nd uncertainties
SystematiclIlCertainty (error)
A �ySlemaIIc ullU'Tlalnty w1ll resuh In aU readIngs being either abol'l' or below the
xcepred value. This uncertainly canrlOl be eliminated by repeaUng readings and
then a\'erJ.glng. Instead S)"stematJc ullU'nalrny can be reduced only by ImplO'o'lng
expertffil"1l1ll1 techniques. Examples ofS)"5!errollc uncertainty are:
• zero error on an instrument
The SGIle readlng lsnol lero bekw"e me:tsufCmcnlS arelaken-5ee Plgure2.46
Oleck before startlng the experlmefll.
• wrongly c:alibr.lled scale
In school labocatones we assume that measuring dC\'ices are correctly o.lIbrated,
and ","ould 001 be expected 10 check the calibration In an experlmt'fll. HCI'I'.ever,
If you have doubts, you can check the callbrallon of all ammeler by COIllledlng
severnl In series In the cln:ult. orofa voltru(>\erby connectlng se\"t'r:al In
parnllel. Rules can be checked by laying severnl «them alongside each other.
Thermomelers can be checked by placing 5Cveml In well-sUrred wal("f. These
checks will nOi enable you to say which oflhe InSlrumen(s are calibrated correctly,
but they wi[J show you Ifthere ls a dlscrepancy.
• rea ction time of experimenter
When timIngs are carrIed out manually. It must be accepted that there will be
a de13y between the e:l:perlmen!l.'rol>serv1ng the event and startIng the timIng
device. ThIs delay, called the reactlQll lime. may be as much as a few tenths of a
5eCOIKI. TO reduce the effect. you should arrnngethat !tw Intervals you are timIng
are much greater than the reaction linK'. !'or example. you should time sufflclent
swlngs ofa pendulum fortht' tolal tlnll' to be ofthe oroer ofat least ten seronds,
so that a reaction time of a few tenths c( a 5eCOIKI is less Important
Figure 2.46
This ammeter hal a zero errOf of aoout - 0 2A
Random uncertainty (error)
Random uncertaInty results In readIngs b(>lngscanered arol.lnd the accepted value
Random uncertalnty may be reduced by repe�tlng a rcadlng and averoglng. andhy
plotting a grnph and drawing a be5I·flt line. Examples c( rnndom errors are'
• readIng a scale. particularly If this Involves the experimenter's Judgement about
Interpo!atlon be!:weenSGIle readings
• t iming oscillations without the use of a ref('rence mark('r. so that timings may not
aJ\\"aYS be made to the same point ofthl' SWing
• taking readings c( a quantity thaI '"arieS WIth lime. Involvtng the dHfkulty c(
reading bOIh a timer scale and anotlll'r meIer .simultaneously
• readIng a scaJe from different angles InlToduce:s a varlable paraJlax error. (In
contrnSl.
If a scale readIng Is a/ways TI\Olde from the same non-normal angle, this
wtJIlntroduce a systematk: error)
35
I
II Measurement techniques
Example
The rummt in a r�slor isto be m&I\.Url!d using an analogue all'VTleter. State one source of
(a) a systenatk uncertainty, (b) a random uncertainty. In both cases. so:;Igesl how the
unoortaintymay be reducl!d.
Ca)
Systemat.ic uncertainty could be a ZI1rO error on the meter. ora wrongly calibratl!d 5Cale
This can be rl!duced by checking for a 2II!r0 reading before starting theexperinlent. or
using two�e,... in series to ched that the readings agree.
(b) Random uncertaintyrould be a parallax error caused by taking readings from different
angles. This can be reduced by the \r.iI!ofa mifror behind the scale and viewing
normany.
Now it's your turn
10 The length of a pendl is m&l5Ufed witl1 a 30cm nJe. Suggest one possible Wllrce of
II
(a) a systematic uocerl:dinty, (b) a random ullCIlrtainty. In ea:d1 case, sug.gest how the
uncertainty may be reduced
Thediameter ofawireistobe mea\.Url!d t o a precisioll of:l:O.Ol mm.
(a) Name a \.Uitable instrumellt
(b) Suggesl a source ofsyslematic uncertainty.
(e) Exptain why it is good practice to average a set of diameter readings. talten spiralty
along the length of the wire.
Combinng uncertainties
There are two simple rules for obtaIning an estImate of tile O\\'rall uncertainty I n a
I
forqUilntitieswhich areaD:ied Of �btractl!dtogjye afir.al result,addthe actUilI
uncNIaintieos.
2 forqUilntitieswhichare mu�iplil!dtogetherOfdividedto give a final resu�, add the
fractJonalunc:ertainties.
Suppose that we wtsh to wain the value of a physical quanUly x by me:Jsuring IWO
OIher quamlJes, y and z. 11K' relatloo belween x. y and z Is known. and Is
If the uocen:a\r(Jes Jny and zare Ay and Az respecll\·cIy. tile uncertainty t..r In x i s
gl\1'1l by
Ax : Ay + t.z
If thequantity x ls given by
the uncertalmy I n x ls agaIn glH'fI by
1
I, and 11 are two currerlts coming into ajunctioo In a drruit. The rurrent I going out of
the junction is given by
Inan experiment, thevallll'S ofll andI2 aredetermined as2.0 �0.lAandl.S� 0.2A
respectiYely. 'Nhat isthe value ofl7 What is the uncertaimy in thisvalue7
2
36
Using the given equation, the value of lis giwn by I_ 2.0 + 1.5 _ 1.5A. The rul�
for combining the uncertainties givl!s tJl_ O.l + 0.2 _ 0.3A. The result for lis thus
U.5 � O.l)A
Inanexperiment, a liquidis lteatedelectricaly, causingthe temperaturetochange
from 20.0 � 0.2°( to 21.5 � 0.5 "C. Find the change of temperature, with its assodatoo
uncertainty.
2.2 Errors ,nd uncertainties
Too cha�of tl!fTlPefature i� 21 . S - 20.0 .. 1.5·C. The rule for combining the
unc:ertainties givesthe uncertainty in the temperaturedJange asO.2 + 0 . s . 0.7°C.
Thel'MUkforthe IOOlp2faturernangl! isthus(1.S:t O.7)°c.
Notethat this second example YJO"w.>that a small difference between two quanl�iM
may have a large uncertainty, even if the uncertainty in measuring each of the quantities
is sma•. This is an important factor in considering the design of expenmems. v.t1ere the
differe�betweentwoquantitie;may introduce anunacceptablylarge error.
Now it's your turn
pposi
12 Two wHiquares and a ruler afe uwd to measure the diameter of � cylinder. The cylinder
is placed between the o;.et-squares, and the set-squares are aligned WIth the ruler. in the
manner of the jaws of calipers. The re.ldings on the ruler at o
te ends of iI diarrK'ter
are 4.15cm and 2.95an. fad! fwlling has an uncertainty of :to.OScm.
{.1 INhatis the diameter of the cylinder7
(bl lNhat is theuncertainty in thediameter7
Now suppose that we wish to find tlx> uncertainty In a quanllty x. whose re13110n to
two measured quantities, y and z. Is
x ; Ayz
where A Is a constant. The uncertaInty In the nloeaSurentent r$ y IS ±toy,
:tGz. The fractional uncertalnty lnxls glven l)y
and
that In Z IS
�+�
�
,
x = y
lb combIne the uncertaIntIes when till' quantilles are rolsed to a pcI'o\'er. for example
X = Ay"�
� o{j') ' b(�)
where A Is a cOflSlant. lite ruie Is
A vakil' of the aca>leration offree falls was determined by measuring the period of
osdlation Tofa�m� pendulumof length/. The reiation betweens, Tand /�
g " 4�
(�)
In tMexperiment. /was mNsured as O.SS :t 0.02m and Twas measured as \,SO", 0.02s.
Findttw;. vallK'ofg,ar1d the uncertainty in thisv�ue.
Substituting in the equatioo,s ,. 41t./{0.SSIl.S(2) . \I,7m S-I. The fractional (.IO(I!rtainoos
are 6111= 0.0210.55 = 0.036 and 6Trr . 0.02l1.SO .. 0.0\3.
Applying the rule to find the friKtional unO!rtainty I n g
� = � + � = 0.036 + 2 ,, 0.D13 . 0.062
6
g
I
T
The actual uncertainty i n g i s given by(valueofg) " (fractional uncertainty ing)
.. 9.7 " 0.0 2 = 0.60m 5-1. The experimental value ol.Q, with its uncertaintv. isthus
(9.7 ", O.6)m s-2.
Note that it is not good practice to determine g from the measurement of the period of a
pendulum of fixed length. It would be much. better to take values of 7'fOf a number of
diffefE!flt lengths I. and to draw a graph of Tl against I. The !Jfadient ofthisgrapf1 is 4i!1g.
Now it's your turn
13 A value ofthe volume vof a cylinder is determined by mNsuring the radius rand the
length L.The reiation between v,rand L is
Inan experiment. rwas measured as 3.30 :t O.OSon. and Lwas measured as
25.4 '" 0.4cm. Find the value of V, and the uncertainty In this value.
37
I
II Measurement techniques
If you flnd II dlfficul{{() de:l1 wlh the fraC{10031 uocertahlty rule. )'OO can easlly
estimate the uncertainty by substkUling extreme values Into the equation
For x " A""Z�, taking account of the U1lU'I1alntleS In y and z. the 1o\\'t'5t 113lue of x Is
glllenby
Xbw "A(y � A>�z� !J.z'j>
and the highest by
X�","A(y -+- AY)-(z+ !J.z'J'
If Xiowand
xb;p are worked out. the uncertainty In the\l;Jlue of x Is g[\'en by
(xbilb - Xbw)fl·
Apply the I'xtrl'lTW' vdllJl' method to the data fof the simple pendulum experiment in the
Elldmple on page
37.
BlKau"" of the form of the I!qudtioo fof 8. the lowest value fof IJ win be obtained if the
lowest value of I and the highest vdlue for Tare substituted. This gives
lfuN ,, 4�{0.5311.521j=9.1m 5-1
The highest value lorg is obtained bysubstitutiflg the highest value for I 3nd the IoWPSt
valU(' for T. This gives
/(h91 = 4]1[1(0.57/1.481) = 10.3m .1
thU5 {g�91 -g�/2 . (10.3 _ 9.1Y2 . O.15m 5-2, as
The uncNtainty in the value 01 gis
before
Now it's your turn
14 Apply the I'xtrl.'lTle 'o'iIlul' method to the data fOf the YOlume ofthe cylinder. on page 37.
If the
expression for the quaBily under considerntlon In\\:II\'eS combinations of
products (or quaUefils) and sums (or dlffere� then the best approach Is the
extreme value Ol{1hod.
•
Methods aYdilabll' forthe meiI'iUfement of length include:
metre rule (rdnge 1 m, reading ulKertainty 1 iMV
micrometer screw gauge (IiIngtl 50mm, reading uncertainty 0.01 mm)
Yernier c:aliper (range 100mm, reading uncertaintyO.1 mm),
• Methods ilIIailable for the meiI'iUfement of mass include
top-pan bi!lanCl'
spring baldnce
levl'r balarKe
• Ml'lhods ilIIailable for Ihe meilsuremer1t of time
stopc:iock (rl'dding uncertaintyO.251
stoP'Natch {readinguncertaintyO.Ols)
Include:
cathode·ray oscilloscope
• Methods iI'Iailable for the meilsurl1ffil1
lf1 of temperature indude'
liquid·in·glass thermometer
thermometer.
thermocouple
• Methodsil'lailableforthe meilsureme!1t ofwrrE!!1 taild potentlaldiiference indude:
analogue
meter
digital meter
multimeter
•
•
.
cathode-riYj osdUosc0p2
Methods available fort� measurement of magnetic flU)( density include the Hall
pm""
AcrurdCY is COllCerned with howcbie a reading Is to its true value.
• Pre<:is.ionis d£.terminedbythesill!ofthe random errorand canbe COlltroned byt�
I'xperinenter.
38
E>amination style questions
• Uncmainty indicates the range01 values within which a measurement is �keIy to Iil!.
• A syslMlalic uncertainty (or systematic error) is often due to instrumental cau5('S,
and results in all readings b@ing abowor below thetruevalue. It cannot be
�iminaled by iM'raging.
• A random unc:ertainty (or random error) is due to the scatter of readings around the
true value. It may be reduced by repNting a reading and averaging, or by plotting a
graph and taking a beit-fit linl'.
• Combininguna>rtdinties:
for expresloions of tm> formx .. y+zOf x_y_:, the Oo'eI'all uncenainly is
lu = 6y+Az
for expre'>Sials of the form x .. AY"Z6, the avera. fractional uncertainty is
!u1x= a(Aylj1 + b(AzJz)
Examination style questions
,
2
You are asked to measure the internal diameter of a glass
capillary tube (diameter about 2mm). You are also to
investigate the uniformity of the tube along its length.
Suggest suitable methods.
The value of the acceleration of free fall varies slightly at
differeot places 0fI t he Earth's surface. Oiscuss whether
t�s meaosthat
a a top·pan balance,
b a spring balance,
a lever balance,
c
snould be re-ca�brated when they Cll"e rno.'td to different
locations.
]
4
tf you needed to, howwould you calibrate a balance?
The shutter 00 a particular camera has settings which
al!ow it to be open for (nominally) I s, 0.5s, 0.255, 0.1255,
0.067s. 0.033s, 0.017s, 0,008s. O.OO4s, 0.002s and
0.001 s. Suggest a method (or method� of calibrating the
exposure times 0Ief this range.
Explain the factors you would consider when de<:iding
whether to use a liquid-in-glass or a thermocouple
thermometer in pill'ticular experimental situatioos.
5ummarise the advantages and disadvantages of analogue
and digital ammeters.
6
Explain how to use a cathode-ray oscilloscope to measure
the characteristics of the sinusoidal output from a signal
generator.
ii
used so as to reduce random errors.
measurements,
{Sj
the acceleriltion of free fall g. Measl.Jrements are made 01
the length La the penOJum and the period rof osc:ilation
The Value5 obtained, with their uncertainties, art' as shown.
T= (1.93 :1: 0.03)s
L = (92:1: 1)cm
meilsurement of
i the period T,
ii the length L.
b The relillionship between T, I. and g is given by
g=�
{I}
{I}
{1j
as 9.7Sl m s-l.
By reference to the measurements of L and T. suggest
wt,r it would oot be correct to quote the value ofg as
{1j
i
{I}
f2}
ii Use your answer in b to determine the absolute
b State howtheinstrument in a is
checked so as to avoid a systl'fl'lilt ic ermr in the
signifkilnt flQUres
Cambridge International AS and A level Physics,
9702121 May/June 2009 Q I
9 A simple pendulum may be used to determine a value for
c The villues of L and Tare used to calculate a value ofg
a State what instrument should be used to measure the
i
for the measurement of the following,
i the diameter of a wire of diameter about I mm
{1j
ii the resistance of a filament lamp
{I}
iii the peak value of an alternating vdtage
{1j
b The milss of a cube of aluminium is found to be 580g
with an uncertainty in the measurement of 109. Each
side of the cube has a length of (6.0 s O. I)cm.
Cilkulilte the density of aluminium with its uncertainty.
Express your answer to an appropriate number of
Using your ilnswers i n a, calculate the percentage
uncertainty in thevalueofg
A metal wire has a cross-section of diameter of
approximateIy 0 .8mm,
diameter of the wire
a Stilte the most appropriate instrument. or inslruments,
a Cilkul<rte the perceotage uncertainty in the
5
7
8
9.7Sl m s-l.
Cambridge International AS and A Level Physics,
9702122 Mayllune 2010 Q I
{I]
uncmainty ing.
Hence rtate the value aR, with its uncerfainty, to an
appropriilte number of significant figures.
{l}
Cambridge Internaional
t AS and A level Physics,
9702122 OctlNew 2009 Q
,
39
I
AS Level
3 Kinematics
By the end of this topic, you will be able to:
(a) define and use distance, displacement, speed,
velOClty and acceJcnllon
(g) wI"e problems using equations which represent
unifonnly �ccelernted motion lo a straIght llne,
(b) use graphical ml'lhods to represent distance,
dlsplacement, speed, llcloclty and acceJeralion
Includlng the motlon ofbodJesfaJJJna l n a
unifomlgrnvKalional f1eJdwlthout air resistance
(h) describe an experlmenl to determine the
acceleratiOn of frec all
f using a failing body
(i) describe and explain motloo due to a unifoml
(c) detemllne dlsplacement from the area under a
velOCIty-lime graph
(d) detennlnevelocHy usJIli the grndlent ofa
dlsplacemenl-tlme�ph
(e) detemllne acceler.ulon uslnathe gradlentofa
�·elodty-tlme graph
(0 derl.'e, from the dcflnUlons of ,'cloeily and
acceleratJon. equatlol\s ...
1
1I(:h
.
represemunlformly
velocl1y lnone dlrectlonand aunlfoml
accelerallon n
I a perpendicular direction
accelerated motlOl\ Ina str:llght Une
Starting points
• Kinematics is a de'icription of how objl'<:ts 1'l'lOYe.
• TlMo motion of obje<::ts can be described in terms of quaotitiMsudt as position,
speed, '.'I'!ocityandaa::eleration.
3.1
Speed, displacement, velocity
and acceleration
Average speed
When talking about mcxlon. we shall dl.>ruSS the way In which the position of a
panicle varies wnh UrnI'. Think about a pantcle moving along a straight Hne. In 3
cenaln time. the panicle will cover a cenaln dtstanre. '1)(' avel'llJ,le speed of the
Table l.1 Ex�mplesof s.peoeds
light
electron � round nucleus
3.0 w l01
2.2 w 1 01
3.0w I0·
jet airliner
panlcle ts deflned as the dtstance mOl-'ed dlvtdedby tl)(' tlmetaken. Wrlll{'11 3S a
\\'ord equltlon. thls ts
2.S w 10l
The unk ofspeed Is the metre per serond (m
s-').
One of lhe most furodame..al of physiGll COfISI:Ints Is the speed of IlghI ln a v:JOJum.
It Is tmpatam because I Is used tn IhedefutJ:1on ofllie metre. and because. acrordlngto
thelheoryof reL1tlvly, II defines an upper lImll lO allalnabie sjX'eds. The rangt' ofspe€(is
tilal you are likely 10 cane acJUS/l ts enonnou� some are sumnur!sed In 'P.IbIe 3.1.
w�g speed
40
It Is tmportant IO recogntse that speed has a meanlngonty If II Is qUOled relJlh'e to
1 ,, 10-'
a fixed refereoce. In most cases, speeds are quaed rebtlVe 10 the surface of the Eanh,
which _ although It Is IIlO'o'tng rel;lI1\'e 10 the SOIu 5ySlem - Is often taken to be fixed
3.1
Speed, displacement, velocity and acceleration
Thus, when I'I't' say that a bin:! can ny al a cenaln 31't'rage
speed. we are relating Us
flying
speed 10 the F.3nh. Howe\'t'r. a passenger on a ferry may .see that a .5elgulJ,
parallel lothe boal, appears lobe pracrlGllly stationary. lfthi.sls the case.the .5elguU'S
speed relatl\'t' 10 the boaI ls zero. Ho
..
..
e
n
�r,
If the speed cJ: the boot through the water
ls 8m s-', lhen the speed of the seagull relalh't' to ICarth Is 3150801 s-'. When talking
:abouI relatl\'t' speeds we must also be careful aboul directions. Il lS easy If the moUons
are In lhe sallll' direction, as In lhe example of the ferry and the seagull, l1le addlUoo
oln'lodtyn'ctors ls ronsldered In 1bplc I (page 8).
1
Thl' radius of thl' E.arth is6.4 " \O'im;one rl'VOlution about its axis takes 24 hours
(8.6 " 10'<;j. Cairulatl' thl' aW'f<lg1' 5pI'!ed ofa point 011 lhe Equator rl'latiW' to tM
(rotre ofthi' Earth.
In 24 MUfS, the point on thl' I'quatorwmpietes ooe revolution and tra'Jeis a di5lancl' of
2
2!! ><thl' Earth's fadius. that is2!! ><6.4 " 1()6_4.0 >< 101m
Thl' a�agl' 'ipl'!!d is (di5tancl' rnov!!CI)t(timelakefl),
or 4.0 >< 101/8.6>< 10'",4.7>< 102m 5-1.
Howfdfd�sa (ycli51Ifavel in l l miflutesif his average speed is22km h-'7
Firsteonvl'rtthl' avl'fage � ifl km h- ' to a value ifl m S-'.
22km(2.2>< IO'm)in l hour(3.6 >< 10ls) isan average speed of6.1 m s-'. I' miflutl's is
66Os. SineI' average spl'ed is (distance movedMtime talefl), the distance moved is givefl
by (avl'ragl' spI'ed) >< (timl' takoo), or 6.1 >< 660 _ 4000m
1
Note the importance ofwori:.ing in consistent uflits: this is why the average spllE'd and
the time were converted to m :o' and s respectrvely.
A train is travellirlg al a speed of 25 m 5-' along a straight track. A boy walks along the
corridor in a carriagl' towards thl' rl!ar of the train, at a 'ipI'!!d of I m s-' reiatiW' to the
train. What is his speed relatiW' to Earth?
In OOI' secood. thl' train travels25m forwards along the track. In the same time the
boy moves 1 m towards the rear ofthe train. so he has moved 24m along the trade
Hisspl'ed reiatiYeto Earthisthus 25 - 1 _ 24mt-1.
Now it's your turn
1
The s.peed of an elKlron in orbit about the nucleus of a hydrogen atom is
2.2 >< l06m s-'. tt takes 1.5 x lO-'tis for the electron to comp!ete one orl:it. Calculate
tMradiusoftheorbit
2 The a�age s.peed of an dirlilll'r on a domestic f�ght Is 220m s-', How long wiM ittaloP
to fly b@twem two airpats on a flight path 700km long?
1 Two Cdfs afl' t ravel�ng in !hi' saml' dirKlion on a long, Slraight road. The one in front
has an averagl' speed of 2 5 m 5-' rl'latiwto Earth; the other's is 31 m s-'. alsofE'latM>
to Earth. What isthl' speed ofthl' 5I!Cood Cdf relatiw to thefirst when it is OW'rtaking7
Speed and velocity
In ordinary language. there Is no dl(f('wnce betwtX'n ttle !ennS
s/x!{!d and w!oclty.
Is
Howe",r, In physics thew Is an Imponam distinction IX�WtX'fl tlle two. Velocity
used to represent a vector quantlty: the magnitude clltow fase a partk:1e Is moving alit:!
the dIrection In whk:h U Is moving. Speed does f\C( hlVe an associated direction.
scalarquantlty(see lbpk: 1 page
7).
II Is a
So far, we have talked about the tolal distance Irnvelled by a body along lIS actual
p:lIh. Uk!> spt'Cd. dlSlaoce Is a scalar quanlUy. bec'Juse we do nee have to
specify the
direction n
I which the dIstance Is travelled. HO\\'e\·er. In dennlng I'l'iodly we Introduce
a quaTtlJry called d
isplaremenl. Dlspiacenll'Tlt d a partk:1e 15 Ils change cI posklort.
llle dlspiacemeTtl Is the distance travelled In a scrnlght line n
I a specified direction
from the Slartlng paint to the flnIshlng palm. consider a cycliSl travelilng 500m due
east along a strnight rood, and then turning round and coming oock 300m.
dlSlance trnl'elled Is 800m, bui the dlspbrement IS only
The 100ai
200m due e:lSl, 510ce the
cycilst has ended up 200m from the Slanlng paint.
41
B
Kinematics
The anrage \'elocity Is deflned as the displacement dIVided by lhe time taken.
Bec2use diSlaOCl." and displacement are different quanilies. the a\wage speed c;I
motion wlU sometimes be different from the magnitude c;llhe a\'eragt' \'eloclly. If the
time taken for the qdl9:'s trip in the example abo\'e Is 12Os. the a\'erase speed Is
8O<VI2O ;6.7rn s-', whereas till' rnagnlrude cl the average \'clodty Is
200/120" I.7rn s-'. lbls rnay :<iefill confUlilng, but the difficulty arises only when
the molion involves :l change cidlrectlon and w" tllke lin l\vcrage V;Jtue. If we are
Interested in describing till' ITKXlon of a partICle at 3 particular moment In time, the
speed at that moment Is the SlIme 3S the magnitude of tI-.e velocity at tilat moment.
\l'e now need to deftlll" ....'erage veDdty more precisely, In terms of a mathematical
equation, instead of our previous word equatIOn. Suppose that at time I, a partICle Is
al a polm x, Of] the x-axis (Figure ".0. At 3 latC!' tln-.e II. ti-.e panicle has moved 10 X"
The displacement (the change In posItloo) IS (;Xl - x,), and the time taken Is (IJ _ I,).
Flgure l,1
The al'er:J.ge l'elocity v is then
The bar over Ii Is the symbol meaning 'average'. AS a shoftlllnd. we can wrile (xl
x,)
as Ax, where t,. (the Greek Glpitalletter dek3) means 'tile change In', Similarly. IJ _ I, Is
wrtten as M. This gtves us
If x, were Ie5s than x" (;X,- x,) and Ax would be neglItl\'e. This would mean thai
the particle ilad 1IlO''ed 10 till' 1ef1. In5lead cilOthe right as In "lgure 3-l. The stgn
oftill' displxemem gin's the direction ci partICle motlOll. If Ax Is negaU\·e. then the
a\'er:J.ge ,'eIocly v is also negathl', The 51gn c{lhe ,'eIoc!ty. as well as lhe Sgn c{the
dlsp!aceIJll'N:, irxllcales the dlrectlon c{the panlCle"s motlon. Thls ls beell.l5l.' both
dlsp!ace1Jll'N: and "eIoctty are ,·ectcr quantitJes.
Describing motion by graphs
PosiliOfl-timegraphs
Figure ".2 Is a gr:J.ph of po51tlon x against tlmelfor� panlcle 'ncr.'lng ln a stralght llll.l"
This curve g"'es � complete description of the motion of the panicle, \l:re em see from
the graph that the particle starts at the origin ° (;11 which x" 0) at (till(' I " O. Prom °
tOA the graph is a str:J.lght line: the panICle !S CO\'Cflng equal distances In equal periods
of tinl('. This represents a period of Im!fOml l.(1oclly. TIl(' avcrage velocity during
thls time ls (x, - O)/(t, - O). C!early. this Ls the gfadicnt of tll(' .5Ir:J.Ight-lh-.e part of Ill('
gr:J.ph beiWeffi 0 and A. Between A and B the panICle IS slowing down. because tl-.e
distances travelled In equal periods of time are BClllng smaller, Tile aver:J.ge vcloclly
durlngthLs period Ls(;X,- X,y(t, - I\). On thc graph. thLs Ls represented by Ill(' gradient
c;I the Slralght line joining A and B. At B. for a monl('fit. til('
particle is at rest. and after
B It ilas reversed II:s directlOll aoo Is heading IXIck towards the origin. Between B and
Flgur. l,Z
C the a\'eragevelodcy ls(;Xj - x2>l<tJ - li). BeCau5l.'XJ Is less than Xl. thls is a negative
quantlty. IOO\caUnglhe rev�l cidlrectlOll.
Clicula!lng the a\"l'ragt' l'eIocIty of the partICle O\-er the reiatlVe/y long in\t'TVals I"
(j, - IJ and 0, - Ii) \\1U nol, 11o\\"\.'ver. gh'e us the complete description c{ the
motlon. 1b describe the moUon exactly. we need to know the particle's \'eIoc1y
at el'ery lnSlant. We Inuoduce the Idea cl instanlanOOtlS \·elocity. 1b define
Instantan.eous \-ekx::ly we make the interv.lls of tlnl(' O\'er Willett we measure
the 3\"1'r3ge ''elocky shorter 3nd shoner. Thl'! has the effect of approximating the
curved dlsplacement-tlme graph by 3 !ier1eS c{ short straight-line segments
"
3.1 Speed, displacement, velocity and acceleration
lllf" applUl{ lmatloo becomes bet!er the shorter the Ume lntClVal. as llIu!ilralOO In
figure 3.3. Io.venrually, In the case o(extll'mely lifIlall Ume Intervals (Il1:ltlle matlcally
we would say 'Infinltesl mally small'), the Slralglx-llne seg ment has the S:llne direction
as the t angent t o the run't'. This Umlllng case gh'es the Instantaneous \'t'Iodly as the
gr3dlent o(the tangent 10 !he dlspiacemenl-llme rurve.
Flgure l.l
Displacement-time and velocity-time graphs
Figure 3.4
Is a sketch graph 5h.owlng how the dJsplacenlenl of a Clr. IrJI'elllng along a
Slralght lest track. vactes wllh lime.
We Interpret this graph In a descripllve way by noting
!hat between 0 and A !he dl.ltances travelled In equal lmervals of Ume are progresstvely
locreaslng: !hat Is, tile velocltyls locreaslng asthe Clf Is acceJerallng. Bcl:weenA and B
!he dl'it:lTlCeS for equal lime Intervals are decreaslng; tile Clf Is slowing down. Finally.
!here Is no change In position. elfen though lime passes. so Ihe Clf mU!il be at rest. we
can use Figure 3A to ded.Jce the detalls o(lhe way In wilich tlle Cl(s n
l !ilal'llaneous
\'eIocIly vwrleswllh lime. lb do lhls, \\� draw langelllS lO Ille CUI'\'l' In FIgure 3.4 at
regular lnIerv:IJso(llme, and measure the 5Iope ofeacl1 langenl lo cb!alnvaJueso(
Flgure l.4
L'. 1be plot of v against I gln� the gr3ph n
I Figure
.lS, This cooflmlS our descrlplln'
inlerprelaIion: the ''eIocIly locrea.se!i from Zl'IOlOa ITI:lxlmum '':lJue. and thendecn'ases
to rem again. We \\1U look at lhls example In more detaU 00 pages 47-48, where we shall
see lhal thearea under the ,'CIorty-tlme I!f1IPh In Figure 3.5 gl'ies the dJsplarement x.
Acceleration
We ha\'e used the "uro acceIemtfrlg In descr1b1ng lhe Increase In "eIodly o(the car tn
the prevlous sectlon.. Aa:eIer3Uon \sa measure of the rnle al which the \'eIodly ofthe
particle Is changlng. A,'erage acceleration Is def!ned by tile y,uro t'ql,l.1ti oo
Flgure l.5
The untt of accelerallon Is the untt o( veloctty (the n'lClre persecond) dlv\d('d by the
untt o(lIme (the second), glvlng the metre per (secondj which Is represented as m S--'.
"�
A
,
Flgure l.6
In �ymbols, this equation Is
wtlere I', andv, are the veloclUes at Ume I, and l , respectll'Cly. To obtain tt\e
in5tanlant'Ou5 acceleration. we take extremely small time Intervals, I'.lS!: a5 we did
when defining lrutantaneous velocity. Bealuse It In,'oll't'S a change In velocity (a vector
quantlly), acceleration Is also a ,"ector quantly, we neoo tospec tf)' bo\h itSll1:lgnlude
,
andits dlrectkln.
,
We can deduce the acceIer.ltkln of a panICle from Is "eIod;y-tinle gr3ph by dmwtng
a tangent to the curve and ftndlng the slope athe tangent. I'Igufe ".6sho'1\'li the reru.
a doing thls for tile car's motloo de:scrlbed by I'lgure l4 (the displacement_lime graph)
and Plgure ".5 (the ,'CIocIly-tlme gr3ph). 11Je car acceIer.lles at a conSlanl mte Ix1ween
o and A. and then decelemtes (that Is. sIov>'S dawn) unlfocmly between A and B.
43
B Kinematics
T..blll l,� E�.lIl'Ipll!S of �ce!er"lioI'Is
aueleratlonlm,-'
dueto circul�r motion
9 .. 1()l6
I w lOl
"
t.lmilyc�
�tEqu�IDr, due to
2
3 w 10-l
rotloon of E�rth
6 .. 10-'1
An acceleration "1th a "ery famlllaT value IS the accc\erntion a free fall near
the
F..anh's surface(see page 45} this Is 9.81 m S-'. otten approximated to 10m s-l. 1b
lIlusu:l.te the rnnge of values you may come across. some accelerations are summarised
In T:!ble j.2.
1 A 'IpOrts car accelerates along a waighttest
is ilSaverage acceleration?
track from rest to 70km h-' in 6.3s. What
First cornoertthedata intoconsistmtunits. 70km {7.0 w lO'1rn}in l hour{3.6 w IOlslis
1 9 m s-'. Sinceaveragea{celeration is(changeof velocity)l(timetaken), theil«eleration
is I916.3 =l.Om s--l.
2 A railway train, traveiflng along a straight trad:, takes 1.5
10 om to rest from
minutes c e
What is its average acceleration wh�e braking?
11Skm h-' is 31.9m s-l, and 1.5 minutes is 90s. The average acceleration is (change of
a speed of 1 ISkm h-I.
velocityY{timetaken) = -31.9190 .. -0.35m l-J.
the a{celeratiofl is a m!qatrve ql><lntitybeciluse the change of velocity is
fleqatrve: the final velodty is less than the initial. A negative acceleratiofl is often called
a dea!leratiofl.
Note that
Now it's your turn
" A sprinter, startiflg from the blocks, reaches his full speed of 9.0m S-I ifl 1.55. What is
his average il«eieratioo?
5 A car is t raveiling at a speed of25m s-'.Atthis speed, it is capabjeofaccelerating
at l.8m S-l. How long would it take to ac(('!erate from 25m S-I 10 the speed limit of
3I m s-l ?
6 At afl average speed of 24km h-', how many kilometres will a cydist travel in
75 minutes7
7 Aflaircrah travels 1600km in 2.5 hours. What is its average speed, ifl m S-I]
' [)06 a car speed�ter rl!9isterspeedorvelocity7 b;plalfl.
9 Afl aircrah traW!ls 1400km at a speed of 700km h-I, and then rullS ifll oa headwind
that reduces its speed over the ground to SOOkm h-I for the I\e)(l 800km. What is the
total tirrK> for the flightlWhat is the average speed of the atrcrahl
10 A 'IpOrts{ar can stop in 6.ls from a speed of 110km h-I. What is its accelerationl
11 Canthe veloc:ity ofa partide dtange ifits speed isconstaflt7 Can the speec:l ofa partide
change if its velocity is ronstant1 lf the answer to either question is 'yes', give eumples.
Uniformly accelerated motion
Having defined displacement, "e!oclly and accelerntlon. we slull use the deflnitlons to
dertve a series of equ atlofls, called the ilfllcmfllfC eqllatfo,Js, which em be lIS('(! to glve
a complete description of the motion of a particle in a strolghl line, The mathematics
,,111 be simplified If we deal with situatIOnS In whICh lhe acceleroUon dot>s nol vary
with time; that Is, the acrelerntiOfi Is ufllform (or COIlSlant). This approximation applies
for many practical cases. However. there a� tWO Important types of molton for which
the kInematic equatlOfls do fl()( apply: clrculJrm()(lOn and the oocillatory mOlIOfl called
simple harmonic m()(iOn. We shall deal with these scpar:ttely Ifl lbplc 7 afld TOpIc 13
Think about a panicle movIng along a straight line wJth corutant accelerntlon (I.
Suppose that llSlnttlal veiocl1y. attlme t = O. I.'l Il. Aftera funher ltme llts veloci.tyhas
increased to v. Frornthe deflnltloo ofacrelerutlOn as (cltange In �'I'Iocly)Allme takeo).
we llave a = (v - IIYt or. re-arranglng,
From the deflnllon of avernge \"eIocity ii as (dlStan<:e trlwellcd)AUrne lakefV, over the
Ilnll;' IIhi'" dl!lan<:e trowelled s "111 be gh'en by the al·Cfltge velocity muklplied by the
Ilme laken, or
44
3.1
Speed, displacement, velocity and acceleration
lllf" a\'eT3gevelocly v ls wrlnen ln rernls o(thelnllal \·cIoctty r. and flnal \·eIocllY IJ 3S
U+IJ
IJ "' �
_
and.
using tlie previous equation fIJ ",
Substltullng thIs we han"
lllf" Iight-IlaOO side oflhls equation Is the sum o(t\\O terms, The
U/ term Is the
distance the partlclt' \\oukl lla\'e U1I,"cl1ed In ume I If l llad been rrawillng wlh :1
con,;ulll spee<! u, and tlie iat" term ts tlie 3ddlllOll:i.1 distance tra\'elled as a reilIlt d
the aa:eleratlon,
The equation re!aUng the flll3l \,elocly I', the In.lalveloclty u, the aa:e!eratlon a and
the dlstance travelied sis
If yeti wish 10 see how this Is oiJt:llned from pre\1CuS equallons, see the Maths Nete below,
From V= II + fII,
I>lulllplylng bol:h Sldes by UI and expanding the tenns,
1 = ( v - INa
Substllute lhIS ln J = III+
s = U(P- u)fa +
2as = 2111J- 21t" + ,il - 2I1v+ II'
�II'
IJ v = II' + 2as
i'a<v - II)'!tfI
lllf" Ihe equatloos relating tlie varIOus qu:;mtlle:S which define the monon dthe
panlde In a stralght line In uniformly accelerated rTIO(lon are
IJ.U+aI
+1af1
s.l1t-�f1
vl.u2+2as
s. (U;IJJt
S .. III
;
In tlll"Se equatlons u is thelnltlal wlodty, lJlS the flnalve!od:y, (lls the ;l('(l"lerntlon, s is
tlie dlstance travelled, aoo Ils the tlme tak.ro.ll'le a\'erage veloct:y ills glven by (U IJ) .
In solving problems Involving idnemallCS, It IS Important to uoderstaOO the sKualloo
befIJe you tryto substltute numerlall V:llues lmoan cquatloo.ldt'nllfy lhe quantlty
you want to know, and then make a list dthe quantities you know already, ThIs
shouk:! make 11 obvloos whIch equatIOn Is to be
used.
Free fall acceleration
A very common example of uniformly accl."k."rnted mOlkln Is when a body falls freely
near the Eanh·s surface. BecauSl" of the gravitatIOnal al1r:tctlon of the F.;lrth, all obje<:ts
fall with the same uniform accelerntton. ThIs acceleratIon Is called ttle accelerarion
of free fali, and Is represemed by the symbol g. II ll:1sa value of9.81 m s-', and Is
directed oo...'nVo':lrds. For completeness. we oug./lf to qualify thIs statement by sayIng
th:!.t thefaUmuSl:beln theab5enceofalr resistallCe.bl.rtln most sltuatlons thlscanbe
Flgur.l.7 Strobo-ftuhphotog'� of
obJl!Ch W1lrHf.lll
1be acceleration <I free faJl may be dl."rermlrll"d by an experlnler1l In which
the time d faU I of a bOOy between twO pointS a distance s apart Is measured.
45
B Kinematics
If
the lxxIy falls from rest, we Gin use the second cJ the equatlorls for uniformly
acceler:ned mOlIon In the form
R ", "b/t'
10 alcubte the value or& Nelle that. because the time cJ fall IS likely 10 be onty a
few tenths cJ a second, precise timing 10 ooe-hundredth cJ a second Is required.
e
An expertmerv. lnlulvtng the swltrnlng cJlIgln �tes by tile failing obtect has been
described In lbpIc 2 (Rgu� 224). 1be UghI gateS are connected to an eledronlc timer.
t
t
S
lly
�n:�heJaczrn���;I
ap=::�:� =�. ::g�
:�:t:::YU; ;:.fu
This Idea "'':IS a consequence cJtlle effect of air resistance on tight �swlth a large
surface area, such as feathers. HOI\�wr. Galiloo Galllel 0S64-1642) su�ed that. In
the absence of reslstance, all lxxlies "'U1ki fall wkh the s:lITle comtanl acceler:ltlon
He shoI>."ed IIl.1thematlcally that. for a body falling from
reSI. the distance tr:ll'elled Is
proportional to the square cJthe time. Galileo tel;tOO the relltloo experimentally by
liming the fall of objects from vartous lewiS of the Le�tnln8 Tc"'''er cJ Pis:! (Figure 3.9).
This Is the relatlOll we have derl\'oo a s s ", 111 +
For a body Slanlng from reSl., lI = 0
and s=
Figure 3.' leanlngTowetof PIS�
t(jr2.
tat'. That Is, the dlsunce Is proponloml t o Ume squared.
Figure 3.8 wlilro in hilltudy
Examples
1
A car irxrN'>e1 rts spero from 25 m ,1 to 31 m S-1 with a unifOfm acceleration of
1.8m S-l. How far does ittrdvel while acceleratirl91
In this pro�em we want to �nowthe distarxe s. We know the initial speed
u .. 25m s-1, thefinal s.peed " .. 3 1 m s-I, andthe acc\!lerahon(l. 1.8ms-l
Wl!need anl!quation lin�ingswith ll, ,,and";thisis
,,1 : u 1 + 2as
2
Substrtuting thl! values, lWhilVI! 3 1 1 . 2 S1 + 2 " 1.85.
Re-arranging, .s : (311- 2S1:K2 ,, 1.8) . 91m.
ThI!avl!foJgl!ao::ell!fation ofa sprinter lromthetimeol leaving thl!blod::s to reaching
hI!!" maximum speed 019.0m S-1 is 6.0m s-2. for IlCMI long does she ao::elefale? What
distance does shl! cover in this time?
In thl! first part of this probll!m, WI! want to know the time I. We know the Initial speed
u= O.thl!finalspeed II", 9.0m s-1, andthl! acceleration " . 6.0m s-l.We need an
equation �nking ,with u, "and a; this is
Sub'itrtutingthe vdiues. WI! hilW!9.0 . 0 + 6.0f. Re-arranging. / . g.0J6.0 . 1.5s.
46
3.1
Speed, displacement, velocity and acceleration
For the serood part of the problem. we want to knowthe distance s. We know the
initial speed II = 0, the final speed 1/ . 9.0m s-1, and the accelel'ationo. 6.0m s-l; we
h�alsojustfoundthetime / .. l.5s. There lsa choice of equations linking Jwithll. t,.
a and I. We can use
Substitutingthevalues, 5 . 0 +
� ,, 6. 0 ,, (1.5)l . 6,8m,
Another relevant equation � Ii .. t:.xJ1J.1. Here the average l'eIocity ;:; is giYen by
Ii =(u + 1iV2 =4.Sm 5-1. tuJAI is thesame as slI. so4.5 .sI1.5. and
s .. 4.S .. 1.S = 6.8m asbefore.
1 A cricketer throws a ball vertic:aUy upward irlto the air with an in ial velocity of
18.0m S-I. How high doe5 the ball g01 How long is it before it returns to the cric1aole(s
hands1
it
In the first p.lrt of the proMrJ, WI! want to know the distance s. We know the initial
II£>Iodty u = 18.0m S-1 upwards and the aaeleration (I . s. . 9.81 m 5-1 downwards. At
the r.ighest point the ball is momentarify at rest, so the final vel!Xity II. O. The equation
lin�in g s with 'I, vand(lis
/.J= /I1+ 2as
Subs!ituting the values, 0 ", (18.0)l+ 2(-9.81)s. Thuss.-(18.0)212(-9,8 1 ) . 16.5m
Note that hefe the ball has an upward velocity but a downward acc�efation, and that
at the highest point the velodty is Zl!r0 but the accelefation is not ZefO
In the second part we want to know the time I for the bali's up-and-down flight. We
know /I and <I, and also the overall displacement s. 0, as the ball returns to the same
point atwt1ich it was thrown. The equation to use is
l fm 1
s= II +
Substituting the values, 0 " 18.01 + J<-9.81lr. Doing some algebra, 1(36,0 - 9.811)= O.
Th£.re are two soiutlons, , = O an d , . 36.0I9.81 . 3.7s. The / . O value corresponds lD
the time wtH>n the displacement was zero when the bal was on the point of leaving the
crid:ele(s hands.The answer required here is 1.7s.
Now it's your turn
12 An airliner must reach a speed of 11Oms-' to take off. If the available length of the
runway is 2.4km and the alKraft accelerates uniformfy ftom rest at one end. what
minimum aa::eleration must be ava�able ifit is ID takeoff1
U A �ing motomt paSse5 a traffk police officer on a stationary mo\orqde. The police
officer immedialefygives chase: his uniform aa::eleration is 4.0m s-J. and by thl! time he
drilW5 level with the motorist he is travening at 30m S-I . How long does it take for the
poIi<e officerlD {at{h the{ar1 Iftheta( continUl!s to tfavel ata stl!ady � duringthe
chase. what is that speed1
14 A cricket ball is thrown vertiGlify upwards with a speed of IS.0m ,', lNhat is its velocity
wnenit first passesthrough a point8.0m above thecricketl!r'shands1
Graphs of the kinematic equations
Il ls oflen useful 10
repres nt
e
the mOllon of � particle g/'lphlcally, Instead ofby means
of a series of equaUons. In Ihls sealon we brtng logether the g/'lphs whk:h correspond
to the equallons we haH" already d{'flved. \\'e Shall see thaI there are
some
Important
links bel:v.een lhe graphs.
FlrSI. think aboul a particle moving
COnstant H"lodty
r
In a
straight une with constant \·e!oclty.
means Ih3l lhe particle cO\'ers equal
of tIme. A g ap h of displacement x agaInst time
distances In equal lnterv:lls
l IS thus a straight line, as In
Ftgure 3.10. Here the partlcle has started at x = O aoo at Ume l = O.
The slope of
the graph Is equal to th e magnitude of the velocity. since. from the definition of
a\"t'rage velocity, Ii = (X1 - X,)/(11 - I,)
st nta
the a\'t.'T3g1" 1"t'lodty and the In
Flgure l.10
descrtblng the graph
Is x = tt.
a
'" fu/l!.I. ae-a.use
this graph Is a straight line,
neous ,'eloctty are the same, The equation
47
B Kinematics
bout a particle lIVo'lng In a !H"tghl line Wllh COI'I5tant acceleration. The
Now think a
panicle'S velocity will change by equal amouOls In equal lnten"ls oftlme. A graph of
the magnllude /Jdthe veloctty against time I "111 be a straight line. as In
Here the pank:Jloo has Slaned \\-1tll \'eIocIty
FlgUIl' 3.11
" at time I = O. The slope of the graph Is
The graph Is a straighi line showing that
The equatIOn describing tile graph Is /J = // + at.
equal tothe magnitude athe acreIerntion.
the acceier3tton Is a COIlSlllnL
An Important fearure a 11K' l"eIodty-lllTll.' graph Is that we can deduce the
displacCll"lCN: a the panicle by cak:ulatlng the �re3 Delween the graph and the
between appropriate ilmlls of
l.axl:5,
tlme. su� we want 10 ctxaln the dlsplacemeot of
the pank:Jloo between times I, and /1 10 Figure 3.11. 8elween these times thea\'erage v
vek:ldty ls repn'serv.ed bylhe horlZontaIIlneAB. l1le area between thegr:tph andthe
I_axis Is equal to the area d the rectangle wtn;e top edge
"
,
Flgure l,11
This area Is vAl. BuI, by the deflnllJoo
Is AB, or a\'er:tge velocity ii.
of' avernge ve\oclly (ii = fJ.x/fli), vAl Is equai to
the dlspiacemelllt.x durlngthe time IOll."rVai tJ.
\X'e can deduce Ihe grnph d dlsplacemffil
s agalnSi time I from Ihe \'elocIly_tlme
graphby cakuialing thearea be!:ween lhegrnph and lhe l-llxls for a suoxesslon
of values of I. As shown In Figure 3.11, we can spilt ttlC area up Into :l. numberof'
rectangles. The displacement al a C{'fI�ln lime IS then juSl tl1C sum of the areas of the
h
rectangles up 10 t at lime. I'lgure 3.12 .'ihows the result of ploUing the displacement
s de'lermln ed In this way agalnSl lime I. II Is a CUfve Wlttl a slope which Increases
the higher the value of I. IndlcatlngtlL1t the partICle Is accelcr:ttlng, The slope at
a pantrularlinlC g!ve5 thcmagnlllL{\{> o(IIIC InSiantallE.'OUS ve!ocl1y, The equatlQn
descrlbtng FIgUIl'3.12 IS S = ZII +4arz.
Example
F'lIJ re 3.13. US!?
The di'>pta.c£>ment-time graph fOfa car on a straight test track is shown in
u
thisgraphto draw velodty-tirooand aa:ell!fation-titrM! graphsfor thl!test rl.lll
Flgurel,12
It �
o
Flgure l,1l 0IspLlcement-tlll'le gr.lph
2
4
6
10
8
12
14
16
We MVI! already root this graph when we disrussed tM COfIC!?pts of velocity ar.d
acceleration (Figure 3.4, page 43). In Rgure 3.13 it Ms bEEn re-drawn to scale, and figures
MY!! �n put on the dj>;placement and time a�es. We find the magnitude of the velocity
bymearuringthegradientofthe displiKl!fTll'!l1t-time graph.Asan exampM!, a tangentto
thl!graphhilS been drawn at t = 6.0s.T� stOpl! ofthistangent is I 8 m s-I, lfthe proc('SS is
repl!ated at diffe-reot times, too following velocities are determined
I�m s-'
6
11
18
14
)0
20
10
0
F ...re 3.14. Ched::: some of too
These values are plottl!d !Xl th-e Y!!locity-time graph of ig
vallJe5 bydrawing tangents yourself.
transparent rule-r.
Hint: 'M'len drawirlg tangents, uS!? a mirror ora
Rgure 3. 4 shows two straight-�ne portions.. In ia y from r _ 0 to /_ lOs, the car is
accell!fating l.llliformly, andfrom r .. 10stor. 16sit is decll!lerating. Too aa:e!eration is
1
it n ,
given by a = Al'l!J.r", 3�10 ", 3 m s-lupto r . 'Os. 8eyond r _ l0sthe acceleration is
_ m
.
min
n
t ha t t car ls
le a ing
0J6= 5 .,1 (The
_3
48
ussig !ihows
oo
dece l t .)
3.1
Speed, displacement, velocity and acceleration
Flgur. 3.14 vetodty-�megr.lpll
1 :\ !III!IIII!!!!:' �
TllI! Ma!�ration--timegraphi5 plotted ifl Rgure 3.1S.
i '.
i
!!lIWtiIt
_,
"'1'
eo"',.
finally. we can coofirm that the area under a wlocity-tlme grapf1
The area under the line ifl Figure 3.14 i5
116,
mels
tl
gives Ille dispjacl!fTlMl.
(iJ< 1O J< 30)+(i J< 6J< 30),. 240m
he va� ofs alt= 16son Figure 3.13
t
Now it's your turn
lS l n a t..slof a spor!scarona slraighttrack. thefolowirg readings of velOOtYIIWl!l'''
obtained at thetim..s I stated.
r/ms-'
0
1 5 23 28 32
.....
(_)
35 31 38
....
Ongrapf1 p.1pO'r, draw a locity-timegraphanduseittodet min.. the
Ma!iO'ration ofth.. car at tilTM! / . Ss.
(b) Findalsoth.. total di5lao<.. tra lled betweeo l . O andr. 30s.
.....
Note:Th..,;e figure!i ref..r to a uSI! of l'lOn-uniform acce!eration. wNct1 is
more reali5lkthan Ih.. PfeYious ..xample. Howev«, tile same rules apply: the
;KulO'ratkm is giW'fl by thO' slope of the velocity-tlme graph al the relevant time.
and thO' di5lanc.. Iravelled can be found from the area u der tile graph.
n
Two-dimensional motion under a constant force
Oi
So far we h.ave been dealing with. m lon along a Slrnlghl line; thaI Is. OIle­
dImensional moUon. We will now think about the motion of partIcles moving In
ion
paths In ('wo dImens s. We shall need to make use of Ideas we have already
learnt regarding vectors In Topic I. The partIcular example we shall take Is where
a particle mo\'es In a plane under the action of a constant Force. An example Is
(F u
ba .
eight. For the eleclron. lhe conSiant force Is the fon:e provided by
the m(>l1on of a ball Ihrown at an angle 10 the \'('rtlail ig re }16). or an electron
moving at an ang\e 10 an eleclTlc fleld. ln tOO case of lhe
ll the conSlanl fOfce
Figur. 3.16 Codeter bowbrgtlle
bolD
Il
aCiing on ls Its w
lheelectrlc flekl.
49
B Kinematics
....
"..'
This lopic Is Glen called projectile
of this molirn.. He
molion. Gaillco flrstg:lI'e an aCC\lr:lIe anal}'sls
did so by spIIulng Ille motion up 1010 lIS \'Cftlcal and hcl"lzonIal
componenls, and ronslderlng lhe!;e sep:;lr.llely. TIle
key Is lhat llle IWO components can
be consklered Independently.
5llbject to
(Its welghO. As before. air reslstanre 11111 be neglected. We
will analyse the moUon In terms cf the horlzorul and I'ertlcal components of I·e!ocl)'.
AS an example, think aboul a parUdl' sent off In a hcl"lzoolal direction and
a l'enlGll gravlaUonal flJU'
The panicle Is projected at Ume I" 0 al the OI1:gln 01 a system ofx. y ro-onilnales
(P1gUR'" 3.17) \11th l'eIocly u" In the x-<i[n>aIOO. ThInk nrst aboul lhe panicle's Il.'fIlGll
motion (In the y-dlrecrlon). Throughout ille motion, II. Ius an acce\er.lIton ofg(the
Figure 1.17
acceleration of free fall) [n the )'-<ilrecrlon. TIle In[lIal l'a!Ue of lhe venlcal compa1ffil
of 1'eIoc1l.}' Is u)' ''
O. Tbe venlcal component IOCR'"ases ronUnuously under the uniform
acceler.lUon g. Using IJ = U -+ aI. Is V3lue II� at ume lis given by II)' '' gl, Also al tJme I.
gl\"('n by Y "
Now for the hortzontal
lhe venlGll dlspl.:u:emenly OOwnv.'lIrds
Is
�r2.
motion On the x.<Jlrecrlon): heR'" Ihe accelernllon Is zero. so the hOrizontal componefll
of velocll.}' rematns con51am al
u.... At ume l lhe horizontal dtsplacement x Is gtven by
x " /1,/. To find the I'elodly of the panlck' �I �n}' tlrne I. tile two components l'.. and
Ii)' must be added I'ectortally. The dlrecUon of the result�nt vector Is the direction of
fIlQ(lon of the panlde. The curl'e Imced OUI by a p�rtlcle subject to a constant force In
one dlrecrlon l s a panl>o13.
Figure 1.18 W�t"'"Jets 'rom .1 garden
ijlflnldefshOWlllg .1 pi1t.1boI.1·shol� SPliIY
If the panlde had been SCIll off with I'{'loclt}' 1131 an angle 6to the horlzontaJ, as
In FlguR'" }.19. the only difference 10 Ihe anal},slS of tl"IC m,:(1on Is that the lnklal
y_component of velocity Is II sin 6. In the (>x,1mplc lIlustmted 10 flguR'" 3.19. this Is
upwards. Because of the downwards acrelerntloO i/o tl"IC y-component of velocity
decreases 10 zero. at which time the panlcl(> IS �t the crest of ks path. and then
Figure 1,19
Increases In magnitude again bul Ihls !lme In the opposke directIon. The path Is again
a parnboia.
For the panlcuLarGisc ofa panIcle projeCted "1th velocity II at an angle 6tolhe
IIorIrontal from a poI!"W. on levei grOllnd (FIgure 3.20). tt"IC rnnge Rls del'lned aSlhe
dlslaoce frorn the polm of
projection tothe polnt at II'hlchthc particle reaches the
ground agam. We can shoI\' tliat R ls gl\'en by
R=
Figure 1.20
50
(u1sm29)
g
For dl.'l:alls, see the folaths Note oppolilte.
3.1
SUppc6l' IMtthe parliclels protected frQlll theortgln
tcrprel lhe rnrJge Ras being tlie
Y Is again zero. The equallon which links dlsplxemeru. Initial
speed. accelernuOfI and time 155= !II + iar. Adapting this for
(x = O. y = O). We Gin ln
horIZOflIal dlstance:( trn\'elied al the time 1 when the \'lI.1ue d
the venlc:1I componen: 0( the fO()(\on. we have
o = (u s[n 9)t -
�rl
Speed, displacement, velocity and acceleration
value d 1 with the hof1zorual componenl of\'t'Ioclly U CO!i 8to
flnd the dISlllnce xlrn\·elled (the rnnge ll). Thls ls
X= R = (u 006 8)1=
(21/1 sin 8006 8J1s
There Is a tr1gonOfnetnc relationship sin 28= 2 sIn 900tS 8, use
of which puts the range expression In the requIred form
R = (rrSin "lB)/J!,
have liS IlU.xlmum v';Ilue for a given
The 1"'"0 soIUllOnS 0( thls eqwUon are 1 = 0 and 1 = au sin 6)A/'.
We Gin !iet" that R will
Is wilen It retums 10 lhe ground at y = O. We use thls secood
or 6= 45". The vlllue cJ IllIs IlU.XlmUffi r:1I'lgt' Is R..u = rjlh/
Ttte I = O Glsels"''llen lhe par1lclewas�ed; the second
speed d profe'CliOO u wllen sin 28= 1. lh31 IS wilen 29= 90",
Examples
1
A stone is thrown from!hetopofa vertica l cliff. 45m hlgh above le..-e l groond, wit h a n
initiaI Yelocily of 15m s-' i n a hOlizOlltal direction {F� ure 3.21 ). How klngdoos it take to
reach the ground? How far lrom the base 01 the cliff is it Vllhen it reaclIes the groond?
To find the time 1 for wI1kh the stOlle is in the air. WOI"k with the l'ertkal compo�t
of the motion. forwtlkh we mow that the initial componeflt of velocity is zero. the
dtspjac"meflty= 4Sm. and the acceillfatiOll a is 9.81 m S-I. The eqlliltiOll Iif1�if19 theS!:!
isy= �tl. Substituting the valuei. we haW4S .. J " 9.8112. This giws
I .. ,J(2 )( 45/9.81) = �.05
FIgurel.l1
Forthe second par1 of tlle que5tioo. we need to find the hoI'izontal distancextraYeliro
inthetimel. BecauS!:!the horizontai componentofthemotion is not acceler�ting, xis
gillen simply byx= U I. Substituting the y�lues, we h�ye.l'. 1 5 " �.O. 4Sm
2
2.0)( I01 m s-1
Flgure l.l2
..
An elecuon, \ravening with a velocity of 2.0 " 10' m S-1 In a horizont�1 direcllon, "nte�
a uniform electrk field. Thisfield giW'S theelecuon � constantacceillfation of
5.0 � 10,s m s-1 1n a direction p"rpendicular to Its original YeIoc�y(F"gurt' 3.22). Tht'
field ex\"nds for a horirontal distance of6Omm. What is the magnitude and direction
of the wlocity of the electron when it Il!aWs the f1eidl
� horirontal motion ofthe elecuon is not a(("lerated. ThetimE! lspo;ontbyth" eloctron
inthe field is gillen byt= xJlI.... 6Q" 1 0-312.0 " 101_3.0 " l0""9s. When theeleclron
I!nt«'i the fi"ld, its wrtiGIl wmponenI of W!Iocity is zero; in timE! f, it has been acceillf"
atoo to ",. = ar :o 5.0 " 10' s " 3.0" 1 0 -9 .. 1.5 " 101ms-'. When theelectn::>n Ieav{'S
the field. it ha s a horizontal compooo;ontof wlocity tl._2.0" 10'm s-', unchang ro
from the initial value II... Theverti<.ll component is 11, . 1 .5 " 107m S-I. The I"t'su�nt
J
ll is giv"n by
J(II, / + 11/) = ,fi{2.0 )( 101)2 +(1.S )( 101)2) .. 2.S ,, '0'm s-1
w ocity
11=
The direction of this r"sultan! velocity maKes a n angle 810 the horizontal. where (Ii!
given by tan (1= lIyl1l" = 1.5 " 10112.0" 10'.The angle 8is �1·.
Flgure l.2l
Now it's your turn
16 A ball is thrown horizontaltyfrom the top of a to'M!r 30m high and lands 1 5 m from its
di'
rst
baS{! (Figure 3.23). What is the ball's i nitial spee
17 A football is kicked on lewl ground at a -.elooty of 1 5 m s-' at an ang1e of 30' to the
orirontal (Figure 3.24). How far away is the fi
bourKe7
h
18 A c�r accelefates from 5.0m s-, to 20m s-1 in 6.0s. AsstJiTVng uniform acceleration,
is
how far does it trawl in th time?
19 If a raindrop wert' to f�n from a height of 1
gl"Olln d if therewere noairre!iistancel
Flgure l.24
kin. with what vekdy would it hit the
W!hides IllYOived In accideflts by th" I"nglh of
th" marks made by skidding IyrI!5 on the road surface. It is known that the maximum
20 Traflk police can estimate the speed of
o
deceleration that a car can attain when braKing on a ormal road surfac" is about
9m s-1. ln one acddent, tile tyre-marbWf!rt' found to be 125m long. Estimate the
spo;oro of the whicit' before braking.
51
B Kinematics
21 On a theme pari:: ride, a cage is traYeWing ulM'ards at constant speeo<!, As it pass.es a
platform alongside, a passengerdrop!i coinA throughthe cage floor,At exactlyt�
same lime, a person standing on the platform drops coin 8 from t� platform.
Ia) 'Nhichcoin, AorB (ifeither), reac:hesthe ground first?
(b) 'Nhich (if eithN) has the grl!ilt� 5peed on Impact?
U W�liam Tel was faced with the agonising task
of shooting an apple placed on his son
Jenvny'"s�ad
AssulTK'thatWdliam i s placi'd 2Sm from Jerrvny; hiscrossbowfires � bohwith an initial
spero of 4Sms-1• TM aos'ibawand �pple areon thesarne horizontal �ne. At what
angle tothehorizont.JI!ihouIdW�tiam aimso thatthe bohhitstheapple?
2] TM position ofa sports car on a straight test track is monitored by taking a series of
photograpils at fixed time intervais. The fonowing reoord of position xwas obtainoo at
theruti'd tiITK'Sf.
0
0, 5
1.0
\ ,5
2.0
2,5
3.0
3, 5
4. 0
4.5
50
0
0.4
1. 8
4,2
7.7
12.4
1 8.3
25 .5
33,9
43.5
54.3
On graph paper, draw a graph ofxagaif1stl. Useyourgrapt1 toobt.Jin valuesfort�
�ocity /! ofthecarata f1umberofvalues ofl. Draw a s.e<ond grapt1 ofll against r. From
this graph, what um you deduce about the acceleration of lhe car?
• Speed is a scalar qUiffitity and is described by magrlitude only. Velocity is a vector
qUoilntity and r�uires magnitude and direction.
• DispiacOOleOt isthe distanc:e traYelled in a straight line I n a specified dire<tion and is
a vector quantity.
• Average speed is defined by: (distance travelk!d}/(tirnetaken)
• Average velocity is deflfled by: (di�acernent)J{time taken) or 6x/�
• The instantaneousvelodty istM iM!fagevelocitY /l'll!ilsuroooveran inflflitesimaly
shon time interval.
• Average aa:eleration is defined by: (change in wIocity)/(time taken) or fll{1lt
• Aa::eI�ation i s a vector. lnstantaneous acceleration isthe average accelerallon
tnedSUred CM'r an infinitesim.any shori time interval.
• TM gradient of a displacement-time graph gi...es
the wIodty.
• TM gradimtof a Yl'locity-time graph giYl'S the acceleration.
• TM iJ(1!iI between a velooty-time graph and the time axis gives the displaa>ment.
• The equations for a body moving in a straight line with uniform a.cceI.eration iJ(e:
,, = u + at
�rl
j-ar1
;
s = lIf +
s = z·r_
I� = u1 + 2as
{u lIlr
s=
• Objects falling freely f">I!iIrthe surface of the Earth in the absence of air resistance,
experiena> the same acceleratioo, the acceieration of free fali,g, which hast� value
g = 9.81m s-1
• The motion of projectiles is anaiysed in terms of two independer1t motioflSat right
angles. The horizontal compofli'flt of the motion is at a constant velocity, while the
vertical moHon is rubj«t to a constant acceleratioog.
"
Examination style questions
Examination style questions
1
TtlI' bal is ttvown with a horizontal speed of 8.2m S-L.
Ttli' sid\' of the building is vertical. AI point P on the path
of the bal, the bal is distance x from the buildng and is
In ill driving manual, it is suggested that. when driving at
13m 5-1 (about 45km per hour), a driver 5hoUd always
keep ill minimum of two car lengths between the driver's
car and the one in front.
rnoviog at an angle of 60° to the horizontal. Air resistancl'
is negligibll'.
a Suggest a scitntific justiflCa1ion fOf' this safety tip,
2
making reasonable assumptions about the magnitudes
of any quantities � need.
b How would you e�pe<t the length of this 'exdusion
zone' to depend on speed for �ds higher than
a For the ball at point P,
i show that thevertical component of its veIodty is
U rn s-11
A student, standing 01'1 the platform at ill railway station,
notic:esthat the first two carriages of an arriving train
pass her in 2.0s, and the next two in 2.4s. The train is
deceleratingunifoffi1o/. Each carriage is 20m long. When
has fallen,
ii detl'rmine the horizontal dist.Y\Ce x.
b The path of the ban in a, with an initial horizontal
speed of 8.2m S-I, is shown again in Fig. 3.26.
�2m�,
W
ii determinethevertical distance through whichthe bal
f2]
f2]
the Irain stops, the student is opposite the last carriagl!.
How many carriagi!sare there in Ihetrain7
3
A ball is to be kicked so that, at the highest point of its
path, it just clears a horizontal cross-baron a pair of goal­
�ts.The 9round js levei andthe cross-baris2.5m high
The ball is kicked from ground level with an initial speed of
8.0m s-l.
• Calculate tne angleof projection oftheballand the
distance of the point wnere the ball was kicked frorn
thegoal·!ine
b A1so c.llculate the horizontal velocity of the ball as it
4
paS5es CJverthe aoss-bar.
( Forhowlong istheball inthe air bef� it reaches the
ground on the far side of the aoss-bar7
An athlete cornpetiog in the ion9 iump lecwes the ground
at an angle of 28° and rnakes a jump of 7.40m.
Flg. l.26
On a COJ7fof Fig. 3.26, sketch the rlI'W path of the ball for
the ball having an initial horizontal S9\'ed
i
which theatNete took off.
b tf the athlete had been able to increase this speed by
S%,ooat percentagedifference would thishave rnade
to the length of thejump7
A hunter, armed with a bow and arrow, takl's direct aim
ata monkey hangiog from the branch of a trl'\'. At thl'
• Calculate the speed at
5
inslant that the humer releases the arrow, the monkl'Y
takl'S avoidiog action by releasing its hold on the branch
8y 5elting up the relevant equations for the motion of
thl' monkey and the motion of the arrow, show lnal thl'
ii
gL"l'aterthanB.2ms" aod with negligibleair
resistance (label this path G)
I'qUaltoB.2ms·, but ....,;th air resistance(labeIthis
path A)
f2]
f2]
Cambridge Internation;J! AS and A Level Physks.
9702121 OctlNov 2010 Q 2
7
A 51udl'nt has been asked to determine the linear
aa:l'll'ration of a loy car as it mOYeS down a slope. HI' 5ets
up thl'apparatus as snown in Fig. 3.27.
monkey was mistaken in its strategy
6
A ball is thrown horizontally from thl' top of a building, as
shown in Fig. 3.25
6.2 m s-L
Flg.l.27
The time I to move from rest through a distan-ce it is found
for different values of a. A graph of d (y-axis) is pIottl'd
agail"6t rl (x·axis) as shown in Fig. 3.28.
flg.l.2S
53
B Kinematics
a Determine
i
i
the5peed uoflhe car before the brakes
are applied,
thetimeinteNal between th e hazard .appearing
and the brakes being applied.
{2}
{2}
b The legal �d �mit on the road is 60 km per hour.
Use both
of your answers in II to convnent on the
standard of the driving of the car.
OJ
Cambridge InternationalAS and A level �s,
9702102 oalNqv 2008 Q }.
9
a Define
i llekxlly,
ii acwkJml/oll.
b A car of mass lS00 kg travels along a straight
horizontal road.
Thevariationwith time l of the displacemenl xof the
car is shown in Fig. 3.30
Flg.l.28
a Theory suggests that the graph isa straight line
through theongin.
Name the feature on Fig. 3.28 that indicates the
presence 01
i raMomerror,
ii systematic errOf.
b
{7j
{7j
{2}
i Determine the gradient of the lne of the
graph in Fig. 3.28.
{2J
ii Useyour answer to i to cakulatelhe ac�ationof
thetoydoNnthe slope. E�yourworti1g.
{3]
Cambridge InternationalAS andA /evp/ f'frysics,
9702102 Mayllu� 2004 Q 3
8
A car is traW!lling along a straight road at speed v. A
hazard suddenly appears in front of the tar. In the time
interval between the hazard appearing and the brakes
on the car coming into operation, the car moves forward
a distance of 29.3 m. With the brakes app�ed, the front
wheels of the car leaW! skid marks on the road that are
12.8m Jong, as iliustrated in Fig. 3.29.
""·" II:,
I
FIg.l.10
_
n�=----+-'
�
==-I
..
FIg.l.29
r=::l
It is estimated that, during thesk.id,lhe magnitude of
the deceleration of the car is 0.85& where[J is the
acceleration of free fall.
54
i Use Fig. 3.30 to describe qualilaliveiy the W!iocity
of Ihecar dunng the first six seconds of the motion
shown. GiW! reasons for your answers.
OJ
ii Calculate the average velocily during the time interval
I = O t o l = 1.5s.
11}
iii Showthat the averageaccelerationbelween r = 1.55
and / = 4 . 0 s is-7.2 m r.
{2}
iv Calculate the average force acting on the car
betwee n l = 1.5 s and / = 4.0s.
{2}
Cambridge International AS and A level Physics,
9702123 OctlNqv 20/3 Q 3
AS Level
4 Dynamics
�
By the end of this topic, you will be able to:
(a) understand lhat mass Is the property ofa body
that res[s!S chanae In mot\on
(b) recall the relatIonship F= rna, and solve problems
using U, apprecl3t1ng that acreteratJon and force
are always In Ule same direction
(e) define and use linear momentum as the product
ofmassand \'clocUy
(d) define and use forre as the rate of change of
momentum
(e) stale and app!y each of NeWlolI's laws otmOlion
(a) describe and use the concept of weight as the
effect of II aravUatlonal field on II mass and recall
that the wel�:lU 01 a body Is equal to the product
of Its mass and the 1IOcreler:ltion of free fall
4.3
(b) deKJibe qualitatively the mollon of bodies f2lling
lna unifonn gr.lI'ltatlonaJ field wlth alr reslstance
<a) state the prinCIple o(collScrv;IIJon of momemum
(b) apply the prinCIple of OOfIserllatlon of momenrum
10 solve sImple problems Including elastic and
lnelastlC n
l teractlons between bodies In both one
and two dlmenslollS
(c) recognise Ihal, for � perfectly elastic collision, the
relative speed of approach Is equal to the relalil'e
speed 0( separation
(d) understand thai. while momentum of a system Is
always consen'ed In Interactions between bodies.
some change n
I kinetic enerll'Y may take place
Starting points
• Motion of an object can be described in terms a/ displacement. velocity and
acceleration
• A force is required to make a
4.1
body accelerate.
Relationships involving force and mass
When }UU push a trolley In a supennarkel: a-pull a case behlnd }uu at an alrpc:l1. you are
exl"rtlng a toru.-. When you h:!mmer In a nail, a force Is bclng excrted. When you drop
a book and � falls to tlll" fbor. thl' 1xxJk Is illJlng l:.oecalllie oflhefon:eof gr.lvly. \l'hen
you lean agaln.lt a wall or sit on a malr. you are exerting a force. I'orces can change tlll"
shape or dimensions d bodies. You can crush a drinks can I)}' sqUl'I'".£lng k and applying
a force, you can strefch a rubbertxmd by pulling II. In everyday life. we ha\'e a good
under.;tandlng otwhat lsllX"am by force aoo the sltuatlorui lnwhk:h forces al'(" lnvolved.
In phy.5lcs thl' klea ot toru.- Is used to add detail to lhe descriptlorui cI IllC'I'lng objects.
AS with aU physical quantities, a method 0( measuring force must be established.
One way 0( doing this Is to make use of the bct thai forces on change the dimensions
clbodles In a reproducible way. 11 takes the same force to stretch a spring by the same
change In length (provided the .'ipflng Is llO( ()\'('fSlrC'lche<i I)}'applylng a very large
force). This principiI.' Is used In the spring balance. A scale sOOws how much the spring
has been extended. and the scale can be oUbr.l1Cd In ienns 0( force. Lahor.ltory .'ipflng
balances are often oiled newton balances. because the I'\ewton Is the St unit 0( face.
Forres are l'ectOr quantilles: ther have magnlU(\e as \\'ell as direction. A number 0(
forces acting on a body are often .shown by melllS of a force diagram drawn to scale,
In which the forces are reprel£'flled by lines 0( lenglh proportional to the magnitude of
the force, and In tlll" appropriate direction (see TOpIc I). The combined e/fed of se\'('flIl
f(r('('S acting on a body Is kno"'nas the resullanl forre.
55
II Dynamics
Force and motion
The Greek philosopher Arlsl:ak> believed lhal lhe naltlr.ll state d a I:NxIy W:lS 3. Sl3Ie
of reS!. and lhal a fooce was necessary 10 mlke II rtlO\·e and 10 keep II mov1ng. This
argument requl� thaI the gA.'"\IIE'r !he rooct'. lhe gre;aterlhe speed oflhe body.
Nearly I\\U lhousand )"l'31"S laler. GaUIeo queslloood Ihls Idea. He suggested that
Figure 4.1
motion at a ronstaB speed could be JUS! as natural 3 stale 3S the Slate d res!. He
Introduced an under:onoolng 01" the l'Ifect 01" friction on
motion.
Imagine a heavy bm. being pwhed along a rough floor al COIlStanl speed
(PIgure 4.1). l1lls may uke a consider.lble rorre. The forre required C2n be reduced
If the floor Is made smoolh and polished, arKI reduced even more If a lubrlc:anl, roc
exantpie grease,ls appiledbetween thl'baX and the f1oor. We can lmaglne a slruatlon
where, whl'n frlclion Is reduced t03 \'anl'lhlngly small Vlllue. the forre required 10 push
lhe box al COIlSlaB speed Is also vanishingly srnall.
Galileo reallsed that the foru> or frlClIon W3.S a force thaI oPf.'OS'.'d lhe pushing force
When the box Is moving at constant speed. tho:.> pushing force Is exactly equal to the
frlCllonal force, bUl In the opposite dlre.::tlon, soII131 Iho:.>re ls a net forre of zero acllng
on the txDi:. In the skll3tlon ofvanishingly small frlcllon, lllC box will oontlnue to move
wlth constant speed. because there Is no force 10 slow It down.
N ewton's laws of motion
Isaac Newton (1642-1727) used Gallleo·s Ideas 10 produce a theory of motion,
expressed In his three laws of mOl:ion. The I1r5t law of mOlion rt'-SU.les GaUIeo·s
Flgur. 4.2 1� Newton
deduction about the natural state ora body.
EWfY body CDf1tinul!'> in its 'itate of rl!'>t, or with uniform velocity. unless acted on � a
resullanlion:e.
Thl'l law tells us what a force does: II dlslurm lhe stale of res! or \·eIocll)· 01" a body.
"I1Ie properry ol" a body toS!ay In a �me ofrelil or uniform velocity IS C211ed inerlia.
Newtorl"S second law tells us whal happens If a force Is exerted on a body. 1\ causes
the \·eIoclty to change. A force exerre d on a body at res! makes II ffiO\'l' _ II gI\'l'S II a
Velocity. A force exened on a mov1ng body may make lIS speed Increase or decrease,
or change lis dlrectJon o( moIlon. A change In speed
or \"CIocly Is acce!er:ltlon.
Newton's .second law relates the magnltude athts occe\er.l tton 10 lhe fom' applied. II
also IBroduces ttle Idea oI"the mass 0( a body. Mass Is a measure o(the lne-rtla oI"a
body to change In ,·eIocity. The blggerlhe IIUSS. ttle more d(fficult II Is to change its
state of rest or velocly. A slmplltled form a Nev.·IOIl"S second law Is
For a body of ronstant mass, ilsaul'li!ratiOfl is directiy proportional to the resoltant ion:e
applied to iI.
The dlrecllon of the accelemllon Isln the dlrecllon 01" the reSlt!tant force. In a l'o'Ol"d
equallon the relation between fort:e and aC<X'k.>mtlon Is
fort:� = ma 55 X iK(ele(ation
and In symbols
where F Is the resultani fDlU'. In Is the IruISS and a Is the acceler.ltlon. Here we han'
made the conSiani of proportionality equal 10 unIty (thaI Is. \\'l' use an equals sign
I1Ithel" than a proportlon3ltry sigr1) by choosing quantllies with units which wtJl gln' us
this simple relation. In 51 unlts, the force F Is In newtons. the mass In In Idlogr:1ms and
Itle acceleratlon a In metres (secondsyl.
OnefK'WIon is defined astl\@lon:ewhic:hwil give a lkg rnilSS an accell!fillion ofl m � in
tile direction of theforc:e.
56
4.1
Relationships involving force and mass
When you push a .\t.Ipermarket trolley. the trolley experiences a force (Figure 43). The
trolley applies an equal and opposlle fOfU" on ano:her body - )'OU. Newton uflClergood
that the body on which lhe fOfU' Is exerted applIeS aOO(hcr force oock 0"1 the body
which Is applying the force. when body A applIeS a force on body B then body B
applies an equal and opposIe
I fOlU' Ofl body A. Newton"S third law relates these two
fa=
Figure 4.]
WhMF"o'{'I"OI"l@bodyexmsa foKeon another, t he second body er.ertsan equaiand
owoo;i1l' fo� on ttH' fm.
Very cti.en this law Is stated as:
10 every action, there Is an
equal and oppOllile reacHon.
portant pol'" that the action force
and the reaction force act on df./Ji!n!l/l objeCts. lb lake Ihe ex:lmple of lhe supennarket
BUI this statement does nOl highlight the \-ery Im
trolley, the action force exerted by you on the troll(')' Is equal and opposite to the
reaction forre exened by the trolley on you.
Newton's third law has applications In l'\'ery brlnch cI everyday life. We w:Jlk
because clthls law. Wh('fl you take a step fOfW..rd. your foot presses against the gR;lUnd.
TIle ground then exens an equal and opposite force on )'OU. This Is lile force, 0"1 you,
which propels you In your path. Sj:xlce rock{'(S won.: because clthe law Cl'tgure 4.4).
lb expel the exhaust gases from the rockel:. the rocket exerts a force on the gases. The
gases exen an equal and opposite force on the rocket. accelemUng U forward.
1
Flgur. 4.4Sp�e rocket �ndl
An object of mass l.S Kg is to oo iJCcek;!rated at 2.2 m S-2. What force is reql.lired1
Fmm Newton'sserond law, F.. rna _ I.5" 2.2 _l.lN
2 A carof mass l.5 tonnes(1.5" 1()lkg), !rilllelling at80km h-1, istobe SlOppedin lIs.
What fon:e is requirl!dl
Theaa::eleration ofthecar tim be obtainedfrom ,,_II+ (ll(seeTopi{l).The initial
speed ll is 80km h-l, or 22m s-1. The final speed vls O. Then (l __22111 _ _2.0m -r1.
This is negatiVl! because ttH' Cilr is decelerating.
By Newton's '>{'{ond law, F� rna .. 1.5 " 1 ()l " 2.0. 1.0 " 101N.
Now it's your turn
1 A fo� of5.0N isawlied toa body of mass3.0Kg. What is the a{(elelation of the
body)
2 A 51.one of mass 50g is a{(eteratl!d from a catapult toa speed of 8.0m ,1 from m5I.
t:Nel a d istdlKe of30cm. 'NhataveragefOl"ce is applied byttH'rubberofthe catapultl
Momentum
We shall now Introduce a quantity called momentum. and see how NcwtOl1's laws are
reL1ted to It.
The rnorT1!!ntum 01 a partkle is definl!d as the product of its mass and its velodly.
II'IC1I'Il('{Itu m= IrIdSS " veIcxity
andln symbois
11leunlt ofmonle("(um islhe unlt of mass times 100 unl of\"eb:!ly; that Is.
An alemath-e unll is the nev.1on M'COfId
kg m s-'.
(N.$ Momentum. lUre "eIodIy, IS a Il'CIor
quantlly. lis cornpIeI:e naDll' Is linear momentum, to distinguish II frem :mgular
momentum, which does not COflU'fn us here.
57
II Dynamics
Newton's first law Slates thaI ewry body
CQIlUnucs In a stale of rest, or with
uniform veloclly, unless acted on by � resultant force, \1{'e Cln express this law In
terms d momentum, If a body maln:�lns lIS uniform \'eloclty, lIS momentum IS
unchanged. If a body relI\3lns at rest. :.Ig3ln IS momentum (zero) does not dlange.
Thus, an altemaUn' statement d the nrSl law IS that the nlOI1lCT1tum of a partlell'
relI\3lns conSiant uruess an external resultant force act S Orl the partlde, Asan
equation
p .. COflSlant
This IS a spectalGlse, for a slngle partk.1e, da \'cry Important COI1S('f\";Itlon law: IhI'
prtnctple of consen"3Uon of momentum. ThIS \\'O!'d 'conservation' here ITems trut the
quanUty rernalnS COllSl3Il1.
Newton's second law Is
expressed In terms of momemum, \l'e already haw 11
In a form which relates the force acting on a body to the product of the mass and
the aa:eler:J.tlOll of the body. Remember that thc acceleration of 3 body IS the rate
of change of Its velocity. The product ofmJss and aa:elcratlOll then Is lu.'it the mass
limes the rate of clung<' of wlodly. I'Or � body a constant mass. this IS lu.'it the same
as the rate of change
of (ma.lS x velocity). But (mass x velocity) Is momentum, so the
of 11lOIncntum. Thus,
product of ma.lS and acceleration Is Idffitlcal to the r:tte of change
Newton's second law IS Slated as
The reSIJltant force acting on a
body i� proportional to the rate of change of its mOffil/nlum
The conSiant of proportlonallcy Is made equal to unity as described on page
the second lawust'd n
I problem.'lOlvlng ls
The rewltanl force
56. Hence
actWg on" body j� equal to the rate of change of momentlm.
Expressed In terms d symbols
for conSlant massm
F_ I1(mI')lIlJ = m (tu'lAJ) = ma
NOll' that Frepresents the resultant force acting on tile body,
ContinuIng with the Idea of force bellIg equal to r:tte of change a momentum,
the third law relallng to acUon and reaction forces becomes: the rate of change of
ITlOIllf'ntum due to the actiofl force on one body Is equal and oppOOte to the rate
ofchange of momentum due to the reaction forcc on tIle other body. The action
and reaction fon:es act on each body for the same lime (At). Hence Ffll is equal and
opposke for each body. TheJefore Whffi lWO bodies c:<ert acllon and reaction forces on
each OIher, their changes d momentum are equal and opposite.
4.2 Weight
We $JW In Topic 3 that all oblectS released n e ar lite surface of the Ilarth fall
wllh the same acceierallon (the acceleration of frec fall) Ifalr resI5tance Gin be
neQlected. The force causlng thl$ JccelerntlOrl IS the grnvltatlonal attmcUon ofthe
Earth on the object, or the force ot gmvlty. The force ot gravity Which a(ts on an
object IS called the weight of the object. we can apply Newton's second law to tile
weigh!. For a body of ma5S m failing "1th the acceleration ot free fall JJ, the weight
Wls glven by
The SI unit of force Is the nE'''1on
(N). ThIS IS �Iso tile Unit of wetg:hl. l1w;-' Weight of
an � Is obtained by mu.lp/ylng IS mass In kilograms by lhe ;JCCeIerat\ofl a free
58
4.2 Weight
f:J1l. 9.8lm s-'. Thus a mass done kilogram haoS a ..
v
elghl d9.8IN. Bec:luse welgtf Is
a fom." and fom." I s a ,-ector, we ougtl lO be a\\':lre dlhe dlroctlon of the weight dan
obtect. U Is towards the centre d the Eanh. BeCauSe weighi al\\';l)'s has this direction.
we do na: need to specify dlre::tlon t'\l'ry ume we g/'.'e the magnitude d the weight d
,'*,,,
How do we me3.'ll1re mass and welgtl? If)'OU hang an obtect fro," a newtoo
balanet', )'OU are measuring lis weight (Figure 4S). The unkllO\\'fI 9,'elghl d the
obtect Is balanced bya force provided by lhe sprlng ln lhe batanet'. From a previous
callbratlon,lhls force Is related 10 the ex((,'nSlon of the spring. There Is thepo5lilbtJ.y
of confuSIon here. Labor:J{(xy newton balances maY, Inck>ed, be Clllbr:lIed In newtons.
But all rommerdal spring bal:mces - for example, the IXllances Jt fro. and vegetable
Figure 4.5 A newton bal.1nce
ooumers In supermarkets - are calibrated III k
i lograms, SUCh baL:lflce5 are really
measuring the weight dthe fruit and n�get:lblcs, but the SC':lle reading Is In nuss Ulllts,
because there Is no distinction benn�en mass and weight In everyday life. 11le averaRl"
shopper thinks 0{ Skg of mangoes as having a weight of 5kg. In fact. Ihe mass of Skg
has a weight of 49N.
The\\urd 'balaoce' In the spnng balance md In IIIC laboratory lop_pan balance
rebtes 10 the balance d folt:es. [n each Clse. the unknown force (the \'.elghO Is
equalled by a folt:e which Is known through calltlmtlon.
A way dcomparlng masses Is to use a be:lm balance, or lever balance
(,<;ee
FIgure 2.17) Here the weight ofthe 00100 IS balanced against the welghl of some
masses, which have prevlou5ly been calibrated In mass unlls. TlIC word 'ballnce' here
refers 10 Ihe equilibrium dthe beam: when tile be:lm Is horizontal. tlIC rnClIl1em of the
welghl on one side of tile pivot Is equal and oppo6lte 10 tile mornent on Ihe other side
ofllle pIVOl. Because welgtl Is glven by the product of mlSS and tlIC accelernlion of
free fall, the equal.y of me weights means lhal lhe masses Jre also equal
We ha�l' lntroduced the ldea d weightby lhlnklng allOllt a n obtect i n fTeerali.
But
objects at rest also ha,l' ..e,i
ght : lhe gravlwlonaJ auractlon on a IXIOk Is the
same whether. Is falling or whether It IS resting on a table. The fact thaI the book
Isal rest tellsu$. by Ne\\1oll's flrst law, Ihal lhe re:sult:lnt force acrlng on It IS zero. So
there must be allOlher force acting on the boo!< which exactly balances Its weight.
[n Figure 4.6 lhe table eltl'rts an Up"':lrds force 00 the
book. This fon:e Is equal In
magn.ude 10 lhe weight but oppo/lIte In direction. n Is a normal oontact force:
'CORacr' because • occurs due to the COlllaCl between
book and table, and 'normal'
bec'ause It acts perpen<!lcularly to the plane ofcontacr
Figure 4.6 A book res�ng on a table forces
on!he book. (Theforc�s�tlnlhe�me
Yl'rtiCOlI line, but are separatecl slightly for
darity,)
The book remains at rest 011 the table because the "'elglll Wof the l)(XIk downwards
Is exactly balanced by the normal COlllact force R exencd by the table on tlIC book.
The VI'Ctor sum dthese forces Is zero, so the book IS In equilibrium. A very common
ml.51ake Is to stale Ihat 'By Newton's third law. Wis
equal to R. But these IWO forces
are both acting on the book. and Glnna: be related by the thIrd law. Third_law forces
always act on d
(ffim/
m bodies.
To see the application d the third law. Ihlnk about the normal contact fon:e R. This
Is an upwards fom." CXfi1ed by the /able. The reaction to this Is J downwards fon:e Il'
t'Xf"ned by the book. By Newlons Ihlrd law. lhese forces are eqwl and opposile. This
s.uatlon tS lllustrated In FIgure 4,7.
59
II Dynamics
Figure 4.7 A book reSbng on .. tilb1e ilCuon
:t�dlOll fon:es. Xlang ..t the POint 01
Havlngconsldered thl' action and readlon forces between I)()(jj,; alld table. we ought
book. regarded as an action
thl' book Is 001. on the uble. Thls ls 001. so easy. because there doe:s
to think about the reaction foro.> to the weigB ci the
fom.', e\'et\ when
not seem to be an om1ous s«ood force. But remember that thl' w'e1ght is due to til!"
gravlUtlonal anradiofJ of the E.:Inh on theboolt. lftllC f!:Irt'l �nrads thebook, thl'
book also anracts the E.:Inh. This gl1l\itatlOOa' force ci the book on the lollfth is the
reaction force. We can test whether lhe 1",0 forces do. Indeed. ad on different bodies.
boc>k) aclS 00 Ille booIt. The reacUon force (the
book) acts on the Eaftll. Thus, the condition that acllon
Is s(ulsfled.
The actlon fofU' (the welghl of the
attractlon of the E.:Irth tOlhe
and readlon forces should ad on different bodies
Non-uniform motion
We have ntentloned that. In ntOiSl situations. air resistance can be neglected. In fact,
there are some applications In which this reslSlance
t
Is mOSl lmportant. One IiUCh case
Is the faH cia parachutlst. where air It'SIsIaoce plays a vital port. The velocly of a txxIy
failing through a resistive nuld (a liquid or a gail cloes OOl. lncrea5C Illdefinkely, but
t>\'ffiI:ually reJches a maximum velocly. called t''Ie terminal velocity. The fCll:e due to
air R'SIstaoce locR"ases "'1th speed. \l-'hen thiS reststlve force has reoched a Il:llue equal
and opposlle to the wetght of the fulling body. the body no longer accelerates and
continues at uniform \'eIodly. lllIs is a case c( mOlioo wlh noo-unllbrm acreIl'r:ltlon.
The aa:eler.Jtkln Slarts ctf wlh a v.llue cig. but cIecrease.s to zero at the time when the
terminal \'elocity ts achlen'd. Thus, raindrops and parachutists �re normally tral't'lling
Ftgure 4.8 A �riIChullst ..bout to liKld
at a a:nSl3Jlt speed by the time !hey approoch the ground (l'Igure 4.8).
Problem solving
In dealing wlh problems involving Newton'S laws, 513ft by drav.1ng a general sketch
cl
thl' situation. Then COI15kJer exh body In your skelch. Shaw all the fOlreS acting 00
that body, both known fOlreS and unknown fOlreS )OO may be trying to find. Here it
is a real help to try to draw the arrowswhich represent the fOlreS In appro:dmately thl'
mrrect direction and approximately to scale. Illbel each fOrce wlh Its nugnitude or
with a symbol If you do net know the magnlude. f'or each force, )00 must know the
cause of the fofU'(gravKy, frk:tloo. andso orV. and)oo must also knO'W Qfl what ootect
that force acts and by what cbjeoct It Is
CX('fted. ThlS ]aI)l'.lied diagram Is referred to as
a free-OOdy diagram. because It detaches the bodr from the other:; In the sitlJ.1tion
Having established all the forces actlng on tilt> body. you can use Newton·s .second law
loftnd unknawn quantltles. This procedurt> Is IlIuSirated ln the example which fol!aws
on page 61.
Newton's second law equates the resultant force acting on a body \0 the produa of
Its mass and Its acceleration. In some problems, the �)'Slem cl bodies Is In equlllbrium.
They are al rest, or are moving In a straight line With unlfoon speed. tn this case, the
acceler:r.tlonls zero. .'lOthe resultanl forCf.' ISalso zero. ln Olhercases. tl'le resuhnt
force Is 00l ZCfO and the obtects In the system are accelenUng.
WhIchever case applies, you shoukl remember that
forces are vectors. You will
probably have to resol"e the force:s IntO twO components at rlgltt angles. and then
apply the second law to each 5et of components .separ;;uely. Problems can citen be
slmpllfled by making a good choice of dtrealons for resolution, You will end up wllh
� set c( equationS, based on the application of Newton'S second Jaw, which must be
so!\'I'.d to detennlne the unknown quanllty.
60
4.3 The principle of conservation of momentum
1
A box of mass 5.0 kg is pulled along a horizontal floor by a force Pof 25N, applied at
an angle of 20° to tlw horizontal (figure 4.9). A frictional force Fof20N acts paraDelto
the floor.
Cabliate ltM!aa:eleration oftlwbox.
ThI! fr-.body diagram is� in Rgure4.9. Resolving the forces parallel to ItM! floor,
the componrotofltM! pullingfoKe, acting lD the left, ls2S cos 20_ 23.SN
Thl!frictionalfoKe, acting totheright, is20N.
ThI! rPSUltant force to the left is thus 23.5 - 20.0 .. 3.5N.
From Newton\ second law, a .. Ffm .. 3.515.0 .. 0.10m s-l.
What is the magnitude ofthe momentum of an a·pat1lde of mass 6.6 " lQ-l1kg
travemng with a 5peed of 2.0 " 10lm s-l]
p = IIIV = 6.6 " 10-21 " 2.0 ,, HY .. 1.)" 10-,tkg m 5-1
Flgur.4.9
Now it's your turn
Figure 4.10
)
..
A pe"'IOIl gardening pushes a liM'IllTlO'N(!rof mass 18kg at constant speed. To do this
rl!qLires a fOKe Pof 80N directed along the handle, which isat an angleoi 40" ID the
horimntal (Rgure 4.10�
(a) Calcuiate tlwhorizontal retardingfOt"ceFonthemower
(b) tfthis retardingforce were wnstant, wh.ltiorce, applied alongltM! handk>,wouId
aca-lerate the mower from rest to 1.2m s-' in 2.051
What is the magnitude of the momentum ofan electron of mass 9.1 " ,0-l' kg
travel�ng with a 5peed of75 " 106m s-'1
4.3 The principle of conservation
of momentum
0-: "-0
Figure 4.1' Systemof!W{) p<lf�d�
We have already seen that N<"Wton's nr51 IJW stJtes th�t the rflOlnentum of a single
panicle Is COI\st3m, If no external force acts on the p:trtk:le. Now think about a �ystem
G tl'o"O p:tnlcles (Figure 4.10. We allow these p:tfllCles toexen some son of force on
each Olher: lt coold bt' gmvlt:lllonaJ attmcllon or,lf the p:trtk:les were chargecl. ltrouki
bt' eledAJ6latlc allmctlofl or repul.lIDri.
These two panicles are Isolated from the rest of tile universe, and experience no
outside forn's at all. If the nf51 partICle excns a foo:e FOIl tile second. Newton's third
law tells us that the seorond exerts a force -FOIl lile first. The I1\IIlUS sign Indicates that
the forces are In opposIte directions. As"� 53\\' In lhe last 5t'Ct1on, we Clll express
this Jaw In terms G change of momeTlturTL The change fA mon�um of the second
panicle as a �u. G the force e-"",ned 00 II bylhe first Is equal and opposite to the
chang!' G nxmenrum of the fif51 panicle as a reSldl dthe force exened en It by the
!;e("Ofld. Thus, the changes of moment:um fA lhe IndIVidual partll:k's Clncel out. and tile
momentum of the system G two panicles rel113!nS COIlS!a�. The panicles have merely
exchanged some IllO!Jlemum.
61
II Dynamics
The sltllalloo Is expressed by lhe
equation
P"P,+P2 =conslilnf
P Is IhI' 100ai momentum, and p, and h are the loolvXIuaJ momenlll.
We IDJkl extend this Idea 10 a system alhree. bolt. Of finally any nuni:ler,/ d panicles.
where
II no external loree ads on iI sy.>\em, the totill momentum of till? system remains constant,
or i� con�rwd
A system
on which no external forre :KU IS otten C3lled an tso/(uoo �'Swm. lbe f:Jct that
the 100ai TllO!Il!'fNm
l
d an IlioIated system l'l oonSlanl l'l the principle of consernlion
of momentum. It is
a direct COI1.'lequence ci NewtOO's third l:Iw or motion
Collisions
o
'.
o
"
FIgure 4.12 C�hSlOnbel\Wen tWOp;lrlldes
We now use the prIoc!ple of ronservatiOn d m�um 10 analylie a s�em consisting
d t"''O coIlkling partlcles. (If you WlInt a real example to think about. try snooker balls,)
Qxu;idertvr'O particles A and B makIng a dlroo. head-on ooIl1s1on. Partk:Ie A has mass
In, and is rncwlng wIth velocRy II, In the dlrealOO from left to right; I� has mass III;. and
t
has veloctty Il, In the dlrealon from t1gh to k{, (I'Igure 4.1�. AS vcloctty is a I'/.'CIOI'
quanttty. this Is the 5amt" as saying th:tt the �tloclty IS -Ill from left to right. The panicles
collide. After the collision the)' have velocitieS -v, and iiI respectll'ely In the dlmcUon from
left to rtght. That Is. roh panicles are moving back along tltelr dlroolons d approoch
Acrordlng to the principle d corJSl'rv:l1iOn d montentUm. the 100ai momentum of
this Isolated sySlE'm remains constant. whatever happens as a result of the Inter.Jctlon
of
the particles. Thus. the total momellium before Ute ooIlisloo muSi be equal to the
total rnornenI:um after the collision. The momenlum before the coIlJslon Is
JII,u, - m,u,
-m,fI, + m,t'1
Bec2uselotal n}(){J\{'ftumls conserved
K!1()IIl.ing the ffiaS'ieS G the panicles JOO the \'eIodtles before roIUskIn. this equation
wQllId 311ol\' us IOcak:ulate the relation between the \'eIodtles afler the collision.
lbe way to appro:lch
collision problems l'l as follOl\'S
t
• Draw a labelled diagram showIng he ooIl1dlng bodies before collision. Dnw a
sepante diagram showing the sltuatlon after the colllslon. 1'ake care 10 define the
or the total mom{'fllum before Ute coUislon, remembering
that momentum Is a vector quantity. SImilarly, flnd Ute total momentum aft("r the
• ObtaIn an expressIon f
colHsion. taking the same reference dlrecUon.
colliSIon to 1I1e momentum afterwards
• Then equate the momentum before 111("
Example
A (a(1noo ol mass l.5tonnes (l.S>< 1011:g}IireSa Ca!1non-ball ol mass S.OI:g (FiglJ(e 4.13)
The �peed withwhirn the baIl IeaYesthecannon is70m S'"1 relaliw to lilI? Earth. 'IIr'ila1 is
Ih-I:> inilial speedol recoil oltoocannon?
Flgur. 4.13
62
4.3 The principle of conservation of momentum
Too SystOOl under consideration is the Cilnnon and the cannon-ball. The total momentum
of tOO system before firing is zero. 80cause the total momentum of an isolated systOOl is
({lnstant, the total momentum after firing must also be zero. That is. the momentum of
the Cilnnon-billI,which is 5.0" 70 .. 350kg m s-' to the right, rnust beexactlybalancro
bytOO rnornmtum ofthe cannon. If tOO in�ial speed of recoil is II, the momentum of the
Cilnnon is 1500vtothe k!ft. ThU5, 15ex>v. 350 and " . O.2lms·l.
Now ii 's your turn
5
An ic2-skaterof mass 8Okg, initi.lRyat rest, pushes his panner, of mass 65 kg, away
irom him so \hat stH> rTlO'o'e5 with an initial speed of l.5m s-'.What is the initial speed
of the first skater after this milnoouvre1
Momentum and impulse
Ills now useful 10 Introduce a quamlty called impulse and rebl{' It to a change In
Ifa OlOSlant force FaCls o n a b odyfora timelll, theimpulseofthe force is 9i'o'en byFtJ
The
unit of Impulse is given by the unit of (OfC(.'. th{' newton. multiplied by the unit
the second: It Is the !)('wton second (N S).
the (or«' acting on a body Is equa.l to
of tlme,
We know from Newton's second law that
the r:lte of change of momemum of the body. We have already
('xpreS'ied thIs as
the equatkln
F = l!.p/M
e obtaln
Ifwe muklply bOl:hsldesofthis equatlOn byM. ..
v
F!ll .. !lp
We have already defined FM as the Impulse of lhe force.
Th{' righI-hand
side of the
equatIOn (flP) Is the change In the monX'ntum ofthe body. SO. Nt>Wlon's !ieCond
law tells us that the impul.'H' ofa force is equal to the chanae in momentum
II Is useful for deallng with forces Ihal aCI O'o'ef a shorl lntefV;)j of lime. as In a
collision. The forces between colliding bodieS are seldom constant throughout the
collision, hut the equaUOlI can be applied to obtain lnfOflTlatlon about the 3Ver:lge
force aCllng
Note th31 the Idea of ill1(Jlllse- explalns whytll<'re lsan a.ertllt"·e unit for
ITl()II"lfrum.
fi
On page- 57 we Imroduced the kg m 5" and the N s as pos5ible units
Tlle kg m 5" is the IogIGII un•. the one you �rrl\"e allf you take
momenrum :IS being the prOOUCI of mass and ,"CIocky. The N s comes from the
im pulse--momenrum equa.tlOfl: I! Is the unl! of Impulse, and be-causc impulse is equal
to clunge of momemum. I! Is also 3 unit for momentum
for momentum.
Example
Soml!lennil playerl can S{!rvelhe bali al a lpeed of55m s" . Thetennisbali hasa maIl
of 60g. In an experiment, it is determined Ihat the b.lll is in (ontact with lhe rack"t fol
25ms during the serve (Figure 4.14). Calculate the average fOI(e exerted by Ihe racket
on Ihe ball
The change in momentum ofthe ball as a result of the \eM! is 0.060 ,, 55 .. 3.3kg m .,'. By
the impulse-momef1tum equatioo, the change in momentum is equal to troe impulse ofthe
fon::I!. Sincl!impulse i<>lhI! produCI offOfC@iIr'ldti1Tlf!, FI . 3.3Ns.
Herl! lis O.025s;lhusF: 3.3..u.02S .. 130N.
Now it's your turn
Figure 4.14
6 A golfer hits a ball of mass 45g al a speed of 40m s" (Agure 4.15). The golf dub is
in contac:t'Nilh ttH> ball for 3.0ms. Calculate the average fon::e e)(erted by the dub on
the ball.
63
II Dynamics
o
",
0"
"
Elastic and inelastic collisions
In some collisions. klne'llc energy Is ronSl'fved as well as momentum. By the
conservation of kinetic energy, we mean thm the tOlal kinetic energy of th.e
collkllng bodIes before collision Is the same as the tOlal kInetIc energy afterwards. ThIs
means that no energy Is klIiI: In tiK' permanent deformation 0( the coIlkilng bodies, or
as heat and sound. lllere Is a tramiformatlon of energy during the collision: while the
coIlkilng bodies are In romact 5O!IIe of tile kinetic energy Is transformed Into elastic
paentlal energy. but as the bodies separate, l IS transformed Into kinetic energy again.
Using the same TlOlatlon for the n13MeS and speeds of the colliding p;ln!des as In
Figunt 4.1& COllision between two p�rttdes
the section on CoUIskJIls on page 62 (see tlgure 4.16). the total lo;lf}C(ic energy ofthe
panldes before collision Is
Tbetotal ldnetic energyafterwardsts
If thl! collision is I!lastk, the metic I'TX'Tgy before cofision is equal tothe Uletic energy
aftl!rcollision
NOIe that because energy Is a scalar, the dlre<:tlOOS of lllOllon of the panlctes are nOl
Indlc:ued by the signs of the var10us terms.
equation Is useful because It gIves
velOCIties, In addition to that obtained from the
In solving p!dJlems about ela;t1C colliSIonS. IhlS
anOlher relation between masses and
prIncIple of conservatlOll of momemum.
When the veloctty dlrectlofls are as deflned In I'lgure4.16. appllcaUQIl of the two
conservation conditions shmvs that
1I, "' II, o; /.', "' V,
for a perfe«Jy elastic collision. That Is. the relatl\t> speed ofapprooch
(u, ... 1Ij) ls equal
10 the relative speed of separation (v, ... vlJ. NOte that thls ll5Cful relation applies Ol�
foraperfoctiyeiastfc colltsWn.
ElastlC colUs\ons occur In the coIllsloo.S of atans and mo'ecu!es. We shall see In
ll)pk: 10 that one 0(the most Important aslil..Impdons In the kinetic theory ofgases Is
64
4,3 The principle of conservation of momentum
that the coIUsIoru; of the gas molecules wltll the walls a tilt' container are perfectly
t'lasllc. l-k)lI"ever, in 1afl!'."T SGlIt' collisions, such as II10se a sl'lOOker balls, coIIl5lons
cmnol be perfectly elastic. (Tilt' 'click' d snooker balls on Impact Ind\c2les lhal a \'eT}"
small m.cUO(l oftlit' 100ai E.'Ilt'I"g}' oftlit' s)'stl'"fll has been tr:msfon"ntod IIlIO sound,)
Nevertht'les5. we c:lten make tllt' assumptIOn that soch a coIllskln Is perfectl)' elastic.
Col�s inv.nichtlK' total kineticeneryy isnotthesame before andaftertheeYeTlt are
called inelastic.
lOlal energy must, at course, be conser-·ed. But In an InelasCic collisIOn the klneUc
energy that does TlOI re-app!'ar In the SOIme form IS mmsformed [1110 Ileat, sound and
OI.lIt'r fonns d energy. In an extreme Gl5OE', all the klOClIc energy may be 1cr;I. A lump
of IIlOdt'lIlng clay dropped on to the floor does 001 bounce. All the kInetIc energy It
possessed lust before ll[tUng the floor lias been transformed Into the work done [n
tlauenlng the lump, and (a much smaller amoon1;) Into the sound energy emilled as
a 'squelch'.
A1thoogh kinetic energy may or may not be conserved In a ooilision, momentllm is alway;
conSol'Nl'!l, and so is total eoergy.
The tmlh of this statement may nOl be entirely ool'lous, especially when considerIng
exantples slIch as the lump of modelling clay whiCh ....'as dropped on to the floor.
Surely the clay had momenlllm /Ust before Ihe colliSlOfl wIth the floor, bllt h.ad ro:>
mornenlllm afterW1mis? True! But for tilt' �ystem of Ule IlImp of clay alone, external
forces (tilt' attraction of the Earth on the clay, and tilt' force exerted by lilt' floor on
the clay on Impact) were acting. When external forces act, the COIl5OE'IY,ltlcl1 principle
does no! apply. We need to consider � system In which no e:l:lernal forces act. SUch a
system Is lhe lump cA modeIJing day and lilt' Eanll. While lhe day f.lUsI0W3rns the
floor, gravltatlorul.l anractlon ",til 31<10 pull the Earth to'oV:lrns the cia}', COnserv:ation
cA TIlOffiI"ltl um Gin be applied In that thl.> lOIal momentum cA day and l'..:Irth remains
constant throughout the process: before l he coIlI�, and after It. T he effects of the
tramier athe da}''s momentum to tllt' fanh are 001 noticeable due to the dIfference
lnmassofthelwoobjects.
1
FIgur.4,17
o
o
o
o
AV1OOker baIl A lT1O'o'e'i withspeedu,. directlytow-aros a sWni!ar balIB whidt i s at rest
(Figur!' 4.1n. Th!' coI�sion is l'Iastic. What ar!' the spI'eds v", and v, afw the collision?
It is <:onwnierlt to tak!' th!' direction from left b:l right as tlK' diTKtion of po�itive
mommtum. 1ftlK' mass of a billianl ball is "" the total momentum 01 the syst�m belor!'
the collision is mil,.. By the prindpl.e of conservatioo of momentum. the total
mommtum aftl'f colliskm is the 5ilme as that before, or
The collio;ion is periectly elastic, so the total kinetic energy befot� the collision is the
same as that afterwards, Of
SoI"';ng thl'Sol' eqllationsgive5 I'", ,, O and vlI " II,,,. Thatis, baIi A corroes t o a comp!ei!'
standstill, and ball B nlOVl"> off wilh. the 5ilme speed as Ihal with wtlict1 ball A stnKk.
it. (AnothI'f 50lutioni<;possibie algebraically: v"" lj,. and t'I " O. ThiSCOfr�stoa
noo-mHisbn. Boall A is stili lTlO'oing with its iMia! speed, and ball B i� still at rest. In u ses
wt1ern algebra givl"> ustwo�le solutions. weneedto decic!e wtlidt � is
physic:allyappropriat!'J
2 A particle of mass til mak.es a glancing oonision with a sim'ar particle. also of mass til,
which isat resl (FigllT!' 4.18). Thewl�sionis elastic. Afterthecol�sion theparticles
move off at angles f and/l State the equations that relate
(a) tlK' xwmponents of the momenlum of the particles,
(b) tlK' ycornponents of the momentum of the particles,
(c) tlK' kineticenergy ofthe particles.
65
II Dynamics
From thl'
ation of ITlOITlI'ntum:
oonSl'rv
:�� ��a;�r::�ic:';':;��!�ant
(.1) mu= mt'l00s ,+nlvf:-fJ'iP
as the
coIIi� is I'Iastic.
HI'na!.1lnltl. }-tN1I12 + }-tlllll
Flgure4.1a
Now it's your turn
A trolll'Y A movl'S with spei'd IIA towards a trolii')' B of !'qual mass which is at fest
(Figure 4.19}. The trolleys stidtogether and movl' off as one w;th spee<! fi B
(a) Detl'rmine llA.B
7
...
(b) What fra
i n of thl' initial �inetic er.ergy of trolley A is converted into other forms
in this inelaslic collision?
ct o
Figure 4.19
mass
• The forCl! of friction Oppo5eS motion.
fli and its
• The line<lr momentumpof a body is defini'd as the product 01 its
wkKity II. In symbols: p =
. Momentum has units kg m ,1 Of N s. lt is a vector
quantity.
mil
• NI'w1on·s laws of motion are:
_ First law: Every body continues in its slate of lest, Of with unilorm Vi'lO(ity. unle-ss
acted upon bya resultantforCl!.
_ SKond law: The resultantforce iKting on a body is proportional to the r te of
change of its moml'ntum (this is used to define force). In symbols: P t:.p/Af.
a
..
_ Thirdlaw: Whenone bodyexertsafoo:eon anothefbody. thesecond body
an
Ihe first body.
Ni'Wtoo' first an
of motion
a so bE! stated in tl'fms 01 fl'IOfT\efltum
_ Rrst
Thl' moml'nlum of b y remains constant unli'SS an extl'ma
on thl' body:p= constant
_ Third law: Whl'n two bodil'!
acti n and fl'aClioo forces on Nch Ihl'ir
IfSl u�tsareused F = !JfJIM
•
I'xerts an equal d o-pposrtl' force on
s
d third I_s
can l
liM':
a od
acts
exert
•
"
o
l lOKI'
OIh1'r.
changl'S of ITlOlTll'lu
lt mareequaiand oppositl'.
lfthe mas.o;is conslant, iheresultantforce isequaltomass" actelerationOfF. ,IW.
whi'reforcl'Fisin newtons. mass In is in IdIograms and acceleration a is in m p
Examination style questions
• T"" aa:eleration offreefan g provides the �nk between the mass IN and the
weight Wofa body. w .. mg
• The prindple of ronseNation of momentum states that the total momenrum of
an i50lated system is constant. An isolated system Is one on which no extemal
rl!Sullanlfor{l!acts
• In rolisions bi!tween bodies, appiicationof the principle of con5el'Valion of
momentum sh� that the total momentum ofthe system before the celision is
equal 10 the total momentum after the (elision.
• An elastic ronision Is one in which the total kinetic energy remains the constant
In thlssituation,the relati¥l! � of approach lsequaltothe rel�tive � of
separation.
• An ineiasticcolWsionisoru1inwhichthe total l:ln.eticenergy is notthesame befae
and after the !!YI!!1t.
• Although metic energy mayor may not be conserved in a col�sion. momentum is
always conS('rved, an-d 50istotalel1i!rgy.
• The impuls.e of a for{l! Fi5the product of the forc.e and the time 6/ fo/which it acts'
impulse = FtJ
• The impuls.e of a for{l! acting on a body is equal to the cnange of momentum of the
body: FM = tJp
• Theunilofimpulse is N s.
Examination style questions
1
A net force of 9SN accelerates an objecl at 1.9m s-z
8 What is your mass? What is your weight?
Calculate the mass of the object.
2
A parachute trainee jumps I/om a platform 3.0m high.
When he reaches the ground, ne bends t-is I:nees to
cushion the fall. His torso decelerates OYer a distance of
0.6Sm. Calculate·
9 An atomic: nucleus at rest emits an (I-particle 01 mass 4 u.
The 5ptI!d 01 the (I-particle is loond to be 5.6 )( 106m S�l.
a the spt't'd of the traineejllSt bef� ne reaches the
,�
3
4
5
b the deceleration 01 his torso,
c the a�rage foo:e e)(erted on his tOl"SQ (01 mass 4S kg)
I7)' his legs durir"IQ the deceleration.
If the acceleration of a body is rerQ, does this mean that
no forces acI 00 it?
A railway engine pulls two carriages of equal mass with
uniform acceleration. The tensioo in the coupling between
theengine and the firstcarriage isT. �duce the tension
in the coupling between the first and second carriages.
Calculate the magnitude of the momentum of a car of
mas5 1.5 tonnes ( I . 5 >< 101kg) travellingala speedof
22m s-1•
6
When a certain space rocket istaking ofi, the propellanl
ga:oes are expelled at a rate 01 900kg S-l and speed of
40km s-1. Calculate thethrust ontne rocket
7
An insect of mass 4.5mg, flyir"IQwith a speedol
0.12m s-l, erKOUntersa spider'sweb, whidl bfing; itto
rest in 2.0ms. Calculate the average lorce exerted by the
in5eCt 00 the web.
Calculate the speed with which the daughter nucleus, of
mass 218u, recoiis.
10 A hecwy particle of mass Inl' moving with speed 14, mak.es
a head-on collision with a light particle of mass Inz, which
is initially at rest. The collision is perlectlyelastic, and mz
is very much less than m1• Describe the motion of the
partides after the collision.
11 A light body and a heavy body have the same momentum.
Which has the greater I:inetic energy?
12 A 45g ball with speed of 12m s-1 hits a wa!1 at an angle
of 30· (see Fig. 4.20). The ball rebolllds with the same
speed and angle. The contact time of the ball with lhe
wall i 5 1 S ms. Calculate:
a the change in momentum of the ball.
b the impulse of the ball,
c the lorce exerted on the ball by the wall.
.,,'
l
��
��
wall
Fig. 4.20
67
II Dynamics
1 3 A bullet of mass 12g is fired horizontally from a gun with
a velocity of 180m s-'. It hits. and be<:omes embedded
in, a block. of wood of mass 2000 g. whkh is freely
suspended by long strings, as shown in Fig. 4.21.
b At time I = I" trolley A collides elastkally with a fixed
spring and rebounds. (Compression and eKpansion
of the spring take a negligibly short tme.) Troley A
catdlesupwithtrolley Bat time l = 'z·
i
CalaAatethe velocity d trolley Abetween l = I, and
1 = 12,
ii Find an � forI2 in termsoftl'
with trolley B at time ,z the
dip operates so as to �nk. them 1I9CIin. this time without
the spring between them, so that they move together
with velocity ". Calculate the common velocity /J in
c When trolley A catches up
terms of u.
d Initially the troHeys were at rest and the total
momentum of the system was zero. However. the
ans�r to c shows that the total momentum after
t= tz is not zero. Discuss this result INith reference to
the principle 01 conservation of momentum
17 A ball of mass /II make s a perfectly elastic head-on
collision with a second ball, of mass M. initially at rest. The
second ball moves olf INith half the original speed of the
Fig. 4.21
first ball.
Calculate:
a i
the magnitude of the momentum of the bullet just
before it enters the block,
ii the magnitude of the initiaj veiodty of the block and
bullet afterimpac1.
ii the kinelic energy of the
block and embedded bullet
irnrnediatelyafterthe�.
a Express Min termsof m
b Determine the fraction of the original kinetk energy
relained by the ball of mass m after the collision.
Oefine force.
/Il
ii State Newton's third law of motion.
/3l
b Two spheres approach one another along a line joining
theif tentres, as illustrated in Fig. 4.23.
18 a i
b Deduce the maJIimum height abo.-e the equilibrit.m
position to whkh the
after impact.
bIod: and embedded bullet rise
14 A n uc\eus A of mass 222 u is movirl9 at a speed of
350m 5""'. While moving. it emits an a-particle of mass
4u. Afterthe emission, itis determinedthatthe daughter
nucleus, of mass 218 u, is moving with speed 300m s-' in
the original direction of the par\'nt nucleus. Calculate the
speed of the a-particle.
15 A 5Clfety feature of modem ca� is the air-bag, YAlich,
in the event of a coIlisioo, inflates and is intended to
decrease the risk of se.-ious injury. Use the concept of
impulse to explain why an air-bag might have this effect
1 6 Two frictionless trolleys A and B. of mass IfI and 31f1
respectively. are on a horizontal track (Fig. 4.22}. lnitially
they are clipp ed together by a device which incorporates a
spring, compr\'ssed between the trolleys. At tim e t = 0 the
clip is released. The velocity of trolley B is II to the right
Fig. 4.22
• Cakulate thevelocityof troiley A as thetroleys lT\OYe
apart.
68
Flg. 4.21
When they coIl;de. the average force acting 0f1 sphere A is
F" and the average force acting on sph ere B is FB
The lorces act fortime I" on sphere A and time 'Bon
sphereB.
i
State the r\'lationship between
I F"andFfI,
2 1" and 1B
{1j
{1j
ii Use your ans�rs in i to show that the change in
momentum 01 spher\' A is equal in magnitude and
oppWte in direction 10 the change in momentt.m of
sphereB.
{I]
c For the spheres in b, the variation with time of the
momentum of sphere A before, dll'ing and after the
tollision with sphere S isshown in Fig. 4.24.
Examination style questions
Fig. 4.24
The momentum of sphere B before the ,oIli5ion is also
shown on Fig. 4.24.
Copf and oompItte Fig. 4.24 to show the variation with
time of the momentum of sphere B during and after the
wllision with sphere A.
[3J
Cdmbridge InternationalAS andA
LevPI Physics,
9702122 May/June 2010 Q 3
19
A ball is thrown against a ytrtk�1 w�11. The path of the ball
is shown in Fig. 4.25.
lS.O ms-l
Fig. 4.25
The ball is thrown from S with an initial velocity of
15.0m s-1 at 60.0° to the horizontal. Assurne that air
resistarKe is ntgIigibie.
a ForthebailatS, cakulate
i
ii
its horizontal oompontnt of vtIocity.
its Ye\'"OCaI rompont<1tofvelodty.
b The horizontal distaoc:e from s to the wall is 9.9Sm.
{If
{If
The ball hits the wall at Pwith a velocity that is at right
aogIe5 to the wall. The bal rtbounds to a point F that is
6.1Sm from the wall. u5ing yoor �nsWl'rs in a.
i cakulate the vertKal heiQht gained by the baH when it
tr�l5from StoP,
{IJ
ii show that the time ta�en for the ball to travel from S
to pis l.33 s,
{If
iii show that the velocity of the ball immediately after
reboundinglrom the wall is about 4.6m S-1
{If
, Themassofthebal l i s 6 0 " lO-J�g.
i Calculate the change in momentum of the ball as it
rebounds from the wall.
[2]
ii State andexplain whether thewlli5ion isel�stkor
ine�tH:.
{If
Cambridge Intemarional AS and A level Physics,
970212/ !ktINov 2011 Q 3
20 A 5ITh111 ball is thrown horizontally with � speed of
4.0m 5-1. It falls ttvough a vtft;cal height of 1.96m before
bouncing olf a horizontal plate, �s illllStr�ted in Fig. 4.26.
"
II Dynamics
-C£J-.
c
FI9·4.26
a for thebal,as it hitsthehorizontalplate,
i state the magnitude of the horizontal component of
its velocity,
{Ij
�2m�.
{Ij
show that the vertical component ofthe velocity is
b The components of the velocity in a are both vectors.
Copy and complete Fig. 4.27 to draw a vector diagram,
to scale, to determine the velocity 01 the ball as it hits
the horizontal plate
Fig. 4.27
70
er bouncng on the plate. the ball rises 10 a vertical
heighl of O.98m
i CalaJlate the vertical componenl of thevelocity of
ii
Air resistanoe is negligible
ii
Aft
{3}
the baI as it leaves the plate.
/2J
The ball of mass 34 9 isin contact with the platefor a
time of O.12s.
Useyou'" answer in (i and the data in a ii to
cakWte, for the ball as it bounces on the plate,
1 the dlange in momentum.
{3}
2 the magnitude of the average f� exerted by the
�e on the ball due to this momentum dlange. f2}
(ambridge InternationalAS and A level Physics.
9702/22 Oct/Nov 2009 0 3
AS Level
5 Forces, density and pressure
By the end of this topic, you will be able to:
5.1
(a) describe the force on a mass I n :l uniform
5.3
(a) Slate and apply the principle of momenl.S
and
(c) use a vector trIangle to represent ooplanarforces
(b) understand the origin of the uplhrust acting on a
(e)
body In :l fluid
show a qoalkall\'e undct'standlng of frictional
forces and viscous forces Including air resistance
(d) understand IImt the we1aht
of a body may be
SA
in equilibrium
(a) deflne anduse densky
(b) deflne and use pressure
(e)
derl"c, from the deflnlllonsof pres�"Ure and
density. ttie equallontJ.p = pgtJ.h
taken as actlnl/ al a slntlle point known as the
(d) use lhe equatlOn ilp = pgilh
cenlre ofgravUy
5.2
no resultant force
no resultant torque, a system Is In equilibrium
(b) undentand Ihal, when there Is
gml/natlonal field and on:a charge I n :l unifOrm
electrlcfleld
(a) deflne and apply the mometU ofa force
(b) unclersland Utat a couple !sa pair of forces that
lends lo produce rolatlononly
(c) deflne and apply the torque ofa couple
Starting points
• Understand the concept of weight a5 the effect of a gravitational lield.
• The use of vector triomglC5 to add vectors.
• For zero resultant force, the velocity of a body does not change (Newton's first law).
5.1
Types of force
lbe we
ight of a Ixxly is an example of the force acting 00 � IlUSS In wllat Is called a
field of force. Nearlhe Sllrfaced the I!.oJnh. the gravll311on:11 field is �pproxlmately
O)IlSlant and uniform. This means that In calculations we am take the same value d
g, wllatever the posJtlon on thesurfaceof the l!.oJrth orfor� short dl:;tance (o:;JITlpared
wllh the Eanh·s radtus) aba.'e I.
There are <xher sons offlelds of force. An Imponant example Is an electriC fleld. An
electric charge experiences a force In an {'lectrlC flcld. T/){'re are slgnlf\cant slmllarilies
between the behaviour of a mass In a Bnwllalloflal fleld and an {'Iectrlc chal'g{' Inan
{'le<:trlc flekl. We shall explore thN,e slmilarllleS In lbplc 17.
frictional forces are Important In conskk'rlng the motion of � body (see lbplc 4).
We use the term viSCOU5 torce to describe the frictional force In a fluid (a Uquld or
a IPs). The propeny of the flUid determining thiS force IS the vi�cosity of the fluld
An example or such a force Is air reslslance. In Topic 4 we O)Ilsldefed the fact that
parachutistseventualtyfall ..-ith a con5lant. terminal veloclly because of air r{'slslance.
Wllen a n obj!'« ls lmmersed i n a fluid. I appe:tfli toy,'eIgh less than when Ina \'aCUum.
Il l s easier lo Ufi 1ar&e SkJn!'SuOOer water tl'lan when they are out dthew:lIer. lbe reason
for thl<i l s th:ll lmlJll'l'Sioo lnthf' fluld plO\1desan uplhrust or boopncy 1On:t>.
5.2 Moment of a force
When a force acts 011 an 00jecI, thf' force may
cause the object to !llO\'(' In a Slf3lghl
llne. Hrouklalso cause tlJl' � lO tumor spln (rolate).
71
II Forces, density and pressure
,.
r ge o
flgure S.1 Tu nm
"
ha
re
ffea n a metre u!
Think about a
metre rule Iw>Id In the hand atone end so l
t' "
���
(FIgure 5.1)' If a ,,-eight Is bung from the ruler we em feel a turning effect
Of
ICa
band .....bere the metre rule Is
pl\"OlOO. Keeplng lhe ""Ighl and Its dlstaoce along
can be cbanged
by holding Ihe ruler al an angle
becomes smaller
as tile rule approaches the
r.;
The tuming effect ofa force called the
moment
In
to
er
moment
of a fOR:e depl'nds on the magnitude of Ule force and also on the
The
precisely.
the simple expl'rlment abo\-\'. we saw UIll Ihe
depl'nded on the angJe of the ruler
the bOfIZOntal.
Une of acrlon of the fon:e from the ptvot varied
l
when finding
fincllnglhemomefltof.fofce
the turnlng efJect
the horI7.OI1lal. Tbe turning effect
dlstance of the fOR:e from the pl\ut or fulcrum. ThiS dlSlance
Figure 5.1
on tile
Is ITICI\"OO funber from
Increased It
ct at lile
lile rulecortSlant,
10
v t l pos�lon,
of the force
ruler. Tbe turning effect Increases If tbe we1g./ll lS
tile hand a!ong the ruler. The tumlng effect a s
)
t the rule Is tloI1zontal
the moment ofa force Is the per
must
moment
this nleant tbat tIle
be deflned
of tbe force
Vllrylng
angJe
(see I'tgUre 5.4\. Tlle distance reqUired
p I
end cular distance dofthe Une of
:ldloo of the force from the pMX.
Themomerttofaforceiidelinedaitheproducroftheforceaodtheperpendicu�rdistaoce
thi! i e of action ofthe force
of
from the piYot.
n
Referring to figure 5.2. the force has magnlude F and acts at a potnl distance I from
the pivot. Then, when the ruier lsat angle 810 the horlzonlal.
momeIUO/jora! = F >< d
= F' I./cos 8
Since force Is measured In neMon'l and distance Is measured In melres. the unll of the
�
,
5N
jI
,, �.
65'
Figure 5.)
�B
�c�
ITIOIllE'nt of a fon:e Is
newton-metre (N m).
Example
In Figure 5.3, a �ght rod AS of length 45011 is at A sOlhal lhe rod makes an angle of
65° to the verticaL A Yl'rtical force of 15N acrson the rod at Cakulatethe rTlOITlIlf\t of
thl!forreabolItthi!l!fldA.
of force .. fo((e ><
.. 15><O.45sin65
{Remember thatthe distance mustbe in metres.)
.. 6.1 N m
held
B.
moment
�rpendicu/ardistanC9 from {)Not
Now it's your turn
1 Refening to Figure 5.3. calculate the momenl of the force aboul A for a vertical fOft:e of
25Nwit�therodatilfl angleof 30" lhe vertical.
10
Couples
the
the
When a screwdrtn'r Is used. we apply a turning effect to
apply one fooce
to
the handle because this would mean
...
handle. We do not
scr
e drtl·er WQuld mo'."t.'
sideways. Rather, we apply two foocesaequal size bu! �e direction on opposite
51c1es of tile handle (5ee figure 5.4).
72
5.J EQUilib'ium of forces
A couple consists oftwo for(l'S, Kjual in
of action do notcoincide.
magnitude
§
F
ra lel
tv.u pa
in direction
v.noo;e ines
l
t u qu u
handle at
F g re 5.5 or
T e of a co ple
Flgure 5.4Twofol{@s Olcting
;zs a (oupll>
bllt opposite
of magnitude l' acting as shovm In figure 55 Ofl
oppo;;lte ends of a diameter of a dl.IC cl radIUS r. Ilach force produces a moment about
COnsider the
l
fola'S. each
the centre of the disc of magnllude Frlna clock
wlse dlrectlon. 111e tOlal momelU
about the celUre Is F x
2r or F x perfJemf/culardrsrmrre tx.'frrwll rOOjon::l.'$.
Although a turning effect Is produced, thIS turning effecl ls nOl calk><! a moment
because It Is product'd by I\\-U fOfces. nQ( Ofle. rnS1e�d. tltlstumtng eff('Cll� referred
10 as a torque. The
unll
of torque Is th(,.'lIme as Ihm oflhe moment ofa forre. I.e.
ThelOfQUe a rouple Ihe product of ooe of the fofces and the perpe dicu ar distance
01
is
n
l
betwl!en tNlora;os.
It Is Inreresllng 10 fl()(e Ihat. In englneertng, lhe llgllllleS5 0fnuls and boilS Is often
aWhee!
Figure 5.6 TightenlOg
nut
rKjulri'S the appkcatron of a lOfque
Slated as the maxImum torque
to
be used when screwing up
tile nut Ofltlle
bolt
Spcmfll'fi used fcr this purpao;e are called torque wrenches because they have a SG11e
mag
on them to Indicate the Krque ti1;at Is beIng applied.
sta tr s
Calculate the torque produc:ed by two forces, eadl of
nitud lON, actWtg rr oppo5ite
directionswiththeirlinesof action�pilfate d by a dlstanceof2San.
e
torque . forcex �paration dforces
.. ]O x O.2 5 (di
Now it's your turn
2
e e)
ncein m
N m.
The torque produced by a pelWfl u i g a soewdriver Is 0.18 This tOfQl.le is applil!d
tolhehandleofdiam@ter4.0cm. Calcutatetheforceappliecttothehandle
5.3
sn
Equilibrium of forces
The principle of moments
A metre rule may be balanced Ofl a plVOI. so that the rule Is hOr1wntal. Hanging a
re
weight on tile rule \\-111 make the rule rotatt' about tile plVOI. �1cwlng the welgtu to the
other side
tile
ofthe ph-U1 ",111 make
rule rotate III tile opposite dIrection. Hallglng
sides of the ph-UI as
In figure S.7 meallS that the ruler nta}·
mai n horizontal. In thls hortzorta
po:sItlon, tht'"re Is TlO resultanr rumlng effed and so the tOlal IlImltlg effect of the forces
In the clockwise d1rectlOfl equals the lOIal rumltlg effed til the anUc\ kw se direction.
weights Ofl both
SOCM'n
raate clockWise, or an!ldockwlse. cr It may
You can check this \-ery e:asUywlth the app;tr.ltus of figure 5.7.
oc t
l
13
l
II Forces, density and pressure
Flgur. 5.1
Wilen 3 body bas no lendency to chJng<> liS speed cJ roI:3lion, II is Slid 10 be in
rolational equililH'ium.
for a bodyto be In rotational equilibf'lJm, the
sum of the dockwi..e momrots aboot aflY point must equal tile SlIm of the antidockwiSol'
�tsaboutthat5ilme poiflt.
ThI! prindple of moments states that,
Example
Some wejghll are hUflg from a light rod AU as shOWl1 ifl Figure 5.8. The rod is pivoted
Cakulate the magflitude of the force Frequired to balaflce the rod horizontally
�---L._
2.5N
Flgu,. 5.8
Sum of dodwise moments
,.(0.25 x 1.2) + O.35F
Sum of antidod:wise moments : 0.40 " 2.5
By thl! principle of moments
(0.25 " 1.2)+ 0.35P: O.40 ,, 2.5
0.35F ", 1.0 -0.3
Now it's your turn
3
SDlTll" weights arl! hung ffom a light rod ABas shOWl1 Ifl Figure 5.9. The rod is pivoted
Calculate the magflilude 01 the force Frequired to balarKEl lhe rod hQfiZ(mtally
Figure 5.9
74
5.J EQUilib'ium of forces
part
Centre of gravity
An oIJte<:t may be made (0 oolanre at a
ICular point. When II is balanced alillis
point. the object does Il()( tum arxi liO ali till' welghl 00 one side d the plV(JI ls
balanced bylheweighlOllthl' otherSide. SUpporllng lhl'obje« al llleplYOl llll":l.llS lhat
the only fOfU" wfllch has to be applied :.lI the pWClt Is one to s«lp tile object falling _
lhal ls, a force equal to the w�lghl ci lhe object. Alhoogh all parts ofille object han'
v,'elght, the whole weight of the 00jecI appe:trs to act at this balance poIn!. This point
Is called the centre of gnvity (e.G.) dille object.
raw:udv s
ng
of glil'lity of an objoo: is the point at ....t1ich the whole weight of the obiKt may
be conYderro to act
nw centre
1be v,elght of a body can be shmvJl as a
le
folC(> acting VCftOUy down
at the
Q'flIre of gr:l.vlty. For:.l. unlfonn body such 3S :.I. ru r. 1he centre of g ity Is al the
geometr\cal cellire.
Equilibrium
s
riangl
The principle of moments gives the conditIOn neceSs.1l)' for a bOldy to IX' In roI:atlonaJ
equilibrium. However, the lxxly couJd llil h�ve 3 resu�anl (Or«' acti on it whIch
would cause U to accelerale llnear1y. Thus. (orcomplcce eqUIlU)rlUffi. there cannot be
any resultam force In anydlre<:tlon.
In lbpIc I we added fOKes (veao�) uSing a vector triangle. When three forces act on
an d:>ject the condition for equlllbrlum IS that tile vector dllgf:lffi for these forces forms
a closed triangle. When four a more fooces act on an � the same prlndples apply.
For equilibrium. the closed vecta t
e then becomes a closed vector polygon.
Fofa body to bI! in equilibrium:
1 TOO'IUIT1of�fon:esin anydirectionmustbe zero.
2 TOO'IUIT1of� rnorroentsoftheforc:l!S aboutany point must be zero.
Example
Th� uniform rod PQ'>hown in Figur� 5.10 is horizontal and in equilibrium.
�'
29N
0
Flgur. S.11
�
i
Q
,
""
FIgure 5.10
The Ylleight of the rod is SON. A foTO! of 29N that acts at et1d Q is 60· to the horizontal.
Theforce at end P is la�led X.Draw a Vi!Ctortriangleto lepfesenttheforces acting onthe
rod and detoonine themagnitude anddirection offOfC�x.
Thefon:es keep therod in equilibriumand OOnceform aclosed �as shoNn inFi9U/'1!S.ll.
A 'lCa� diagram «Ill bI! drawn to '>howthat X is 19N and acts at 50· to the horizontal.
l
Now it's your turn
.. TOO same uniform rod PQ is in equ�ibrium, as in the aboYe example.
(a) (j) Show that the upward forces equal the downward fOKes.
(ii) Sh!7Nthatthe hori:rontal fon:eto�leftequalsthe horizontai forcetothe right
(b) The �ngth of the rod in Fig Xe 5.10 is lOOcm. Determinetheforcexby taking
moments about Q.
75
l
II Forces, density and pressure
5.4 Density and pressure
In this seam we ,,111 bring together denslly �nd pressure 10 show an Important Unit
between them.
T�den5ityof a substance is defirn!d as hmass per un� IIOlume.
p _ mN
Tbe �ymbol for den�y Is p (GreeK
rho) 3I'id IS SI unit IS kg nr'.
Example
An iron s�re of radill'> OJ8m has mass 190kg. Calculate the density of iron.
Rrst cakulate the volume ofthi' 5phere from
v.'¥o�. Thisworks outat O.024ml.
Applicatioo ofthe forrnula fordemitygiws p . 7100kgm..)
J'rE<;sure isdefir.l'dasfolU! pl'funit area,wheretheforce"·acts perpendiaJlarfy totheil'eaA.
p . r1A
The symbol fOf pressure Is p and Its SI u nit IS Ihe pascal CPa), which Is equal to 0111"
newton per square metre (N m-lj
The Unit between pressure and denSity comes when we deal w�h liquids Of with
nulds ln 8fflCraJ. ConSIder a point ata depth h below the surface a a llquld l n a
container. What IS the pressure due to tile liquid? Very Simply. tile pressure 15 C1used by
Figulli 5.12 Columl'lofhqLltd
the weight of the column of liquid :ib<JI."l' a SIruIIl area at that dep(h. as shown In
aboYeth�rUA
Figure 5.11. TlIewetghl. ofthe rolumn ls W= mg= P'I/rs. and the pressure Is
W/A = pW!.
Tbe pressure Is proportiona l to the dl>p.:h below the surface of the liquid. If an external
pressure.
such as atmosphel1c �ure. aas 011 the surface a the liquid. this mUSl be
taKel'l IflIo acroufll lncalrul:nlng the ab!iolute pressure. l1le absolute pres!lUre Is the
sum of the external pressure and the pressure due to the depth be\o1>.' the 6urface of
the liquid.
Example
Calc:ulate the elOCeSSpreS'iUfe owr atm05pheric at a point 1.2m below the sorfaCl! of the
water in a swimmk.g pool. The density of water is 1.0 >< 100kg m-l
This is a stra�htforward calc:ulation from p .Ilfh.
SubstittJting.p= 1.0>< 101>< 9.8 .. 1.2 .. 1.2 .. 104Pa
If th� total pfl!l'iure had been required, thisvallll' would be added to atmospheric; pfl!SSlIre
p�. Takil'lgp�tobe 1.01 >< IOs Pa. the total pressure i s 1.13 >< 10s Pa
Now it's your turn
5
Ca lculate the differef1Cl! in blood pre5SUfe between Ihe top of the head afld the soles
of the fel11 of a student 1.3m tall. slanding upright. Take the deosity of blood as
1.1 >< 1()lkg m-1.
Upthrust
When an object Is Immersed In a tluld
(a liquid or a gas). II appe:lrs to weigh less than
when In a vacuum. Ills e3sier to Uft large stones uoder water Ihan when they are out
of the water. The reaSOfl for this Is that immersIOn I n the nuld prcwldesan uplhru�1
orbuoyancy fooce
We can see the reason for the upl:hTU51 when we think about an object, such as the
cylioder I n Flgure 5.13, ln water. Rernember that the pressure Ina Hquld Increases with
dep(h. Thus, the pres9.Jre at the batom dthe cylloderls greater than the pressure at
the lop 01 the cyUoder. l1t\:s means that there IS a bigger b"ce aC1log up90'ards on the
76
5.4 Density and pressure
base dtlle cylinder. than there
Is acting d
o
...
.'
nwards on tile lOp. The difference In these
forces lsth!> upthrust or buoyaocyforceFtr LOOklngat l'lgure 5.13. wecan ,s.ee that
and, since
p .. pg/t = FlA
F� ·PSAfJtl-hJ=pgAl
.",v
where
I ls th!> length Gthe cyllndl-r. and Vis lts \,olume. The uplhnlSlls slmp/y the
weighl of Ihe liquid disphl(:�d by the Imme15ed ob\«I. This relation Ius been
der"·edfora cylinder, bulllwlllalso app/yto obfeCtS G anysh:lpe,
d IS equal 10 Ille
displaced Is known as Archimedes' principle,
The rule that the upthruot acting on a body Immersed In a nul
weight of the fluid
Example
Flgur. 5.1l0r�lnofIIlQ
buoyancylorce (uptllrusl)
Cakulate (a) the force needed to lift a metal cylindef when in air anc! (bl the force needed
to lift the (ylinder when immer�ed in water.
The density of Ihe metal is 7800kg m-1 and the demilyol water is 1000kg m-3, The
VQlume of the cylinder is O.50ml.
(.I) fon:e needed in air=weightol cylinder .. 0.SO,, 7800 .. 9.81 .. 3.8 .. 100N
(bl force needed in water=weight of cy1inder- upthrun
.. 0.50 .. 7800 .. 9.Bl - 0.S0" 1000" 9.81 .. ].] . IO·N
The differMCe in the values in (al and (b) isthe upthrust 00 the meta! cylinder when
immer�in water.
[The upthrust of the cylinder in airWil!> neglected as the density of aoir is very mU(h less than
that of the metal.)
Now it's your turn
6 Explain v.+Jya boat made of metal is in equilibrium when stationary and fbilling 00 water.
moment of a force is a me.l!iUfe of the turning effect of the force.
The moment ofa force is the product of the fofce and the perpendicular distance of
he �ne of actioo of the for{efrom the pivot.
couple consist<; of two equal forces acting in opposite d�ections whose lifll?S of
action do not (()indde
• ThetorqUl' of a couple isa medSUre ofthetuming effectofthe coupie.
• The torque of a muple is the product of one of the forces and the pefpendkular
distance between the lines of action of the forces.
• The principle of moments states that the sum of the dockwise moments about a
point is equal to the rum of the antidockwise moments about the point
• The {rotre of gravity of a body is the point at which the whole weight of the body
may be considered 10 a{1.
• For a body to be in equilibriIJm
_ thesum ofthe forces in anydiroction must be zero.
_ the sum of the moments of th.e forces about any point must be zero.
• Density P is defined by the equation p .. mlV, where m is the mass of an object and
VisitsllOlume.
• Pfessurep is defined by the equationp .. FIA, where F is the force acting
perpendicularly to aon area A
• Thetotal pressurep at a poinl id a depth h below the rurface of a f1ui:l of densily p
• The
•
t
• A
isp=PA+ pSh.PAbeirg the idrr=pheric pressure; the difference in pressure
between the surfa{e and a point at a depth h isff!/!
• Theupthrust on a bodyirrvner� i n a fluid ls equal totheweightof thefluid
.
displac:ed.
77
I
II Forces, density and pressure
Examination style questions
1
ALniform rodof length 60cm h as a weight of I4N.ltis
pivoted atone endaodheld n a horizontal p05ition �
a thread tied to itsOlher end, iJSshown n Fig. 5.14. The
thread rnakesan angle 01 soowith the horimntal. Calculate:
horizon
a the moment of the weight of the rod about the pivot.
b the tension Tin the ttvtad requirt'd to hold the rocI
tally.
!
6O,m_
p,"
Fig. 5.14
houst
4 A nut isto be tightened to a torque of 16N m. Calcuiate
the force which must be app�ed to the end of a spanner
of Iength 24cm in ordef to produce this torque
5 The water in a storage tank is 15m aboYt a water tap in
the kitchen ofa
. Cakulate the Prt>Ssure of the water
leaving the tap. Density of water = 1.0 I< 10lkg m-l.
6 Showthatthe pressurt' pduttoa �quid ofdensi ty p i'i
proportiooal to the depth h below the surface of the liquid
7
a Define «"f1fre ofgra�ry.
b A uniform rod AB is attached to a vertical wal at A
The rod is held horizontally by a string attached at B
and to point C, as shown in Fig. 5.17.
{2}
2 A ruler is pivoted at its centre of gravity and weights are
string
hungfromthe ruleras shOlNn inFig. 5.1S. Cakulate·
a the total antidockwise moment about the pivot,
b the magnitude of the force F.
Flg.S.H
Flg. S.lS
3 A uniform plank of weight 120N rests on two stools as
shown in Fig. S.16. A weight of SON is placed on the
plank, midway between the stools. Calculate:
,
ro:5riiI
Flg.5.16
78
1
eoN
i Use the resolution of foroes to calculate the �tiCaI
cr.mponent of T.
a the fOfCe acting on the stool al A,
b the fOfce acting on the stool at B.
rt
The angle between the rod and the string al Bis SOO. The
rod has length 1.2 m and weight S.5 N. An objec;tO of
mass M is hlng from the rod at 8. The tension Tin the
string is30N.
t
�
(IJ
ii State the principle of moments.
(1J
iii Use the principle of moments and take moments
about A to show that the weight of the oo;e.:t
O � 19N.
m
iv Hence determine the massMof the object o.
(1J
c Use the concept of equilibrium to explain why a force
must act on the rod at A.
{2}
May/Ju
Cambridge International AS and A level Physics,
9702/22
Expla4n what is meant by centre ofgravity.
b Define moment of a force.
8 a
ne 2013 Q 3
{2]
(1J
c A student is being weighed. The student. of weight Iv,
stands 0.30m from end A of a uniform plar*: AB, as
shown in Fig. 5.18.
EJ<amination styl, questions
10 a
b
{l]
Define the torque of a couple.
l.Sm and weight 2.4N is
A uniform rod of length
shown in Fig. 5.20.
Fig. 5.'8
The plank has v.-eight 80N and length 2.0m.
8.0N rOpliB
A pivot p
supports the plank and is 0.50m from end A
A weight of 70N is moved to balana' the weight of the
Fig. 5.20
student. The plank is in equilibrium when the weight is
O.20m from end \3.
i
Statethetwoconditionsnecessary for lheplanktobe
i
in equilibrium
Oeterminethe weigh1 Wofthe studenl
121
Cambridge mternarionaJ AS and A Level Physics,
9702121 MdylJuflf' 20H Q 3
9
a Distinguish between the moment of a fon::e and the
torque of a couple
i
{3}
iii If only the 70N wetght is IT\OIIt'd, there is a maximum
weight of studentthat can be delermined using the
arrangement shown in Fig. 5.18. Stale and expl�n
one change that can be made to irw::ll!ase this
maximum weight.
The rod is 5Upported on a pin passing through a hole
in it5 centre. Ropes A and 13 provide equal and oppo5ite
force5 0fB.ON.
f2]
f4]
b One type of weighing machine. known as a steelyard, is
illustrated in fig. 5.19.
CalC\Jlate the torque on the rod produced
by ropes A
{1]
{l]
andB.
ii Di5Cuss,briefly, ....tietherthe rod isin equilibrium.
c The rod in b is re!'l'\CM!d from the �n and supported by
I-
rope5 A and 1:1, as shown in fig. 5.21.
1.5m
i
�rA
j-'
�
Flg.S.21
Rope A is now at point I> O.30m from one end of the
rod and rope 1:1 is at the other end.
Flg.5.19
i CalC\Jlate the tension in rope B.
ii CalC\Jlate the tension in rope A.
The two sliding weights can be mOIled independently
at the zero marlt. 0fI the metal rod, the melal rod is
horizontaL The hooll; is 4.8cm from the pivot.
to retlXn the metal rod to the horizontaj position, the
12 N sliding weight is ITIOIed 84cm along the rod and
the 2.5 N weight is ITIOJt'd 72crn.
Calculate the weight of the sackof flour.
OJ
i Suggest why this steelyard would be imprecise when
�ghing objectswithaweighl ofabou!25N.
/fl
i
Cambridge InternarioniJIAS;md A Iew!I Plrysics.
[1]
9701121 Oct/Nov2011 0 2
With no lood on the hook and the sliding weights
A sack of flour is suspended from the hook. In order
{l]
Cambridge Internatiofl(tl AS and A Level Physics,
along the rod.
11 a
Define density.
c A paving slab has a mass of 68 kg and dimensions
{1]
5 0 m m )( 600mm)( 900rnrn.
i
Calwiate the density, in kg m-3, of the material from
whKh the p;wing slab is made
{2]
ii Calculate the mallimum pressure a slab could exert on
the ground when resting on one of its surfaces.
/3J
Cambridge International AS and A level Physics,
9701/11
Oct/Nov lOll Q I pafT$ a;md c
9701101 OctlNfN1008 Q 3
79
I
AS Level
6 Work, energy, power
�
By the end of this topic, you will be able to:
6.1
(:I) give eX�nlple5 of energy In different forms,
6.3
(b) recall and apply the fomlUia J:"k = Wm"z
principiI' of conservation of energy to simple
(c) distinguish between gr.Jllltallorml potential energy
examples
6.2
(a) den,'e, from the equations of J11O(ion, the formula
klnetlcencrgy li k = Y. t,,,,,.z
Its convCfslon and conservation and apply the
and elastlcpotenllal (.-'Tlet'gy
(a) understand the concept of work In ternls of
(d) unders!and and use the relaUonshlp l>et\\'ren
the product ofa force and displacement in the
force and potential energy n
I � unlfonn field to
dlrec1lon ofUle force
solve problems
(b) calculate the work done n
I a number of si1U31ions,
Including the work d<:lJle by ::t gas \\'hlch Is
(e) derl,'e, from the definlrli equatlon W :: rs, the
formulat.Ep:: IngAh for potential eoergy changes
expandlng agalnsl a constant external pressure:
W ; p!:.V
(0
(c) recall and understaoo that the efficiency of a
the system to UIC total enelll:Y n
I put
near the Earth's surface
recall and use the formula t.Ep :: IngAh for
potenUal energychanaes near the Earth's surface
system 15 the l'lItio of useful energy output from
6,4
(b)
energy losses In pnctlcal devk;es and use the
::��:�:: :Su�'7;::��Ia��=��;�:�
(a) define power as work dooe per unll ilme and
(d) show an appreciation for the Implications of
t
P = Pv
ooncept of elflclency toso!\'e problems
Starting points
• Know that there are vari0u5 forms of energy.
• Understand that energy can be wrlYerted
'I'mgiJf'lS /Q worl/ I(l(lay,'
'Wbero
do)'Oll U'orl/?'
1'Vedollfl some U/or/J III lbi! 1J(m/(m,'
'Lots O/IiOril UYIS done l/ftfl18 lbi! box:
1'Vedollemy bomt'1loril,'
from one form to another.
• Machines enable us to do useful work by converting energy from one form to
another,
6.2 Work
The words 'work', 'energy' and 'l>O""'er' are In use In evel)'day Ilngllsh language but
they have a variety of meanings, [n phySiCS, they ha\'e very precise meanings. The
word work has a definite Irn<'fprerallon, The vagucness of the
term 'work' In everyday
speech Gl.ust"s problems for some slUderns when they come to gll'e a pre<:lse sdernlflc
definition of \\urk,
WOO: is d� whefl a
dirl!ction ofthe foo:�
:
force ITlO'II!S the point at whid1 lt acts (the point of application) in the
It Is very Imponam 10 Inctude direction In the deflnltlon of work done, It. Clr can be
pushed halzorUal/y quite eaSily bul,
If the car IS 10 be lifted off [(5 wheels, much men"
work has to be done and a machJne. such as a car-jack. Is used.
Flgur. 6.1The_ght·llftef uses � lotoi
�nergy toMftth�welghts but theyGlnbe
rolled �bng the groundwrth Irttl� effort
80
When a force !ll(Jl.'t'S Is poIm r:lappUcatlon ln the direction r:llhe forre. the fora'
doeswak and the work done by the fora' IS said to be pas/nt¥!. COn\'ersely.
If the
6.2 Work
dlll'Clton ol the forre Is oppot;fre to the dlR'Ctton ol fIlCI\"enlent. \\"000; Is done on tile
force. This work done Is then saki 10 be m'[IUlfu!.
Thls IS IIluslr:lled In Figure 6.2.
linal posilion
inilial posilion
inllial posilion
linalposition
figure 6.2
An atternaUH' name for dl51aoce moved In a partICular direction Is displacement
Dlspiacemenl Is a Vff!Or quanll[y, as Is
force. HCM·C'o·er. work done has no dlll'Cllon.
joUle!S (J).
only magnllude (size), and Is a scalar quamuy. il ls measured In
usefulwoJt; done bylhe
islound uSlllg thecornponent
01 the II!f'lSIa'l III the ropt along the dlTeC\lOl1
01 motion of the shiP
Figure 6.3 llIe
:;malltug-bo�t
Whm a � of one nev.tton lTV'll!S it5pointofapplicationby one metre inthedill!Ctionof
too force, onejoule of work is dooe.
worlldone Injoules =jorce In /lemons >< duumu m«'I!d It/ merres /'1 the
dlr«riou ojthejorce
II foImos that a joule (I) may be S3k1 1O be a newton-metre (N m). If the force and the
displacement are not txxh In the same dlrecllon. then tIle compooent of the force In
the direction oftne dlspJaa>mem must be found. corulder a force FaCllng abnga Une
at an angle 9to the displacement, 3S
soo.'Tl
.. ln FIgure 6.4. Thecanpont'llt dtlle force
aD1g ttle dlR'Ct1on of ttle dlspIJeemem IS F cos 9.
�. . . .
Figure 6.4
\\'OI"k done for displacement X = FCOIl 9 >< x
= Fx o:JS 9
mponent Fsln 90flhe for«' IS at rtglll angles to the dl5piacenlent.
component. no work Is done In
N()(e that the co
Since there Is no dlspL1cemem In the dlrectlon d Ihls
A child tows a loy by means 01 a 51ring itS shown in Figure 6.5.
s
a ie of 2S° with the h-orilontal.
Thelension in the string is 1 . S N and thil striog make an ng
Cak\Jlale the wor� done in moving th-e toy horizontal� through a distance of 265(f1l
tomrkdone= horizonta/�toftemioll " disrancernoved
= 1.5 cos 2 S >< l §.
= 3.6J
Figure 6.5
100
Now it's your turn
1
A boJt weighs 45N. Calrulati' thi'work doni' in lifting the boJt t!vaugh � vertical
height of:
(a) 4.0 m.
(b) 67cm.
81
D Work, energy, power
2 A fDlu of 36N am at an angll! of 55' to the Wft;cal. The fOtCI! m()\/I!S its point of
application by 64an in the direction of the force.
(a) thl! horizontal component ofthl! force,
(b) the vertical compooent of thefon:e.
Calculate the won done by:
Work done by an expanding gas
A bulk:ltng Gin be demolished "1th explcX';I\-es (I'lgure 6.6). When the explosives are
de!onated, large quantities oI'gas at hlgh pressure are produced. AS the gasexpaOOs, lt
<!oeswa-k by breaking d<JI>,'n the lll3sonry. In thl5 sect\on. we wUl der1\"e an equation
for the worK
done when a gas changes Its volume
Conslder a gas cav:alllt'd lna cy!Jnder bymeans a a fr1cllorJless piston ofare3A.
as shown In FIgure 6.7. The pressure p dtlle gas In the cylinder Is equal to the
atmo;;pherlc pressure 0ut5ldt'the cyUnder. Th15 pressure may be thouglu tobe con,;tanl.
'�A
\
I
Figure 6.1
Since pressure =0
\
-:= '
the gas produces a force ''' 00 the p�on given by
F =o ]JA.
When the gas expands at const.1nt pressure-, tile plSlon moves outwards through a
dlStance x.So,
1I'Ori.! dO/w
by thegas =0force >( dtsltmce IIIO/..'(!{1
W=o pAx
HOI"ever. Ax Is the Increase In voiulIll' 0( the gas 4V Hence.
W=opAV
When thevolulIll' d a gas changes at COIlSiant pressure.
When the gas I':qxmds. wa-k Is done by the gas. If the gas commas. then wa-k is
done 0/1 the gas.
"
6.1
Energy
Figure 6.8 It i<; expaooing g;JSI's pushing on the pistons which ause wor!: to be (!one
by tne engine In a car
Remember that the unU of "urk done Is the jOule U). Ttl(' preS5Ure mU5t be In
pascals (pa) or newtons per metre squared
(N m-J') and the change In volume
In metres cubed (mJ).
Example
A 'Sam� of gas has a volume of 75Oan1. The gas expands at a constant pressure of
1.4" tOSPa solhal �svolume becomes 900cml. Calculate the work done by the gas
during the expan�on.
chilngein Kllt.mel>V. (900 - 750)
_ 150crnl
_ 150>< IO-6rnl
lM:Irkdonebygas.pt.V
. (1.4 >< lOS) >« 150 " 10-6)
.21J
Now it's your turn
1
The volume of air in a tyre i5 9.0 >< lO-lrnl. Atmospheric pressure i5 1,0 " 10SPa.
Calculate the work done agaimtthe atmospherli! bythe air
the air expands to a volume
"
of 2.7 ><
whef1 the tyre bursts ar'ld
lO-lml.
High.pressure gas in a spray-can hasa volume of250cml.Thegas escapesinto
the atmos.phere through a nozzle. so that its final volume is four times the volume
of Ihe can. Ca1culate the WOfk done bytne gas. giWl1 tnat atmospheric pressure is
1 .0 >< 1 0s Pa.
6.1
Energy
In order to wind up a sprlng, "urk has to be done becJuse a force must be moved
thlOllgh a distance. When the- spr1ng 15 released. It Cln do work; for eumple. maldng a
chUd's toy
nKM". When the spring Is wound. It stores the abt!!ty to 00 work. Anything
Ih:lt Is ab!e 10 do work Is saki 10 havl" l"Ol'TlJY
Abodywhkh can do work mu5I hiM! energy.
Flgure 6.9The lfl'Ol'lgs�s energy ilS l l lS
stfl!tched. re�Slngthe energy"Slt rl!turnslO
itsori!l'niIIsh�
A body wlh 00 mergy Is unabll" 10 do work. £ne1iY and "uk are both SC2la�. Since
"uk done Is Jlll"asured [n
joules (J). l"nl"fgf IS also measured In 1OU1es, T:IbIe 6.1 lbts
sane lyplca l \"3IuesG l"Ol'f"gY·
83
D Work, energy, power
Table 6,1 Typkall'nergyvallRl
soundof spl'l'ch onl'arior l leCond
.0-'
rnoonlighl onfKl'ior l leCond
burning� m;rtch
1()J
�,rl'amGlI::l>
10'
"n"rgyr"lulol'd ffom l00kgofco�
10'0
Exthquoalu>
lO,g
l'nl'rgy rI'Cl''''!'d on E arth from Itll' Sun in onl' year
l02s
rotabon�l l'nl'rgy of thl' M il kyW;¥j9�laxy
1050
l'ltimat�d l'nl'rgy offormation of tlll' UnM!rsl'
101'C
Energy conversion and conservation
Newspapers sometimes refer to a 'global energy crisIs'. In the near future, there may
well be 3 shonage of fossil fuels. Fossli fuels are sources d chemical energy.
It would
be more accurate to refer to a 'fuel crIsis'. Wh('n cttemical energy Is used. the energy
Is tr:mstormed In to other forms of energy. some of whletl are useful and some of
which are not. Evemually. all the chemical ('nergy Is Illrely to end up as energy that
Is no longer useful to us. For example. wilen petrol Is burned In a car engine, some
of the chernJcal energy IsCOfl\l�ned Into
Into Internal
tile klfl('(1c energy of the or and some Is
(thermal) energy. When the cat stops, lIS kInetic energy Is CQrwened
energy n
I the brakes. The lempernlure of tile br.Ikes Increases and heal
W:lsted as heat
energy IS released. The outcome Is that lhe chemICal energy has been COflI'ened Into
ther use. However,
the UnlwtSe has remained constant. AU enl'rgy changes
the law of conservation of eneray. This law states that
heat energy which dissipates In the 3tlJlOi5phere and Is d no fur
the tOlal energy present In
are 8O""l"rned by
dur10g
A/AS U··
..
eI Phy.;:lcs studies. Son1t' d the more oommon bms aR' llsled In 1lIble 6.2.
Tbere aR' manydlffeR'fll. forms d energy aocl you \\111 meeI 3 Illlmbcfd these
your
Table 6.2 forfr6
ofroergy
grOlVitabon�potl'ntialrol'rgy enerqy dul' to positio nof �massin a gr;wtta�olloliliekl
Itinl'hc: rol'lgy
enerqy du@ o
t molion
�I�sb( po tl'nlial l'llergy
energy ltor!'d dueloslretc:hlng OfcompresSlng anOOj@(t
elI'Clri(�1 l'nergy
enerqy allOCi� tedwiltl movlng chargecarrlerl due to�
po tenliill d�ference
eier:llO'I talic: po tenliaienergy enerqy due to the po sIlion of a charge In an eteclrk liI!id
SO!lnd energy
energy Ir�mfelT!'d from particle to JWtlcle �slOCiate d wrth �
loundw<r-le
eier: tromagnl'lk !<Imalion
energy allOCiate dwilti WOlVe s in the eiectromilC}fletk �lrwn
soIiIrro\ffi1Y
elL>drom agnetic radiation from the SOn
int�nal roergy
!<InWrn 1:ini!1ic;nj potential enef9Y of the molecules i1 an obj!'Ct
ch,miG!lenergy
energyrelNs!'d during chemical rextions
nudearenergy
energy associ;atedwllti partides in the nudel of atorns
ltierm.;alroergy
energylr<HISferr!'d dueto tempe�turedlllerenct
(sonwl.lIT\eS ulled he;rt roergyl
84
6.3 Kinetic energy
Example
Map out theOO£'l'9y dlangpstaking place when a battery is connected toa lamp.
Chemical elll!l'g y--> eiedrical elll!l'g y-+ light energy and Internai enl!fgY
in battery
of the lamp
Now it's your turn
S
Map out the following energy changes:
a child Wlinging on a Wling,
(a)
(b) an aerosoi {dfl producing l!airspray,
(e)
6.3
a klmp of day thrown
into the air which subsequently hits the ground
Kinetic energy
AS an object falls, It loses gravitational
poI:enUal e!'ICf"8y and. In so oolng, It speeds up.
Energy Is assoctated with a movIng object. In fact. we know th�t a moving oOject can
be- made to do work as II .>lows down. !'or example. a moving hammer hits � nail �rxt,
.
as It SlOps. does work to drIve the n�lI lmo J plcre of wood
Kinetic energy is energydJe 10 motion.
COnsider an object of mass
m mavlng willi a oonSiant accelerallon 01. In a dlSlance l.
the cbject accelerates from velodty
(1 motion (see TOpIc 3),
u to velodty t·. TIlerI. by referrIng to the equatlons
vl = u' + 2ns
Figure 6.10 Whe1l the mass f�ns. It g�ins
kJneltc l!lll!rgy �nd drNes the pile IIlto the
ground
By Newton's law
(see TOpIc 4). the fOfU' F gIv:lng rise to the acceleration a Is given by
COmbIning these two equations,
vl = U' + 2
�S
R/NIrr:Jnglng,
mvl = mu'+2Fs
2Fs " mvl _ mu'
Fs "
rmv l - -inm'
By deflnlllon, the term Fs Is the work done by the force mavlng a distance s. Therefore.
"imvl and "imll',
3Ve the unlts of'l'<"Ork done. Of energy (sce TlJplc l). 111e magnllude of
sloce Fs represenl5 'I'<"Ork dOlll'. then thl' other terms In lJ"K' ('(juatlon.
muSl also h
each ofthl'se terms depends on wJoclly ,.;quared and
SO"iIl1I� arxtimll' are tenns
representIng energy whIch depends on velodly (or spee<!). The kInetIc energy E. of an
obje<:t of mass m movIng with speed ri Is glvcn by
FCI" the kinetic energy to be-In jouLes, mass must be- In kilograms and speed In metres
per second.
The full name for the term E. "
imv> Is Im,/SJmforwl kflw'k elw'RY becluse
I Is energy due to an obft'Ct fIlO\1ng in a strnight Une. IIshoold be remembered
th::lt rotating obft'Cts also han' kinetic energy and tillS form a energy Is known as
ro/a/toualkfnettc ellerg)'.
85
D Work, energy, power
Example
Cakulate the kinetic energyola Cilrol mass900kg moving at a speed of 20m s-'. State
the form of energy from which the kinetic energy is derived.
ldneticenergy.�t1I1J
.. � x 90 0 x W
.. 1.8x l0sJ
This energy is deriYed from the dlemical energy of
the fuel.
Now it's your turn
, Calrulate the kinetic energy of a carof mass SOOkg moving a t 100 kilometres per hour
1 A cyde and cyc:ht have a rombined mass of SOkg and ate m oving at S.O m �.I.
Calculate
ial the kinetic energy of the cycle and cycIi�t.
ib} the ooea5e irl kineticenergy fofan iflCTease lll speed of S.Om s·'
Potential energy
Potential energy il the ability of arl object to do'M'Xk as a result of its posilion or shilpe.
We ha�"{' already seen Ihm a \Hllmd-up spring stores energy. This energy Is potential
energy because the sprIng Is straIned. Mor('speclflcally. IheenelllY l1Iay be called
ela5lic(or strain) potenlial energy. El:istlC potCfl tl:i1 energy IsSlored In objects
whlch have hadthelr shape changed elasllcally. l!x:lmples locli.lde stre!ct\ed will'S,
twl5l:ed elastic bands and compressed gases.
Newtons law of gravitation (see 1bpic 8) lells us that all masses attract one another.
We rely on the fooce of gravity 10 keep us on l!anh! When IWO maS5eS are pulled
apan, work is done Ofl lhem and solhey gain gravi tati onal potenti al energy. tfthe
masscs 1IlOI-e doser together. they IoM- gravllational potential energy.
Gr;Mtationai potentiaienergy isenergypossessed byamassduetoits position ina
gravitational field
Ch:!nges ln gravltational potettb.l energY3fe of panlcubrhTljlO(taocefor anCJbtect.
near 10 the Eanh's surface
because we frequently do woo.; ralslng masses and,
OOI1\l'"I"SeIy, the energy stored Is released when the
mass ls lowered again. An obfect c{
ITl3S1l III nearthe Eanh's surface Ilas wetg.ht 1118, where Ills clW! accele1"ation of free fall.
This "\\"l'igh!:
Is the fooce wllh which the !!anh 311ntCiS the maSll t;od the ITl3S1l 311r.Jcts
the Eanh). If the m:J.SIl mOI'es a l.'ertfcal d lstallCt' h.
lI'OtW
Flgure6.11 Tlle c�f5 on the rollerco.mer
have ltored graVlt�tk>1kI1 potential en�rgy
Thisenergy II released al the carl fall
done;force )( dlsumc� mOlw/
When the mass IS r.llsed, lhe work done Is stored as gmlilt(lliomilpotelllfal elll''XY an d
thIs energy can be recovered when IhemaSll falls.
Chang e in !lraYilatiooal potl'ntia lenef!ly� .. mgAh
It Is m
I JXll1ant to remember that, for the energy to be measured In joUles, the maSll m
l
must be In kilograms. the arreleration 8 In metres (second)"" and tile change In hctgh!:
6l! In metres.
Notice thaI a zero point c{gravlta{1on:I1 poI:CIll131
concerned
"
eoergy IlaS flOt OOen Slaled. We are
wrh cbtl/rge5 In potenUal energy when a mass rtses or falls.
6.J Kineticenergy
1
Map outthe _rgy manges lal:ing place when an object moves from its 1c:MIe'i1 point
to�s hig�slpoinl onthelmdofil W!rticalspring after thespring is streiched.
(maximum) elastic poIential _rgy in �retched spring -+ gravitational potential I'nI!I'gy
and kiooic e�and (reduced) elawe potential eneI'gyoi object (as illTlOYl?S up) ......
(maximum) gravitational potential elll!rgyC2ero klnetk energy 1and elastic potential
energy in the compres5ed spring at its highest point
2 A V!op assistant stads iI Yletfwith 25 tins of beans, Nch of mass 460g (Figure 6.12)
Each tin has to be raised through a distance of 1.8m. Calculate the gravrtational potential
energy gained by the tins of beans, Qiwon lhat the acceleraoon of free fall is9.8m s-l.
total mass raW : 2S " 460 .. l 1 SOOg
: \ l 5 kg
increa5ol! i n p otentialeneryy = "' >< 8 " h
.. 1 1 . 5 ,, 9.8 ,, 1.8
", 200J
Figure 6.12
Now it's your turn
8
The iK.(eleriition offfeefan is9.8m s-l. Calrulatelhe dlang e i n g ravitabor.al po�1iill
energy when'
(a)
iI IJ(>rson of mass 70kg dimbs a cliff
of height 19m,
(b) a book of mass 940g is raiwd verli(311y through a distance of 130cm,
(el
anaircralt oftotal ma�s 2 " S x 101�9 desce!1dsby980m
Efficiency
Machines are used to chan&\, t'"flt'"r&Y (rom CInE' form Into 50l1"le other more useful form
tn mOlS( t'"flt'"r&Y changes some energy IS "lost" as he:lt (thennal) encrg)"" For example
when 3 ball rolls down a slope, the tOlal change In grow.aUonal potential energy Is Il()(
equal to (he gain In klnellc enl'T8Y because heat (tllermal)
energy has been produced
a s a resuhd fl1ctlonalfon:es.
87
l
D Work, energy, power
Efficiency gives a mroslIrI' 0( how TD.JCh 0( (he (<<as energy ma)' be considered
useful aoolsTlOl 'IosI:'
EfJldency may be gI\'I'n el!her as a r<ltlO OI' aS a percentage. Slnce enefiY OlnllOl be
Cf('ated. effideocy Oln fIl"\'l'1' be gmuer th3n I� and 3 'perpetual 1TlCt1on' machine Is
IlOI possIbIe (Flgure6.1}).
Figure 6.11 An attemptto d�sign � madnnf to 9@1 saTI�thlOg
for notharg tv bre�bng the I""" of rons�rvatlOn of �nefQY
Example
A man lifts a weight of 480N through a vertical distance of 35m usirg a rope and �me
pul�. The man pulkon 1M rope with a force of 200N and a lergth of 10.Sm of rope
pa�Sf'� through hi� han<k. Calrulate !hi! efficiency of !hi! pulley system
worl:- done by man " force " di5taoce moved an direction of the force)
" 2 0 0 ,, 10.5
,, 21OOJ
\\O"kdone lifting load " 480 "
3.5
,, 168OJ
asener9Y i�the abilityto dowor1<.and lromthe definitionol efficiency.
effidenq " work got ouftwm put In
,, 168012100
Now it's your turn
9 An electft: heater converts electrical ef\Ilrgy into heat energy. Suggest whythi� process
may � 100% eflident
10 The electric motor of an �levator (�ft) USf'S 6301tJ of electrical energy when raising 1M
�iI1or ard pas�nger�, of total weight 12500N, through a vertical height of 29m.
Cairulate the effidency ofthe eleYator.
88
6.4 Powe,
6.4 Power
o
no! only l'i the availability of useful foons of energy
M:JClilnes such as ,,100 turbines or engines do work f r us wllel1 lhe)' ch:ll'lg(" energy
lnIo 3 useful form. Howe\"t'r,
Important, but also the me at ,,1lk:h II Gin be cOlwertcd from one form to ana:her.
power.
The !'ale ofcUl\"enlng erJef&}' or using ('fl('rg)' IS kflO\\T! as
We han" seen
that eJll'1RY Is the ablll!y 10 do 1;\,00. consider a family en and a
Gr:Joo PrIx ractng car which both conIaln the $;Ime amounl 0( fuel. They are Clp;ible
but the rndng Of Is able \0 Ir.lwl llll.ldl f.J>ter. This
is bec:ruse the engine a the racing Glr em COf\\'('rt the chemlall energy 0( the fuel lrno
0( doing tile Slme amount o(woo,
useful energy at a tD.JCh faster rate. The eJJ8Ine is $;lid 10 be more Jl'Cn\'l'rfuL Pm>.� IS
the rate ddolng woO;:. l'OII.'erls gh'l'n by the rormul:l
The unj( of pc:n'l"r Is the wan (symbol
W) �nd IS ('(Iual 10 a f':l1e a working 0( 1 joule
il
per second. This means 111m a light bulb of power 1 W w l COllvCTt I) of eloxtr1cal
energy to aller fonns of energy (e.!!. light and heal) every second, Some typkal values
of power are shown In Table
6.3.
Table 6.J Valuel of pa.o.rel
�r/W
power to OJX'r�t� a smoll GIO::ujalof
light pow�r from a lord!
�� oulpul
S O " 10-0
4 .. 10-3
Jo
manual l�bourerworking(ontinuou�1y
100
w.lIerbuffalowormg (ontinuou�1y
150
h"dr�
] .. 10l
molor,�engine
SO .. 10l
.. loo
electri:;: tr�n
5
electri:;:ity generating �tltionoutput
2 .. lot
1'OII.'ff, llkeenergy, Isa SC3larqwnUly.
Clre must be taken when referrtng 10 p:I\'·er. 11 Is oommon In e'o'Cf)"day language
to Sly that a strong pe� Is ·po'H'1ful·. In physics, strenglh. Of rom.>, �od pm>.l'"r 3re
no.t the Slml'. Large forres may be exerted wlhoul any mOI'emeoI and thus 1\0 work
Is done and the power Is zero! For exampk>, 3 large rock resUng
on Ihe roun
g
d Is not
mc::o.>lng, yet It Is exenlng a large forre.
Consider 3 forre F whkh moves a dlstanCCx 3t COOSIan velocity IJ In the dlrecUon of
l
the fooce, In Ume I. The ·work done wbythe force Is given by
DividIng boIh sides of thIs equatm by time
1 gIves
� = FT
Now, T Is the rate of doing v'Ofk. I.e. the po,,,er Pand T .. II. Hence,
89
I