3URSHUWLHV RI 0DWWHU
/HDUQLQJ 2EMHFWLYHV
After studying this chapter, a student will be able
to understand
z
VJG EQPEGRV QH RTQRGTVKGU QH OCVVGT
z
VJG GNCUVKE DGJCXKQWT QH VJG OCVGTKCN
z
VJG WUGU QH UVTGUU UVTCKP FKCITCO
z
VJG FKHHGTGPV HCEVQTU CHHGEVKPI GNCUVKEKV[
z
VJG EQPEGRV QH VJTGG OQFWNKK QH GNCUVKEKV[
z
VJG EQPEGRV QH DGPFKPI OQOGPV CPF VJGKT
ENCUUKHKECVKQP
z
VJG EQPEGRV QH WPKHQTO CPF PQPWPKHQTO
DGPFKPI OGVJQFU
z
VJG KORQTVCPV CFXCPVCIGU QH +UJCRG IKTFGTU
z
VJG VYKUVKPI EQWRNG QH C YKTG
z
VJG TKIKFKV[ OQFWNWU QH VJG YKTG WUKPI
VQTUKQPCN RGPFWNWO
1.2
Engineering Physics
,1752'8&7,21
6JG DTCPEJ QH RJ[UKEU YJKEJ FGCNU YKVJ VJG GNCUVKE RTQRGTV[ QH
OCVGTKCNU KU ECNNGF GNCUVKEKV[
6JG NGPIVJ QT UJCRG QT XQNWOG QH C DQF[ ECP DG EJCPIGF FWG VQ VJG
CRRNKECVKQP QH CP GZVGTPCN HQTEG QP KV 6JG HQTEG VJCV RTQFWEGU FGHQTOCVKQP
QH CP DQF[ KU ECNNGF FGHQTOKPI HQTEG #P GSWCN CPF QRRQUKVG HQTEG KU
CEVKPI QP VJG DQF[ VJCV VTKGU VQ DTKPI KVU QTKIKPCN FKOGPUKQP 6JKU HQTEG
KU ECNNGF TGUVQTKPI HQTEG +H VJG FGHQTOKPI HQTEG KU TGOQXGF HTQO VJG DQF[ VJG
TGUVQTKPI HQTEG CEVU QP VJG DQF[ *GPEG VJG DQF[ TGICKPU KVU QTKIKPCN NGPIVJ QT
UJCRG QT UK\G 6JKU RTQRGTV[ QH TGEQXGTKPI VJG QTKIKPCN NGPIVJ QT UJCRG QT UK\G QH
VJG DQF[ CHVGT VJG TGOQXCN QH VJG FGHQTOKPI HQTEG KU ECNNGF GNCUVKE RTQRGTV[
&/$66,),&$7,21 2) (/$67,& 0$7(5,$/6
6JG OCVGTKCNU CTG ENCUUKHKGF KPVQ VYQ V[RGU QP VJG DCUKU QH GNCUVKE RTQRGTV[
K 'NCUVKE OCVGTKCNU
/CVGTKCN YJKEJ TGICKP VJGKT QTKIKPCN NGPIVJ QT UJCRG QT UK\G CHVGT
VJG TGOQXCN QH VJG FGHQTOKPI HQTEG CTG ECNNGF GNCUVKE OCVGTKCNU
'ZCORNG
6JGPGCTGUVCRRTQCEJVQCRGTHGEVN[GNCUVKEDQF[KUSWCTV\HKDTGYJKEJTGEQXGTU
OQUV QH KVU QTKIKPCN UVCVG CHVGT C NCTIG FGHQTOKPI HQTEG KU TGOQXGF
KK 2NCUVKE OCVGTKCNU
/CVGTKCNU YJKEJ FQ PQV TGICKP VJGKT QTKIKPCN NGPIVJ QT UJCRG QT UK\G
CHVGT VJG TGOQXCN QH FGHQTOKPI HQTEG CTG ECNNGF RNCUVKE OCVGTKCN
'ZCORNG
# DQF[ NKMG RWVV[ TGEQXGTU KVU QTKIKPCN UVCVG QPN[ YJGP C UOCNN FGHQTOKPI
HQTEG KU CRRNKGF CPF TGOQXGF #V JKIJ QT NCTIG FGHQTOKPI HQTEG KV FQGU PQV TGICKP
KVU QTKIKPCN UVCVG 6JGTGHQTG KV KU ECNNGF RNCUVKE DQF[
+P IGPGTCN PQ DQF[ KU RGTHGEVN[ GNCUVKE QT RGTHGEVN[ RNCUVKE 6JGTGHQTG C
OCVGTKCN OC[ DG VGTOGF CU GKVJGT OQTG GNCUVKE QT RNCUVKE QPN[ KP EQORCTKUQP YKVJ
QVJGT OCVGTKCNU
Properties of Matter
1.3
675(66 $1' 675$,1
1.3.1 Stress
%QPUKFGT QPG GPF QH C YKTG QH NGPIVJ NNO KU HKZGF CPF VJG QVJGT GPF QH VJG
YKTG KU UWDLGEVGF VQ C NQCF QH OCUU NOO MI &WG VQ VJG CRRNKECVKQP QH VJG NQCF C
FQYPYCTF HQTEG KU CEVKPI QP VJG YKTG CPF JGPEG VJGTG KU CP GNQPICVKQP KP VJG
NGPIVJ NNO QH VJG OCVGTKCN # TGUVQTKPI HQTEG KU CNUQ CEVKPI QP VJG YKTG VJCV VTKGU
VQ DTKPI VJG OCVGTKCN KPVQ KVU QTKIKPCN UVCVG +H VJG NQCF KU TGOQXGF VJG YKTG
TGICKPU KVU QTKIKPCN NGPIVJ NNO
&GHKPKVKQP
Q6JG HQTEG CEVKPI RGT WPKV CTGC QH ETQUUUGEVKQP QH VJG YKTG KU ECNNGF
CU VJG UVTGUUR
 5VTGUU 
(QTEGU
(

#TGCQHVJGETQUUUGEVKQP #
6JG WPKV HQT UVTGUU KU 0O
6JG VGTO QUVTGUU KU CNUQ FGHKPGF CU VJG TGUVQTKPI HQTEG CEVKPI RGT
WPKV CTGC QH ETQUUUGEVKQPR
6[RGU QH 5VTGUU
6JGTG CTG VJTGG V[RGU QH UVTGUUGU
.QPIKVWFKPCN QT .KPGCT UVTGUU
5JGCTKPI QT 6CPIGPVKCN UVTGUU
8QNWOG QT $WNM UVTGUU
.QPIKVWFKPCN UVTGUU
9JGP VJG NGPIVJ QH C DQF[ KU EJCPIGF QP VJG CRRNKECVKQP QH VJG HQTEG CNQPI
VJG NGPIVJ QH VJG DQF[ VJG EQTTGURQPFKPI UVTGUU KU MPQYP CU NQPIKVWFKPCN UVTGUU
QT NKPGCT UVTGUU &WTKPI NQPIKVWFKPCN UVTGUU VJG DQF[ WPFGTIQGU EJCPIGU KP
NGPIVJ DWV PQV KP UJCRG CPF XQNWOG
5JGCTKPI UVTGUU
9JGP VJG UJCRG QH DQF[ KU EJCPIGF FWG VQ VJG CRRNKECVKQP QH C VCPIGPVKCN
HQTEG VQ VJG WRRGT RQTVKQP QH VJG DQF[ MGGRKPI VJG NQYGT RQTVKQP HKZGF VJG
EQTTGURQPFKPI UVTGUU KU MPQYP CU UJGCTKPI UVTGUU QT VCPIGPVKCN UVTGUU 6JG
UJGCTKPI UVTGUU RTQFWEGU EJCPIG KP VJG UJCRG QH VJG DQF[ DWV PQV KP XQNWOG
1.4
Engineering Physics
8QNWOG UVTGUU
9JGP VJG XQNWOG QH C DQF[ KU EJCPIGF FWG VQ VJG CRRNKECVKQP QH VJTGG
OWVWCNN[ RGTRGPFKEWNCT HQTEGU VJG EQTTGURQPFKPI UVTGUU KU ECNNGF XQNWOG UVTGUU
QT DWNM UVTGUU 6JG XQNWOG UVTGUU RTQFWEGU EJCPIG KP XQNWOG DWV PQV KP UJCRG
QH VJG DQF[
1.3.2 Strain
%QPUKFGT QPG GPF QH VJG YKTG QT TQF KU HKZGF CPF QVJGT GPF QH VJG YKTG QT
TQF KU UWDLGEVGF VQ C NQCF QH OCUU NOO MI &WG VQ VJG CRRNKECVKQP QH VJG NQCF
VJGTG KU C EJCPIG KP VJG FKOGPUKQP QH VJG YKTG QT TQF
&GHKPKVKQP
6JG TCVKQ QH VJG EJCPIG KP FKOGPUKQP QH VJG OCVGTKCN VQ VJG QTKIKPCN
FKOGPUKQP KU ECNNGF UVTCKP
5VTCKP
%JCPIGKPFKOGPUKQP
1TKIKPCNFKOGPUKQP
5VTCKP KU C FKOGPUKQPNGUU SWCPVKV[ CPF KV JCU PQ WPKV
6[RGU QH UVTCKP
&GRGPFKPI WRQP VJG V[RG QH EJCPIG KP VJG FKOGPUKQP QH VJG DQF[ NGPIVJ QT
UJCRG QT XQNWOG VJG UVTCKP KU ENCUUKHKGF
6JGTG CTG VJTGG V[RGU QH UVTCKP
.QPIKVWFKPCN UVTCKP QT .KPGCT UVTCKP
8QNWOG UVTCKP QT $WNM UVTCKP
5JGCT UVTCKP QT 6CPIGPVKCN UVTCKP
.QPIKVWFKPCN
UVTCKP
&GHKPKVKQP
6JG TCVKQ QH EJCPIG KP NGPIVJ VQ KVU QTKIKPCN NGPIVJ KU ECNNGF VJG
NQPIKVWFKPCN UVTCKP
.QPIKVWFKPCNUVTCKP
%JCPIGKPNGPIVJ
1TKIKPCNNGPIVJ
'ZRNCPCVKQP
9JGP GNQPICVKQPCN HQTEG CEV CNQPI VJG NGPIVJ QH C YKTG VJGTG KU CP KPETGCUG
KP KVU NGPIVJ KPUVGCF KH EQORTGUUKQPCN HQTEG CEV VJG NGPIVJ FGETGCUGU CU UJQYP
KP (KI Properties of Matter
1.5
Fig. 1.1 Longitudinal strain
+H VJG EJCPIG KPETGCUG QT FGETGCUG KP NGPIVJ KU NNO KP C YKTG QT TQF QH QTKIKPCN
NGPIVJ N.O
.QPIKVWFKPCNUVTCKP
N
.
8QNWOG UVTCKP
&GHKPKVKQP
6JG TCVKQ QH EJCPIG KP XQNWOG VQ KVU QTKIKPCN XQNWOG KU ECNNGF VJG
XQNWOG UVTCKP
8QNWOGUVTCKP
%JCPIGKPXQNWOG
1TKIKPCNXQNWOG
'ZRNCPCVKQP
9JGP CRRNKGF HQTEG CEV PQTOCN VQ VJG UWTHCEG QH C DQF[ KP CNN VJG FKTGEVKQPU
KV WPFGTIQGU C EJCPIG KP XQNWOG CU UJQYP KP (KI Fig. 1.2 Volume Strain
+H NXO KU VJG EJCPIG KP XQNWOG RTQFWEGF KP C DQF[ QH QTKIKPCN XQNWOG N8O
 8QNWOG UVTCKP 
X
8
1.6
Engineering Physics
5JGCT UVTCKP
&GHKPKVKQP
+V KU FGHKPGF CU VJG CPIWNCT FGHQTOCVKQP RTQFWEGF QP VJG DQF[ FWG
VQ VJG CRRNKECVKP QH GZVGTPCN VCPIGPVKCN HQTEG QP KV
'ZRNCPCVKQP
9JGP C HQTEG KU CRRNKGF RCTCNNGN VQ QPG HCEG QH C DQF[ VJG QRRQUKVG UKFG
DGKPI HKZGF VJGTG KU C EJCPIG KP UJCRG DWV PQV KP UK\G QH VJG DQF[ CU UJQYP KP
(KI Fig. 1.3 Shear strain
6JG DQF[ KU UCKF VQ DG UJGCTGF D[ CP CPING  6JG CPING QH UJGCT  KP
TCFKCP KU MPQYP CU UJGCTKPI UVTCKP
5JGCT UVTCKP 
4GNCVKXGFKURNCEGOGPVQHVJGNC[GTUCNQPIVJGHQTEGFKTGEVKQP
2GTRGPFKEWNCTFKUVCPEGVQVJGUVTGUU
CPF 5JGCT UVTCKP 
&GHQTOCVKQP
N

1TKIKPCNNGPIVJ .
YJGTG
NNO KU VJG TGNCVKXG FKURNCEGOGPV QH VJG WRRGT HCEG QH VJG EWDG YKVJ TGURGEV VQ
VJG NQYGT HKZGF HCEG CPF
N.O KU VJG QTKIKPCN NGPIVJ
Properties of Matter
1.7
+22.(·6 /$:
+P 4QDGTV *QQMG RTQRQUGF C TGNCVKQP DGVYGGP UVTGUU CPF UVTCKP YKVJKP
VJG GNCUVKE NKOKV
5VCVGOGPV
*QQMGOU NCY UVCVGU VJCV YKVJKP VJG GNCUVKE NKOKV VJG UVTGUU FGXGNQRGF
KP VJG DQF[ KU FKTGEVN[ RTQRQTVKQPCN VQ UVTCKP RTQFWEGF KP KV
KG 5VTGUU5VTCKP
5VTGUU%QPUVCPV5VTCKP
5VTGUU
%QPUVCPVQTOQFWNWUQHGNCUVKEKV[
5VTCKP
+P QVJGT YQTFU VJG TCVKQ DGVYGGP UVTGUU CPF UVTCKP KU C EQPUVCPV 6JKU
EQPUVCPV QH RTQRQTVKQPCNKV[ KU ECNNGF OQFWNWU QH GNCUVKEKV[ QT EQGHHKEKGPV
QH GNCUVKEKV[
+VU WPKV KU 0O QT 2C
'ZCORNG 5RTKPI DCNCPEG YQTMU QP VJG RTKPEKRNG QH *QQMGOU NCY
(/$67,& $1' 3/$67,& /,0,7
9JGP HQTEGU CTG CRRNKGF VQ DQFKGU GCEJ CPF GXGT[ DQF[ JCU C VGPFGPE[ VQ
QRRQUG VJG HQTEG CPF YKNN VT[ VQ TGICKP KVU QTKIKPCN RQUKVKQP CHVGT VJG TGOQXCN QH
VJG HQTEG 9JGP VJG CRRNKGF HQTEG KU KPETGCUGF DG[QPF VJG OCZKOWO XCNWG VJG
DQF[ FQGU PQV TGICKP KVU QTKIKPCN RQUKVKQP EQORNGVGN[ GXGP CHVGT VJG TGOQXCN QH
VJG GZVGTPCN HQTEG
*GPEG VJG OCZKOWO UVTGUU WRVQ YJKEJ C DQF[ ECP TGEQXGT KVU
QTKIKPCN UJCRG CPF UK\G CHVGT TGOQXKPI VJG GZVGTPCN HQTEGU KU ECNNGF CU
GNCUVKE NKOKV #HVGT GNCUVKE NKOKV VJG DQF[ YKNN DG C NKOKV ECNNGF CU RNCUVKE
NKOKV
1.5.1 Stress - Strain diagram
.GV WU EQPUKFGT C YKTG YKVJ QPG GPF HKZGF CPF ITCFWCNN[ NQCF CV QVJGT GPF
6JG EQTTGURQPFKPI UVTCKP RTQFWEGF HQT GCEJ VKOG KU PQVGF WPVKN VJG YKTG DTGCMU FQYP
CPF JGPEG UVTGUUUVTCKP FKCITCO QDVCKPGF CU UJQYP KP HKI 6JG TGNCVKQPUJKR DGVYGGP UVTGUU CPF UVTCKP KU UVWFKGF D[ FTCYKPI C
ITCRJ DGVYGGP UVTGUU CPF UVTCKP 6JKU ITCRJ KU MPQYP CU UVTGUUUVTCKP
FKCITCO
6JGTG CTG C PWODGT QH KORQTVCPV RQKPVU VQ DG QDUGTXGF KP UWEJ C ITCRJ
1.8
Engineering Physics
Fig. 1.4 Stress - Strain diagram
2TQRQTVKQPCN .KOKV
+PKVKCNN[ YJGP VJG NQCF KU KPETGCUGF VJG UVTGUU KU FKTGEVN[ RTQRQTVKQPCN VQ
UVTCKP *GPEG UVTGUUUVTCKP ITCRJ KU C UVTCKIJV NKPG 6JKU RCTV QH VJG ITCRJ KU
MPQYP CU NKPGCT TGIKQP +P HKI 1# TGRTGUGPVU VJG NKPGCT TGIKQP +P VJG TGIKQP
1# *QQMGOU NCY KU RGTHGEVN[ QDG[GF
6JG UVTGUU CV # KU VJG RTQRQTVKQPCN NKOKV 9KVJKP RTQRQTVKQPCNKV[ NKOKV
1# VJG YKTG TGICKPU KVU QTKIKPCN NGPIVJ QP TGOQXCN QH VJG CRRNKGF NQCF 6JG
YKTG QDG[U *QQMGOU NCY
'NCUVKE NKOKV CPF GNCUVKE TGIKQP
6JG UVTGUU KU HWTVJGT KPETGCUGF VKNN C RQKPV N$O KU TGCEJGF 6JKU RQKPV N$O N[KPI
PGCT # FGPQVGU VJG GNCUVKE NKOKV
7RVQ RQKPV N$O VJG YKTG TGICKPU KVU QTKIKPCN NGPIVJ YJGP VJG NQCF KU TGOQXGF
DWV HTQO # VQ $ VJG UVTGUU KU PQV RTQRQTVKQPCN VQ UVTCKP 6JG TGIKQP QH VJG ITCRJ
HTQO 1 VQ $ KU MPQYP CU GNCUVKE TGIKQP
2GTOCPGPV 5GV
+H VJG UVTGUU KU KPETGCUGF DG[QPF GNCUVKE NKOKV $ VJG UVTCKP KPETGCUGU CPF KV
TGCEJGU RQKPV % 0QY VJG OCVGTKCN YKTG KU RCTVN[ GNCUVKE CPF RCTVN[ RNCUVKE +H
VJG UVTGUU KU TGOQXGF CV VJKU UVCIG KV VCMGU C PGY TQWVG %2 CPF TGVWTPU VQ VJG
QTKIKPCN UVCVG 6JG TGIKQP 12 KU VJG TGUKFWCN UVTCKP CESWKTGF D[ VJG OCVGTKCN CPF
KV KU ECNNGF RGTOCPGPV UGV
Properties of Matter
1.9
;KGNF RQKPV CPF ;KGNF UVTGUU
+P VJG TGIKQP DG[QPF VJG RQKPV % VJGTG KU PQ KPETGCUG KP UVTGUU DWV VJG
UVTCKP XCTKGU TCPFQON[ WRVQ VJG RQKPV & 4GIKQP %& KU KP KTTGIWNCT UJCRG 6JG
KPETGCUG QH UVTCKP VCMG RNCEG HTQO RQKPV % CPF KV KU ECNNGF [KGNF RQKPV 6JG UVTGUU
EQTTGURQPFKPI VQ VJG [KGNF RQKPV KU ECNNGF [KGNF UVTGUU 6JG UWFFGP KPETGCUG QH
UVTCKP IGVU UVQRRGF CV & 2QKPV % KU ECNNGF WRRGT [KGNF RQKPV CPF RQKPV & KU
ECNNGF NQYGT [KGNF RQKPV
2NCUVKE TGIKQP
+H VJG UVTGUU KU ITCFWCNN[ KPETGCUGF DG[QPF RQKPV & VJG UVTCKP KPETGCUGU CPF
KV VCMGU RCVJ &' 6JKU TGIKQP KU ECNNGF RNCUVKE TGIKQP +P VJKU TGIKQP VJG
VJKEMPGUU QH VJG OCVGTKCN FGETGCUGU CPF VJG XQNWOG QH VJG OCVGTKCN TGOCKPU
EQPUVCPV
7NVKOCVG 5VTGPIVJ QT VGPUKNG UVTGPIVJ
2QKPV ' KU VJG OCZKOWO UVTGUU C OCVGTKCN YKTG ECP YKVJUVCPF 6JKU
OCZKOWO FGHQTOKPI HQTEG KU ECNNGF WNVKOCVG UVTGPIVJ QT VGPUKNG UVTGPIVJ
$TGCMKPI UVTGUU $TGCMKPI RQKPV
+P TGIKQP '( VJG UVTCKP KPETGCUGU YKVJQWV HWTVJGT KPETGCUG QH VJG UVTGUU +P
VJKU TGIKQP C PGEM KU HQTOGF KP VJG OCVGTKCN YKTG &WG VQ VJG HQTOCVKQP QH VJG
PGEMVJGOCVGTKCN YKTG DTGCMUGXGPVJQWIJVJGUVTGUUKUFGETGCUGFCPFVJGUVTCKP
KU FGETGCUKPI 6JG UVTGUU EQTTGURQPFKPI VQ RQKPV ( VJCV DTGCMU VJG OCVGTKCN YKTG
KU ECNNGF CU DTGCMKPI UVTGUU
6JG OCVGTKCN YKTG HKPCNN[ DTGCMU CV VJG RQKPV ( CPF VJKU RQKPV ( KU ECNNGF
DTGCMKPI RQKPV
1.5.2 Factors affecting elasticity
6JG HQNNQYKPI HCEVQTU CHHGEV VJG GNCUVKE RTQRGTVKGU QH VJG OCVGTKCN
'HHGEV QH UVTGUU
'HHGEV QH CPPGCNKPI
'HHGEV QH JCOOGTKPI CPF TQNNKPI
'HHGEV QH VGORGTCVWTG
'HHGEV QH KORWTKVKGU
'HHGEV QH ET[UVCNNKPG PCVWTG
1.10
Engineering Physics
'HHGEV QH UVTGUU
9JGP C OCVGTKCN KU UWDLGEVGF VQ C NCTIG EQPUVCPV UVTGUU QT TGRGCVGF PWODGT
QH E[ENGU QH UVTGUUGU VJG GNCUVKE RTQRGTVKGU QH VJG OCVGTKCN CTG ITCFWCNN[ FGETGCUGF
'HHGEV QH CPPGCNKPI
#PPGCNKPI KU VJG RTQEGUU QH JGCVKPI VJG OCVGTKCN CV C RCTVKEWNCT
VGORGTCVWTG CPF ITCFWCNN[ EQQNKPI
9JGP UQNKF OCVGTKCN CTG UWDLGEVGF VQ CPPGCNKPI VJG ET[UVCN ITCKP VGPF VQ
QTKGPV KPVQ QPG RCTVKEWNCT FKTGEVKQP 6JKU HQTO NCTIG ET[UVCN *GPEG VJG GNCUVKE
RTQRGTVKGU QH VJG OCVGTKCNU CTG TGFWEGF
'HHGEV QH JCOOGTKPI CPF TQNNKPI
9JGP ET[UVCNU CTG JCOOGTGF QT TQNNGF VJG ITCKPU CTG TGFWEGF VQ UOCNNGT
WPKVU YKVJ VJG TGUWNV VJCV VJGTG KU CP KPETGCUG KP GNCUVKE RTQRGTVKGU
'HHGEV QH VGORGTCVWTG
#EJCPIGKPVGORGTCVWTGCHHGEVUVJGGNCUVKERTQRGTVKGUQHCOCVGTKCN0QTOCNN[
VJG GNCUVKEKV[ FGETGCUGU YKVJ VJG KPETGCUG QH VGORGTCVWTG 6JKU OC[ FWGVQKPETGCUG
QH ITCKP UK\G YKVJ TKUG QH VGORGTCVWTG
'ZCORNG
%CTDQP HKNCOGPV KU GNCUVKE KP PCVWTG CV TQQO VGORGTCVWTG $WV KV DGEQOGU
RNCUVKE YJGP KV KU JGCVGF VQ C JKIJGT VGORGTCVWTG +P VJG ECUG QH KPXCT UVGGN
GNCUVKEKV[ TGOCKPU WPCHHGEVGF FWG VQ EJCPIG QH VGORGTCVWTG 5KOKNCTN[ C FGETGCUG
KP VGORGTCVWTG YKNN KPETGCUG VJG GNCUVKE RTQRGTV[
'ZCORNG
.GCF KU PQV C XGT[ IQQF GNCUVKE OCVGTKCN $WV VJG GNCUVKE RTQRGTV[ QH NGCF
KPETGCUGU YJGP VJG VGORGTCVWTG KU FGETGCUGF
'HHGEV QH KORWTKVKGU
6JG GNCUVKE RTQRGTV[ QH C OCVGTKCN OC[ KPETGCUG QT FGETGCUG FWG VQ VJG
CFFKVKQP QH KORWTKVKGU 6JG KPETGCUG QT FGETGCUG QH GNCUVKEKV[ FGRGPFU QP VJG V[RG
QH KORWTKV[ CFFGF VQ KV
'ZCORNG
9JGP RQVCUUKWO KU CFFGF VQ IQNF VJG GNCUVKE RTQRGTV[ QH IQNF KPETGCUGU
+H NKVVNG COQWPV QH ECTDQP KU CFFGF VQ OQNVGP KTQP VJG GNCUVKE RTQRGTV[
QH KTQP KPETGCUGU GPQTOQWUN[ +H OQTG ECTDQP KU CFFGF VQ OQNVGP KTQP
VJG GNCUVKE RTQRGTV[ QH KTQP FGETGCUGU
Properties of Matter
1.11
'HHGEV QH ET[UVCNNKPG PCVWTG
6JGGNCUVKERTQRGTV[QHCOCVGTKCNCNUQFGRGPFUQPVJGV[RGQHET[UVCNYJGVJGT
KV KU C UKPING ET[UVCN QT RQN[ET[UVCN (QT UKPING ET[UVCN VJG GNCUVKEKV[ KU OQTG CPF
HQT C RQN[ET[UVCN VJG GNCUVKEKV[ KU NGUU
7+5(( 02'8/,, 2) (/$67,&,7<
%QTTGURQPFKPI VQ VJG VJTGG V[RGU QH UVTCKP VJGTG CTG VJTGG MKPFU QH OQFWNKK
QH GNCUVKEKV[
;QWPIOU OQFWNWU ;
+V EQTTGURQPFU VQ NKPGCT QT VGPUKNG UVTCKP
$WNM OQFWNWU -
QT 8QNWOG OQFWNWU
+V EQTTGURQPFU VQ XQNWOGVTKE UVTCKP
4KIKFKV[ OQFWNWU 0
QT 5JGCT OQFWNWU
+V EQTTGURQPFU VQ UJGCTKPI UVTCKP
;QWPIOU OQFWNWU ;
&GHKPKVKQP
+V KU FGHKPGF CU VJG TCVKQ DGVYGGP NQPIKVWFKPCN UVTGUU VQ NQPIKVWFKPCN
UVTCKP YKVJKP VJG GNCUVKE NKOKVU
KG ;QWPIOUOQFWNWU
.QPIKVWFKPCNUVTGUU
.QPIKVWFKPCNUVTCKP
+VU WPKV KU 0O QT 2CUECN
'ZRNCPCVKQP
%QPUKFGT C YKTG QH NGPIVJ N.O CPF CTGC QH ETQUU UGEVKQP NCO1PG GPF QH VJG
YKTG KU HKZGF KP VJG VQR CPF C NQCF KU CRRNKGF VQ VJG DQVVQO GPF QH VJG YKTG CU
UJQYP KP HKI 6JGTGHQTG VJG HQTEG YJKEJ KU CEVKPI CNQPI VJG NGPIVJ QH VJG YKTG
KU ((OI
6JG CRRNKGF HQTEG RGT WPKV CTGC QH ETQUUUGEVKQP KU MPQYP CU
NQPIKVWFKPCN UVTGUU QT NKPGCT UVTGUU
 .QPIKVWFKPCN UVTGUU 
.QPIKVWFKPCN UVTGUU 
(QTEG
#TGC
(
C
.GV NNO DG VJG GNQPICVKQP KP NGPIVJ QH VJG YKTG FWG VQ VJG CEVKQP QH HQTEG
1.12
Engineering Physics
Fig. 1.5. Young’s modulus
6JGTGHQTG VJG EJCPIG KP NGPIVJ RGT WPKV QTKIKPCN NGPIVJ KU MPQYP CU
NQPIKVWFKPCN UVTCKP QT NKPGCT UVTCKP
 .QPIKVWFKPCN UVTCKP 
 .QPIKVWFKPCN UVTCKP 
%JCPIGKPNGPIVJ
1TKIKPCNNGPIVJ
N
.
 ;QWPIOU OQFWNWU ;
.QPIKVWFKPCNUVTGUU
.QPIKVWFKPCNUVTCKP

(C
N.

(.
CN
5WDUVKVWVKPI VJG XCNWGU QH HQTEG ( KU GSWCN VQ QOIR CPF CTGC QH ETQUU UGEVKQP
NCO QH VJG YKTG KU GSWCN VQ T KP VJG CDQXG GSWCVKQP YG IGV
;QWPIOU OQFWNWU ;
OI.
T N
0OQT 2C
'ZCORNG
# URTKPI UECNG KU WUGF VQ FGVGTOKPG VJG YGKIJV QH C DQF[ D[ OGCUWTKPI VJG
GNQPICVKQP QH VJG URTKPI
Properties of Matter
1.13
$WNM OQFWNWU -
&GHKPKVKQP
+V KU FGHKPGF CU VJG TCVKQ DGVYGGP VJG XQNWOG UVTGUU QT DWNM UVTGUU
VQ VJG XQNWOG UVTCKP QT DWNM UVTCKP YKVJKP VJG GNCUVKE NKOKVU
KG $WNM OQFWNWU 
8QNWOGUVTGUU
8QNWOGUVTCKP
+VU WPKV KU 0O
'ZRNCPCVKQP
.GV WU EQPUKFGT C DQF[ QH XQNWOG N8O CPF CTGC QH ETQUU UGEVKQP NCO CU UJQYP
KP (KI %QPUKFGT C HQTEG N(O KU CRRNKGF PQTOCNN[ CPF WPKHQTON[ VQ VJG UWTHCEG
QH C DQF[ VJGP VJG XQNWOG QH DQF[ EJCPIGU DWV VJGTG KU PQ EJCPIG KP UJCRG QH
VJG DQF[
6JG TCVKQ DGVYGGP VJG WPKHQTO HQTEG CRRNKGF PQTOCN VQ VJG UWTHCEG
QH VJG DQF[ CPF VJG CTGC QH VJG UWTHCEG KU ECNNGF DWNM UVTGUU QT XQNWOG
UVTGUU
Fig. 1.6. Bulk modulus
 8QNWOG UVTGUU 
(QTEG
#TGC
 8QNWOG UVTGUU
(
C

.GV NXO DG VJG EJCPIG KP XQNWOG FWG VQ VJG CEVKQP QH HQTEG 6JGTGHQTG VJG
EJCPIG KP XQNWOG RGT WPKV QTKIKPCN XQNWOG KU MPQYP CU XQNWOG UVTCKP
QT DWNM UVTCKP
1.14
Engineering Physics
8QNWOGUVTCKP
%JCPIGKPXQNWOG
1TKIKPCNXQNWOG
8QNWOG UVTCKP 
X
8
 $WNM OQFWNWU -
8QNWOGUVTGUU
8QNWOGUVTCKP

(C
X8
-
(8
XC
$WNM OQFWNWU -
28
0OQT 2CUECN
X
(
  2 KU VJG CRRNKGF RTGUUWTG C 


$WNM OQFWNWU QH C DQF[ KU UCKF VQ DG KP EQORTGUUKDKNKV[ QH VJG DQF[ CPF VJG
TGEKRTQECN QH VJG DWNM OQFWNWU - KU MPQYP CU EQORTGUUKDKNKV[ QH VJCV OCVGTKCN
# OCVGTKCN KU GCUKN[ EQORTGUUGF KH KV JCU UOCNN DWNM OQFWNWU )CUGU JCXG
OWEJ UOCNNGT DWNM OQFWNWU VJCP UQNKFU 6WPIUVGP JCU DWNM OQFWNWU )2C
TWDDGT JCU DWNM OQFWNWU )2C CPF CKT JCU DWNM OQFWNWU )2C
4KIKFKV[ OQFWNWU 0
&GHKPKVKQP
+V KU FGHKPGF CU VJG TCVKQ DGVYGGP UJGCTKPI UVTGUU VQ VJG UJGCTKPI
UVTCKP YKVJKP VJG GNCUVKE NKOKVU
 4KIKFKV[ OQFWNWU 
5JGCTUVTGUU
5JGCTUVTCKP
+VU WPKV KU 0O
'ZRNCPCVKQP
9JGP HQTEG KU CRRNKGF VCPIGPVKCN VQ VJG UWTHCEG QH C DQF[ KV RTQFWEGU C
EJCPIG KP UJCRG YKVJQWV CP[ EJCPIG KP UK\G 6JG TCVKQ QH VCPIGPVKCN HQTEG
CRRNKGF VQ VJG CTGC QH ETQUUUGEVKQP KU ECNNGF UJGCTKPI UVTGUU
 5JGCTKPI UVTGUU 

6CPIGPVKCNHQTEG
#TGC
(
C
Properties of Matter
1.15
Fig. 1.7. Rigidity modulus
%QPUKFGT C UQNKF EWDG #$%&2345 6JG NQYGT HCEG &%32 KU HKZGF CPF C
VCPIGPVKCN HQTEG N(O KU CRRNKGF QP VJG WRRGT HCEG #$45 CU UJQYP KP HKI 6JG
TGUWNV KU VJCV VJG EWDG IGVU FGHQTOGF KPVQ C TJQODWU UJCRG #$%&2345 KG 6JG NKPG LQKPKPI VJG VYQ HCEGU CTG UJKHVGF VQ CP CPING 
(TQO (KI VCP
##
&#
(QT UOCNNGT XCNWGU QH VCP KU PGCTN[ GSWCN VQ  KG VCP
5JGCTKPI UVTCKP 
4GNCVKXGFKURNCEGOGPVQHVJGNC[GTUCNQPIVJGHQTEGFKTGEVKQP
2GTRGPFKEWNCTFKUVCPEGVQVJGUVTGUUU
5JGCTKPIUVTCKP
##
&#
5JGCTKPIUVTCKP
N
.
YJGTG . KU VJG QTKIKPCN NGPIVJ CPF N KU VJG TGNCVKXG FKURNCEGOGPV QH VJG WRRGT
HCEG QH VJG EWDG YKVJ TGURGEV VQ VJG NQYGT HKZGF HCEG
9G MPQY
4KIKFKV[OQFWNWU0
0
QT
5JGCTKPIUVTGUU
5JGCTKPIUVTCKP
(C
0O

0
(.
0O
CN
1.16
Engineering Physics
0QVG
6JG GSWCVKQP HQT VJG ;QWPIOU OQFWNWUCPFVJGGSWCVKQPHQTVJGUJGCT
OQFWNWU CTG UCOG
1.6.1 Poisson’s ratio 
&GHKPKVKQP
+V KU FGHKPGF CU VJG TCVKQ DGVYGGP VJG NCVGTCN UVTCKP  VQ VJG
NQPIKVWFKPCN UVTCKP  YKVJKP VJG GNCUVKE NKOKVU
 2QKUUQPOU TCVKQ 

.CVGTCNUVTCKP
.QPIKVWFKPCNUVTCKP


'ZRNCPCVKQP
.GV WU EQPUKFGT C YKTG HKZGF CV QPG GPF CPF KU UVTGVEJGF CNQPI VJG QVJGT
GPF CU UJQYP KP HKI &WG VQ VJG HQTEG CRRNKGF VJG NGPIVJ QH VJG YKTG KPETGCUGU
HTQO . VQ .N .GV NNO DG VJG KPETGCUG KP NGPIVJ QH VJG YKTG
Fig. 1.8 Poisson’s ratio
6JG TCVKQ QH EJCPIG KP NGPIVJ QH VJG YKTG VQ VJG QTKIKPCN NGPIVJ QH
VJG YKTG MPQYP CU NKPGCT UVTCKP QT NQPIKVWFKPCN UVTCKP
 .QPIKVWFKPCN UVTCKP 
 .QPIKVWFKPCN UVTCKP 
%JCPIGKPNGPIVJ
1TKIKPCNNGPIVJQHVJGYKTG
N
.
Properties of Matter
1.17
6JG HQTEG CRRNKGF VQ VJG YKTG PQV QPN[ KPETGCUGU VJG NGPIVJ QH VJG YKTG DWV
CNUQ FGETGCUGU VJG FKCOGVGT QH VJG YKTG HTQO & VQ F 6JG TCVKQ QH EJCPIG KP
FKCOGVGT QH VJG YKTG VQ VJG QTKIKPCN FKCOGVGT QH VJG YKTG KU MPQYP CU
NCVGTCN UVTCKP
 .CVGTCN UVTCKP 
 .CVGTCN UVTCKP
%JCPIGKPFKCOGVGT
1TKIKPCNFKCOGVGTQHVJGYKTG

&F
&
9JGTG &F KU VJG FGETGCUG KP FKCOGVGT CPF & KU VJG FKCOGVGT QH VJG
YKTG 6JG PGICVKXG UKIP UJQYU VJCV VJG FKCOGVGT FGETGCUGU
 2QKUUQPOU TCVKQ 
QT 
&F&
N.
.&F
N&
6JG PGICVKXG UKIP KPFKECVGU VJCV VJG NCVGTCN UVTCKP CPF NQPIKVWFKPCN UVTCKP
CTG QRRQUKVG VQ GCEJ QVJGT
2QKUUQPOU TCVKQ JCU PQ WPKVU CPF FKOGPUKQPU 2QKUUQPOU TCVKQ JCU C
XCNWG DGVYGGP CPF HQT CNN OCVGTKCNU
1.6.2 Relationship between three modulii of elasticity
.GV WU EQPUKFGT VJTGG UVTGUUGU 23 CPF 4 CTG CEVKPI RGTRGPFKEWNCT VQ VJG
VJTGG HCEGU #$%&#&*' CPF #$(' QH C WPKV EWDG QH CP KUQVTQRKE OCVGTKCN QH
UJQYP KP (KI Fig. 1.9 Stress in a unit cube of an isotropic material
1.18
Engineering Physics
'CEJ QPG QH VJG UVTGUUGU YKNN RTQFWEG CP GZVGPUKQP KP KVU QYP FKTGEVKQP CPF
C EQORTGUUKQP CNQPI VJG QVJGT VYQ RGTRGPFKEWNCT FKTGEVKQPU +H NO KU VJG GZVGPUKQP
RGT WPKV UVTGUU VJG GNQPICVKQP CNQPI VJG FKTGEVKQP QH 2 YKNN DG 2 +H NO KU VJG
EQPVTCEVKQP RGT WPKV UVTGUU VJGP VJG EQPVTCEVKQP CNQPI VJG FKTGEVKQP QH 2 FWG VQ
VJG QVJGT VYQ UVTGUUGU 3 CPF 4 YKNN DG 3 CPF 4
.GV CNN VJG VJTGG UVTGUUGU CEV UKOWNVCPGQWUN[ QP VJG EWDG
6QVCN GNQPICVKQP CNQPI VJG FKTGEVKQP QH
2C234
6QVCN GNQPICVKQP CNQPI VJG FKTGEVKQP QH
3D324
6QVCN GNQPICVKQP CNQPI VJG FKTGEVKQP QH
4E423
9G ECP GZRTGUU VJG VJTGG EQPUVCPVU ;0 CPF - KP VGTOU QH  CPF  CU
HQNNQYU
%CUG K
+H QPN[ VJG UVTGUU 2 CEVU CPF 34 6JKU KU C ECUG QH NQPIKVWFKPCN UVTGUU
RTQFWEKPI NKPGCT UVTCKP
6JG NKPGCT UVTCKP C2
 ;QWPIOU OQFWNWU ;

;
QT .QPIKVWFKPCNUVTGUU
.KPGCTUVTCKP
2
2

;
%CUG KK
+H VJG UVTGUU 4 CPF 32 6JGP VJG GNQPICVKQP CNQPI VJG FKTGEVKQP QH
2 KU
C22
C2
6JG CPING QH UJGCT C
2
Properties of Matter
1.19
 6JG 4KIKFKV[ OQFWNWU KU IKXGP CU
0

2


2
2
0
QT
5VTGUU
#PINGQHUJGCT


0
%CUG KKK
.GV 234 5KPEG VJG DQF[ KU PQY UWDLGEVGF VQ WPKHQTO UVTGUU KP CNN VJTGG
OWVWCNN[ RGTRGPFKEWNCT FKTGEVKQPU
.KPGCT UVTCKP C222
C2
$WNM UVTCKP
.KPGCTUVTCKPC
C2
6JG $WNM 5VTCKP2
 6JG $WNM OQFWNWU
-
5VTGUU
$WNMUVTCKP

2
2
-
QT 

-
4GNCVKQP DGVYGGP ;0CPF (TQO GSP 
0

0
1.20
Engineering Physics

(TQO GSP -
#FFKPI GSWCVKQP CPF GSWCVKQP 9G JCXG
 
0 -0
0

;
-0

;
0-

(TQO GSP ;
0-0
6JG CDQXG GSWCVKQP IKXGU VJG TGNCVKQP DGVYGGP VJTGG GNCUVKE OQFWNK
;0CPF-
4GNCVKQP DGVYGGP 0- CPF 
5WDVTCEVKPI GSWCVKQP HTQO 

0 -

-0
0-

-0
0-
-0
$WV  
; -0

-0
-0
&KXKFKPI GSWCVKQP D[ GSWCVKQP  -0
-0


-0

-0
$WV

2QKUUQPOUTCVKQ

Properties of Matter
1.21


-0
-0
-0
-0
6JG CDQXG GSWCVKQP IKXGU VJG TGNCVKP DGVYGGP 0- CPF 
4GNCVKKQP DGVYGGP ; 0CPF 
&KXKFKPI GSWCKVQP D[ GSWCVKQP 
 0 ;


;

0




$WV 
;
 
 0

;
0

;

0
6JG CDQXG GSWCVKQP IKXGU VJG TGNCKVQP DGVYGGP ;0CPF
4GNCVKQP DGVYGGP ;-CPF
&KXKFKPI GSWCVKQP D[ GSWCVKQP   
 ;

 - 

$WV

;






;
-
6JG CDQXG GSWCVKQP IKXGU VJG TGNCKQP DGVYGGP ;-CPF
1.22
Engineering Physics
%(1',1* 2) %($06
$GCO
# DGCO KU FGHKPGF CU C TQF QT DCT QH WPKHQTO ETQUU UGEVKQP QH
JQOQIGPGQWU KUQVTQRKE GNCUVKE OCVGTKCNU GKVJGT EKTEWNCT QT TGEVCPIWNCT
YJQUG NGPIVJ KU XGT[ NCTIG EQORCTGF VQ KVU DTGCVJ CPF VJKEMPGUU UQ VJCV
VJG UJGCTKPI UVTGUUGU CV CP[ RQKPV QH VJG TQF CTG XGT[ UOCNN CPF
PGINKIKDNG
7UGU
6JG DGCOU CTG WUWCNN[ UGV KP JQTK\QPVCN RQUKVKQP KP UWEJ C YC[ VJCV VJG[
ECP UWRRQTV JGCX[ NQCFU 6JGTGHQTG KV KU WUGF KP VJG ECUG QH DWKNFKPIU VQ UWRRQTV
VJG TQQHU CPF KP VJG ECUG QH DTKFIGU VQ UWRRQTV JGCX[ XGJKENGU RCUUKPI QXGT VJGO
1.7.1 Assumption
+P QTFGT VQ UVWF[ VJG DGPFKPI QH DGCOU VJG HQNNQYKPI CUUWORVKQP JCU VQ DG
OCFG
6JG NGPIVJ QH VJG DGCO UJQWNF DG NCTIG EQORCTGF VQ QVJGT FKOGPUKQPU
9GKIJV QH VJG DGCO KU NQY KG VJG YGKIJV QH VJG DGCO KU PGINKIKDNG
EQORCTGF VQ VJG YGKIJV QH VJG NQCF
6JGTG KU PQ EJCPIG KP VJG ETQUU UGEVKQP QH VJG DGCO CHVGT DGPFKPI
6JG UJGCTKPI UVTGUU CTG PGINKIKDNG
6JG EWTXCVWTG QH VJG DGCO KU XGT[ UOCNN
1.7.2 Bending moment of a beam and neutral axis
6JG DGCO KU EQPUKFGTGF VQ DG OCFG WR QH C NCTIG PWODGT QH VJKP
RNCPG NC[GTU QPG CDQXG VJG QVJGT CPF CTG ECNNGF CU HKNCOGPVU CU UJQYP KP
HKI Fig. 1.10. Cross section - Uniform beam
.GV VJG DGCO DG UWDLGEVGF VQ FGHQTOKPI HQTEG CV KVU GPFU CU UJQYP KP HKI
&WG VQ VJG FGHQTOKPI HQTEG VJG DGCO DGPFU HKNCOGPVU NKMG #$ KP VJG WRRGT
RCTV QH VJG DGCO CTG GNQPICVGF (KNCOGPVU NKMG %& KP VJG NQYGT RCTV CTG
EQORTGUUGF 6JGTGHQTG VJGTG OWUV DG C HKNCOGPV NKMG '( KP DGVYGGP YJKEJ KU
Properties of Matter
1.23
PGKVJGT GNQPICVGF PQT EQORTGUUGF 5WEJ CU HKNCOGPV KU ECNNGF VJG PGWVTCN HKNCOGPV
6JG CZKU QH VJG DGCO N[KPI QP VJG PGWVTCN HKNCOGPV KU VJG PGWVTCN CZKU
6JG EJCPIG KP VJG NGPIVJ QH CP[ HKNCOGPV KU RTQRQTVKQPCN VQ VJG FKUVCPEG
QH VJG HKNCOGPV HTQO VJG PGWVTCN CZKU
Fig. 1.11. Bending beam neutral axis
1.7.3 Expression for the bending moment
%QPUKFGT C RQTVKQP QH VJG DGCO #$%& VJCV IGVU UNKIJVN[ EWTXGF FWG VQ VJG
CRRNKECVKQPU QH VJG NQCF CU UJQYP KP HKI .GV WU EQPUKFGT C HKNCOGPV '( CV
VJG EGPVTG QH VJG DGCO *GTG VJG HKNCOGPVU NC[GTU N[KPI CDQXG '( IGVU GNQPICVGF
CPF VJG HKNCOGPV N[KPI DGNQY '( IGVU EQORTGUUGF 6JGTGHQTG VJG HKNCOGPV '(
TGOCKPU WPEJCPIGF KU VCMGP CU TGHGTGPEG CZKU ECNNGF CU PGWVTCN CZKU
Fig. 1.12. Bending moment
.GV 23 DG C HKNCOGPV CV C FKUVCPEG Z HTQO VJG PGWVTCN HKNCOGPV +P VJG DGPV
DGCO '( CPF 23 CTG VJG CTEU QH C EKTENG JCXKPI VJGKT EGPVTG CV 1 CPF TCFKK 4
CPF 4Z TGURGEVKXGN[
$GHQTG DGPFKPI VJG DGCO VJG QTKIKPCN NGPIVJ QH VJG HKNCOGPV 23 YKNN DG VJG
UCOG CU VJCV QH '(
1TKIKPCN NGPIVJ '(4
YJGTG  KU VJG CPING UWDVGPFGF D[ VJG PGWVTCN HKNCOGPV CV VJG EGPVTG
1.24
Engineering Physics
9JGP VJG DGCO KU DGPV VJG NGPIVJ QH VJG HKNCOGPV 23 KU GZVGPFGF
234Z
+PETGCUG KP NGPIVJ 23'(
4Z4
4Z4
+PETGCUG KP NGPIVJ Z
9G MPQY
.KPGCT UVTCKP 

+PETGCUGKPNGPIVJ
1TKIKPCNNGPIVJ
Z
4
 .KPGCT UVTCKP 
CZKU
Z
4
6JKU UJQYU VJCV VJG UVTCKP KU RTQRQTVKQPCN VQ VJG FKUVCPEG Z HTQO VJG PGWVTCN
9G MPQY
6JG ;QWPIOU OQFWNWU QH VJG DGCO ;
.KPGCTUVTGUU
.KPGCTUVTCKP
.KPGCTUVTGUU;.KPGCTUVTCKP
;
Z
4
.GV NCO DG VJG CTGC QH ETQUUUGEVKQP QH VJG HKNCOGPV 23
.KPGCT HQTEG CEVKPI QP VJKU HKNCOGPV
.KPGCTUVTGUU#TGCQHETQUUUGEVKQP

;Z
C
4
6JG OQOGPV QH VJKU HQTEG CDQWV '(
(QTEG&KUVCPEG

;ZC
Z
4

;ZC
4
Properties of Matter
1.25
6JG UWO QH VJG OQOGPV QH HQTEGU CEVKPI QP VJG GPVKTG HKNCOGPV KU IKXGP CU

;ZC
4
;
 ZC
4
*GTG +IZ C#- KU ECNNGF CU VJG IGQOGVTKECN OQOGPV QH KPGTVKC YJGTG
#KUVJGVQVCNCTGCQHETQUUUGEVKQPQHVJGDGCOCPF-KUVJGTCFKWUQHVJG)[TCVKQP
#V GSWKNKDTKWO VJG DGPFKPI OQOGPV QH VJG DGCO KU GSWCN VQ VJG TGUVQTKPI
EQWRNG CEVKPI QP VJG DGCO
$GPFKPIOQOGPVQHVJGDGCO4GUVQTKPIEQWRNG
;+I
4
5RGEKCN %CUG
K (QT C DGCO QH TGEVCPIWNCT ETQUUUGEVKQP
+H NDO KU VJG DTGCVJ CPF NFO KU VJG VJKEMPGUU QH VJG DGCO VJGP
#TGC #DFCPF-
+I#-DF
+I
F
F
DF
5WDUVKVWVKPI VJG XCNWG QH +I KP GSWCVKQP ; DF
$GPFKPI OQOGPV HQT C DGCO QH TGEVCPIWNCT ETQUU UGEVKQP  
4 KK (QT C DGCO QH EKTEWNCT ETQUU UGEVKQP
(QT C DGCO QH EKTEWNCT ETQUU UGEVKQP KH NTO KU VJG TCFKWU VJGP
#TGC #TCPF-
+I#-T
T
T
T
5WDUVKVWVKPI VJG XCNWG QH +I KP GSWCVKQP +I
6JG DGPFKPI OQOGPV QH C EKTEWNCT ETQUU UGEVKQP 
;T
4
1.26
Engineering Physics
&$17,/(9(5
# DGCO HKZGF JQTK\QPVCNN[ CV QPG GPF CPF NQCFGF CV VJG QVJGT GPF
KU ECNNGF C ECPVKNGXGT
6JGQT[
%QPUKFGT C DGCO HKZGF CV QPG GPF CPF NQCFGF CV KVU QVJGT HTGG GPF KV DGPFU
CU UJQYP KP (KI &WG VQ VJG NQCF CRRNKGF CV VJG HTGG GPF C EQWRNG KU ETGCVGF
DGVYGGP VJG VYQ HQTEGU
K
(QTEG NQCF9OI CRRNKGF CV VJG HTGG GPF VQYCTFU FQYPYCTF FKTGEVKQP
CPF
KK
4GCEVKQP 4 CEVKPI KP VJG WRYCTF FKTGEVKQP CV VJG UWRRQTVKPI GPF
Fig. 1.13 Cantilever
6JKU GZVGTPCN DGPFKPI EQWRNG VGPFU VQ DGPF VJG DGCO KP VJG ENQEMYKUG
FKTGEVKQP 5KPEG QPG GPF QH VJG DGCO KU HKZGF VJG DGCO ECPPQV TQVCVG 6JGTGHQTG
VJG GZVGTPCN DGPFKPI EQWRNG OWUV DG DCNCPEGF D[ CPQVJGT GSWCN CPF QRRQUKVG
EQWRNG ETGCVGF FWG VQ VJG GNCUVKE PCVWTG QH VJG DQF[ 6JG OQOGPV QH VJKU GNCUVKE
EQWRNG KU ECNNGF KPVGTPCN DGPFKPI OQOGPV
9JGP VJG DGCO KU KP GSWKNKDTKWO
'ZVGTPCN DGPFKPI OQOGPV  +PVGTPCN DGPFKPI OQOGPV
1.8.1 Depression of a cantilever - loaded at its free ends.
%QPUKFGT C ECPVKNGXGT QH NGPIVJ NNO KU HKZGF CV QPG GPF YJGTGCU VJG QVJGT
GPF KU NQCFGF .GV '( DG VJG PGWVTCN CZKU QH VJG ECPVKNGXGT (KI 9JGP VJG
NQCF N9O KU CRRNKGF VJG GPF ( KU FGHNGEVGF VQ ( UQ VJCV VJG PGWVTCN CZKU VCMGU C
PGY RQUKVKQP '(
Properties of Matter
1.27
Fig. 1.14. Cantilever loaded at one end
#UUWOG VJCV VJG YGKIJV QH VJG DGCO FQGU PQV RTQFWEG CP[ DGPFKPI %QPUKFGT
C UGEVKQP 2 KP VJG ECPVKNGXGT CV C FKUVCPEG QH NZO HTQO VJG HKZGF GPF +V KU CV C
FKUVCPEG NZ HTQO VJG NQCFGF GPF ( KG 6JG FKUVCPEG 2(̀ 2(NZ
6JG DGPFKPI OQOGPV RTQFWEGF CV 292(
9NZ
6JG TGUVQTKPI HQTEG CEVKPI CV 2
9JGTG
;+I
4
+I KU VJG IGQOGVTKECN OQOGPV QH KPGTVKC
; KU VJG ;QWPIOU OQFWNWU QH VJG OCVGTKCN QH VJG DGCO
4 KU VJG TCFKWU QH EWTXCVWTG QH VJG PGWVTCN CZKU CV 2
#V GSWKNKDTKWO
$GPFKPI OQOGPV  4GUVQTKPI HQTEG
9NZ
4
;+I
4
;+I
9NZ
%QPUKFGT CPQVJGT UGEVKQP 3 PGCTGT VQ 2 5KPEG VJG UGEVKQPU 2 CPF 3 NKG XGT[
PGCTGT VJG TCFKWU QH EWTXCVWTG QH VJGUG VYQ UGEVKQPU YKNN DG VJG UCOG
.GV 4 DG VJG TCFKWU QH EWTXCVWTG CPF F DG VJG CPING DGVYGGP VJG TCFKWU
QH EWTXCVWTGU CV 2 CPF 3 +H FZ KU VJG FKUVCPEG DGVYGGP 2 CPF 3 VJGP
1.28
Engineering Physics
6JG CTE NGPIVJ 23FZ4F
 4
FZ
F
5WDUVKVWVKPI GSWCVKQP KP GSWCVKQP YG IGV
;+I
9NZ

F

FZ
F
9NZ
FZ
;+I
&TCY VCPIGPV CV 2 CPF 3 6JG VCPIGPVU YKNN OGGV VJG NKPG (( CV % CPF &
*GPEG VJG FGRTGUUKQP F[ QH 3 DGNQY 2 KU
F[NZF
(TQO GSWCVKQP CPF GSWCVKQP YG IGV
F[NZ
F[
9NZ
;+I
9NZ
FZ
;+I
 6QVCN FGRTGUUKQP [(( CV VJG HTGG GPF QH VJG ECPVKNGXGT ECP DG FGTKXGF
D[ KPVGITCVKPI VJG GSWCVKQP YKVJKP VJG NKOKVU VQ N
N
9
NZFZ
 6QVCN FGRTGUUKQP [
;+I
N
9

NNZZFZ
;+I 
N
9  NZ Z 
 

N Z

;+I 

6QVCN FGRTGUUKQP
[
N 
9  N
N   
;+I 

N
9

;+I  &GRTGUUKQP QH VJG ECPVKNGXGT CV HTGG GPF [
9N
;+I
Properties of Matter
1.29
5RGEKCN ECUGU
K (QT C DGCO QH TGEVCPIWNCT ETQUU UGEVKQP
+P VJG ECUG QH TGEVCPIWNCT ETQUU UGEVKQP
+I
DF
9JGTG NDO KU VJG DTGCVJ CPF F KU VJG VJKEMPGUU QH VJG DGCO 5WDUVKVWVKPI VJG
XCNWG QH +I KP GSWCVKQP YG ECP YTKVG
6JG FGRTGUUKQP RTQFWEGF CV HTGG GPF HQT C TGEVCPIWNCT ETQUU UGEVKQP
[

9N
 DF 
;

 
9N
;DF
[
9N
;DF KK (QT C DGCO QH EKTEWNCT ETQUU UGEVKQP
+H NTO KU VJG TCFKWU QH VJG EKTEWNCT ETQUU UGEVKQP VJGP
+I
T
5WDUVKVWVKPI VJG XCNWG QH +I KP GSWCVKQP YG ECP YTKVG
6JG FGRTGUUKQP RTQFWEGF CV HTGG GPF HQT C EKTEWNCT ETQUU UGEVKQP
[
9N
 T 
;

 
[
9N
;T
1.30
Engineering Physics
1.8.2 Experimental Determination of Young’s Modulus by Cantilever
Depression
&GUETKRVKQP
1PG GPF QH VJG IKXGP ECPVKNGXGT KU HKZGF KP C YCNN CPF VJG QVJGT GPF KU NGHV
HTGG # RKP KU HKZGF XGTVKECNN[ CV VJG HTGG GPF QH VJG DGCO D[ OGCPU QH C XCZ #
NQCFQH9MIKUUWURGPFGFCVVJGHTGGGPFQHVJGECPVKNGXGT#VTCXGNNKPIOKETQUEQRG
KU HQEWUGF QP VJG VKR QH VJG RKP CPF KV KU WUGF VQ HKPF VJG FGRTGUUKQP QH VJG
ECPVKNGXGT
2TQEGFWTG
6JG YGKIJV JCPIGT KU MGRV JCPIGF KP C FGCF NQCF RQUKVKQP 9 KG YKVJQWV
CP[ UNQVVGF YGKIJVU 6JG OKETQUEQRG KU CFLWUVGF UWEJ VJCV VJG JQTK\QPVCN ETQUUYKTG
EQKPEKFGU YKVJ VJG VKR QH VJG KOCIG QH VJG RKP CPF VJG TGCFKPI QP VJG XGTVKECN
UECNG QH VJG OKETQUEQRG KU PQVGF
6JG YGKIJVU CTG CFFGF KP UVGRU QH I VQ VJG YGKIJV JCPIGT 6JG OKETQUEQRG
KU CFLWUVGF GCEJ VKOG UQ VJCV VJG JQTK\QPVCN ETQUUYKTG EQKPEKFGU YKVJ VJG KOCIG
QH VJG VKR QH VJG RKP 6JG TGCFKPI QP VJG XGTVKECN UECNG QH VJG OKETQUEQRG KU PQVGF
KP GCEJ ECUG 6JG QDUGTXCVKQP CTG TGRGCVGF HQT FGETGCUKPI NQCFU CNUQ CPF VJG
TGCFKPIU CTG VCDWNCVGF CU HQNNQYU
5N
0Q
.QCF
:-I
9
9 IOU
9 IOU
9 IOU
9 IOU
9 IOU
/KETQUEQRG TGCFKPIU O
NQCF
NQCF
KPETGCUKPI FGETGCUKPI
OGCP
&GRTGUUKQP [
HQT C NQCF QH /
-I O
/GCP FGRTGUUKQP [O
6JG OGCP FGRTGUUKQP [ CV VJG GPF QH VJG ECPVKNGXGT HQT C NQCF QH / -I KU
HQWPF QWV
Properties of Matter
1.31
6JG ;QWPIOU OQFWNWU QH VJG IKXGP ECPVKNGXGT KU FGVGTOKPGF WUKPI VJG TGNCVKQP
;
9N
[+I
;
OIN
[+I
UWDUVKVWVKPI 9OI YG IGV
YJGTG
[
KU VJG FGRTGUUKQP RTQFWEGF
+I KU VJG IGQOGVTKECN OQOGPV QH KPGTVKC
O KU VJG OCUU CPF
N
KU VJG NGPIVJ QH VJG ECPVKNGXGT
6JG IGQOGVTKECN OQOGPV QH KPGTVKC QH VJG TGEVCPIWNCT DGCO KU IKXGP D[
+I
DF
YJGTG NDO KU VJG DTGCFVJ CPF NFO KU VJG VJKEMPGUU QH VJG DGCO
UWDUVKVWVKPI GSWCVKQP KP GSWCVKQP YG IGV
;

;
OIN
 DF 
[

 
OIN
[DF
IN  / 
 0OQT2CUECN
DF  [ 
$[ UWDUVKVWVKPI CNN VJG XCNWGU VJG ;QWPIOU OQFWNWU QH VJG IKXGP ECPVKNGXGT
KU FGVGTOKPGF
<281*·6 02'8/86 81,)250 %(1',1*
&GHKPKVKQP
9JGP C DGCO KU UWRRQTVGF U[OOGVTKECNN[ QP VYQ MPKHG GFIGU CPF
NQCFGF YKVJ GSWCN YGKIJVU CV GCEJ GPFU VJG DGPF DGCOU HQTO CP CTE QH
C EKTENG 6JG GNGXCVKQP KP VJG DGCO KU RTQFWEGF 6JKU DGPFKPI KU ECNNGF
WPKHQTO DGPFKPI
1.32
Engineering Physics
6JGQT[
%QPUKFGT C DGCO %& UWRRQTVGF U[OOGVTKECNN[ QP VYQ MPKHG GFIGU CV # CPF
$ KP JQTK\QPVCN NGXGN CU UJQYP KP HKI .GV NNO DG VJG FKUVCPEG DGVYGGP VJG
MPKHG GFIGU .GV GSWCN YGKIJVU 9 DG CFFGF VQ VJG DGCO CV VJG GPFU % CPF &
Fig. 1.15. Youngs Modulus - Uniform Bending
.GV %#$&C 9JGTG NCO DG VJG NGPIVJ DGVYGGP VJG MPKHG GFIG CPF VJG
NQCF &WG VQ VJG NQCF CRRNKGF VJG DGCO DGPFU HTQO RQUKVKQP ' VQ ( KPVQ CP CTE
QH C EKTENG CPF RTQFWEGU CU GNGXCVKQP N[O HTQO RQUKVKQP ' VQ (
%QPUKFGT C RQKPV N2O QP VJG ETQUU UGEVKQP QH VJG DGCO #V VJG GSWKNKDTKWO
RQUKVKQP QH VJG UGEVKQP N2%O QH VJG DGCO VYQ GSWCN HQTEGU VJG CRRNKGF NQCF 9 CV
% FQYPYCTF CPF VJG PQTOCN TGCEVKQP 9 CV # WRYCTF CTG CEVKPI KP VJG QRRQUKVG
FKTGEVKQP EQPUVKVWVG C EQWRNG CU UJQYP KP (KI Fig. 1.16
Properties of Matter
1.33
.GV VJG FKUVCPEG 2%CCPF2#C VJGP
6JG GZVGTPCN DGPFKPI OQOGPV CDQWV N2O KU
92%92#
9C9C
9CC
9C
*GTG VJG ENQEMYKUG OQOGPV KU VCMGP CU PGICVKXG CPF CPVKENQEMYKUG OQOGPV
KU VCMGP CU RQUKVKXG
6JKU OWUV DG DCNCPEGF D[ KPVGTPCN DGPFKPI OQOGPV HQT GSWKNKDTKWO
6JG KPVGTPCN DGPFKPI OQOGPV 
9JGTG
;+I
4
; KU ;QWPIOU OQFWNWU QH VJG DGCO
+I KU VJG IGQOGVTKECN OQOGPV QH KPGTVKC
4 KU VJG TCFKWU QH EWTXCVWTG QH VJG CTE %2
7PFGT GSWKNKDTKWO EQPFKVKQP
'ZVGTPCN DGPFKPI OQOGPV  +PVGTPCN DGPFKPI OQOGPV
9C
;+I
4
5KPEG HQT C IKXGP NQCF 9;C+I CPF 4 CTG EQPUVCPVU
*GPEG VJGDGCOKUDGPFKPVQCPCTEQHCEKTENGYKVJUCOGTCFKWUQHEWTXCVWTG
4 CV CNN RQKPVU 6JG DGPFKPI KU VJGP UCKF VQ DG WPKHQTO
+H NNO KU VJG FKUVCPEG DGVYGGP VJG MPKHG GFIGU CPF [ KU VJG GNGXCVKQP QH VJG
OKFRQKPV ' QH VJG DGCO CDQXG VJG MPKHG GFIGU VJGP HTQO VJG RTQRGTV[ QH VJG
EKTENG HKI (TQO VJG #'1
1##''1
UKPEG
1''(
1##''(
1.34
Engineering Physics
Fig. 1.17 Property of the circle
#'1#'(
 1#

 '(
'(
 '(

4
*GTG #'N'([ 1#4
 'SWCVKQP ECP DG YTKVVGP CU
 4

[
N[
 4


 4
N
[
[

 4
N
[=4[?
N
4[[
5KPEG VJG GNGXCVKQP N [O KU XGT[ UOCNN VJGP VJG VGTO [ ECP DG PGINGEVGF
N
 [4
QT [
N
4
Properties of Matter
1.35
N
 4CFKWU QH VJG EWTXCVWTG 4
[
5WDUVKVWVKPI VJG XCNWG QH 4 KP GSWCVKQP YG JCXG
;+I
9C N [
9C
;
;+I[
N
9CN
[+I
(QT C TGEVCPIWNCT DGCO
+I
9JGTG
DF
D
KU VJG DTGCFVJ QH VJG DGCO
F
KU VJG VJKEMPGUU QH VJG DGCO
UWDUVKVWVKPI VJG XCNWG QH +I CPF 9OI KP GSWCVKQP YG IGV
;

OICN
[DF
OICN
[DF
OICN
; [DF
$[ OGCUWTKPI VJG GNGXCVKQP [ CV VJG EGPVTG QH VJG DGCO VJG ;QWPIOU OQFWNWU
; QH VJG DGCO ECP DG ECNEWNCVGF
1.9.1 Experimental Determination of Young’s Modulus by Uniform Bending
&GUETKRVKQP
6JG IKXGP TGEVCPIWNCT DGCO KU U[OOGVTKECNN[ UWRRQTVGF QP VYQ MPKHG GFIGU
# CPF $ KP C JQTK\QPVCN NGXGN .GV NNO DG VJG FKUVCPEG DGVYGGP VJG MPKHG GFIGU
# CPF $ 6YQ YGKIJV JCPIGTU QH GSWCN OCUUGU CTG UWURGPFGF HTQO VJG GPFU QH
VJG DGCO %CPF& UWEJ VJCV #%$&C # RKP KU HKZGF XGTVKECNN[ CV VJG EGPVTG
QH VJG DGCO CU UJQYP KP HKI 1.36
Engineering Physics
Fig. 1.18 Young’s Modulus - Uniform Bending
6JG RKP KU HQEWUUGF D[ C VTCXGNNKPI OKETQUEQRG RNCEGF KP HTQPV QH VJG DGCO
2TQEGFWTG
6CMKPI VJG YGKIJV JCPIGT CV GPFU % CPF & KU MGRV JCPIGF KP C FGCF NQCF
RQUKVKQP 9 KG YKVJQWV CP[ UNQVVGF YGKIJVU 6JG OKETQUEQRG KU CFLWUVGF UWEJ
VJCV VJG JQTK\QPVCN ETQUUYKTG EQKPEKFGU YKVJ VJG VKR QH VJG KOCIG QH VJG RKP CPF
VJG TGCFKPI QP VJG XGTVKECN UECNG QH VJG OKETQUEQRG KU PQVGF
'SWCN YGKIJVU UC[ IO CTG CFFGF VQ VJG JCPIGTU CV GPFU % CPF &
UKOWNVCPGQWUN[ CPF VJG EQTTGURQPFKPI OKETQUEQRG TGCFKPIU CTG PQVGF CHVGT
CFLWUVKPI VJG OKETQUEQRG GCEJ VKOG 6JG QDUGTXCVKQPU CTG TGRGCVGF YJKNG
FGETGCUKPIVJGNQCFKPVJGUCOGGSWCNUVGRU6JGTGCFKPIUCTGVCDWNCVGFCUHQNNQYU
.QCF
5N
0Q -I
9
9
9
9
9
9
/KETQUEQRG 4GCFKPI O
.QCF
.QCF
KPETGCUKPI FGETGCUKPI
/GCP
'NGXCVKQP [ HQT
VJG NQCF QH / -I
O
/GCP GNGXCVKQP [O
Properties of Matter
1.37
6JG OGCP GNGXCVKQP [ CV VJG EGPVTG QH VJG DGCO HQT C NQCF QH / -I KU
HQWPF QWV
6JG NGPIVJ QH VJG DGCO NNO DGVYGGP VJG MPKHG GFIGU CPF VJG FKUVCPEG DGVYGGP
VJG RQKPV QH UWURGPUKQP QH VJG NQCF CPF VJG PGCTGUV MPKHG GFIGU NCO CTG OGCUWTGF
6JG DTGCFVJ D CPF VJKEMPGUU F QH VJG DGCO CTG OGCUWTGF WUKPI XGTPKGT ECNKRGTU
CPF UETGY ICWIG TGURGEVKXGN[ VJGP
9G MPQY VJG GNGXCVKQP [ RTQFWEGF KU
[
9CN
;+I
UWDUVKVWVKPI 9OIYGIGV [
YJGTG
[
OICN
;+I
KU VJG GNGXCVKQP RTQFWEGF
O KU VJG OCUU
N
KU VJG NGPIVJ QH VJG DGCO DGVYGGP VJG MPKHG GFIGU
+I KU VJG IGQOGVTKECN OQOGPV QH KPGTVKC
6JG IGQOGVTKECN OQOGPV QH KPGTVKC QH VJG TGEVCPIWNCT DGCO KU IKXGP D[
DF
9JGTG NDO KU VJG DTGCFVJ CPF NFO KU VJG VJKEMPGUU QH VJG DGCO
+I
5WDUVKVWVKPI GSWCVKQP KP GSWCVKQP 9G IGV
[

[
OICN
;DF
OICN
;DF
OICN
;DF
4GCTTCPIKPI GSWCVKQP YG ECP YTKVG VJG ;QWPIOU OQFWNWU
;
ICN  O 
0OQT2CUECN
 [ 
DF 

$[ UWDUVKVWVKPI CNN VJG XCNWGU VJG ;QWPIOU OQFWFNWU QH VJG OCVGTKCN QH VJG
IKXGP DGCO KU FGVGTOKPGF
1.38
Engineering Physics
12181,)250 %(1',1*
+H VJG DGCO KU NQCF CV KVU OKFRQKPV VJG FGRTGUUKQP RTQFWEGF YKNN PQV
HQTO CP CTE QH C EKTENG 6JKU V[RG QH DGPFKPI KU ECNNGF PQPWPKHQTO DGPFKPI
1.10.1 Determination of Young’s modulus by Non-uniform bending
(Theory and experiments)
6JGQT[
.GV WU EQPUKFGT C DGCO QH NGPIVJ NNO UWRRQTVGF CV VYQ MPKHG GFIGU N#O CPF
N$O CU UJQYP KP HKI Fig. 1.19 Non-uniform bending
# YGKIJV N9O KU CRRNKGF CV VJG OKFRQKPV N'O QH VJG DGCO &WG VQ VJG NQCF
9 CRRNKGF VJG TGCEVKQP CV GCEJ MPKHG GFIG KU GSWCN VQ 9 KP VJG WRYCTF
FKTGEVKQP CPF N[O KU VJG FGRTGUUKQP CV VJG OKFRQKPV N'O
6JG DGPF DGCO KU EQPUKFGTGF VQ DG GSWKXCNGPV VQ VYQ UKPING KPXGTVGF
ECPVKNGXGTU HKZGF CV ' GCEJ QH NGPIVJ N CPF GCEJ NQCFGF CV # CPF $ YKVJ C
YGKIJV 9
+P VJG ECUG QH C ECPVKNGXGT QH NGPIVJ NNO CPF NQCF 9
6JG FGRTGUUKQP 
9N
+I;
*GPEG HQT ECPVKNGXGT QH NGPIVJ N CPF NQCF 9
Properties of Matter
1.39
6JG FGRTGUUKQP KU
9 N 
  

 
[
+I;
9N
[
+I;
QT +H / KU VJG OCUU VJG EQTTGURQPFKPI YGKIJV 9 KU
9/I
+H VJG DGCO KU C TGEVCPIWNCT
DF
YJGTG D KU VJG DTGCVJ CPF F KU VJG VJKEMPGUU QH VJG DGCO
5WDUVKVWVKPI GSWCVKQP CPF KP GSWCVKQP YG IGV
+I
[
1T [
/IN
[
DF
;
/IN
DF;
/IN
DF;
4GCTTCPIKPI GSWCVKQP YG ECP YTKVG
/IN
;QWPIOU /QFWNWU ;
0OQT2CUECN
DF [
$[ UWDUVKVWVKPI CNN VJG XCNWGU VJG ;QWPIOU OQFWNWU QH VJG OCVGTKCN QH VJG
IKXGP DGCO KU FGVGTOKPGF
'ZRGTKOGPV
6JG IKXGP DGCO QH TGEVCPIWNCT ETQUU UGEVKQP KU U[OOGVTKECNN[ UWRRQTVGF QP
VYQ MPKHG GFIGU # CPF $ CU UJQYP KP HKI # YGKIJV JCPIGT KU UWURGPFGF D[ OGCPU QH C NQQR QH VJTGCF HTQO VJG EGPVTG
' # RKP KU HKZGF XGTVKECNN[ CV ' D[ UQOG XCZ # VTCXGNNKPI OKETQUEQRG KU HQEWUUGF
QP VJG VKR QH VJG RKP UWEJ VJCV VJG JQTK\QPVCN ETQUUYKTG EQKPEKFGU YKVJ VJG VKR
QH VJG RKP 6JG KPKVKCN TGCFKPI QP VJG XGTVKECN UECNG QH VJG OKETQUEQRG KU VCMGP
# UWKVCDNG OCUU / KU CFFGF VQ VJG JCPIGT 6JG DGCO KU FGRTGUUGF 6JG
ETQUU YKTG KU CFLWUVGF VQ EQKPEKFG YKVJ VJG VKR QH VJG RKP 6JG TGCFKPI QH VJG
OKETQUEQRG KU PQVGF 6JG FGRTGUUKQP EQTTGURQPFKPI VQ VJG OCUU / KU HQWPF 6JG
GZRGTKOGPV KU TGRGCVGF D[ KPETGCUKPI CPF FGETGCUKPI VJG OCUU UVGR D[ UVGR
1.40
Engineering Physics
Fig. 1.20 Non-uniform Bending
6JG EQTTGURQPFKPI TGCFKPIU CTG VCDWNCVGF 6JG CXGTCIG XCNWG QH FGRTGUUKQP
[ KU HQWPF HTQO VJG QDUGTXCVKQP
.QCF
5N
0Q -I
9
9
9
9
9
9
/KETQUEQRG 4GCFKPI O
.QCF
.QCF
KPETGCUKPI FGETGCUKPI
/GCP
'NGXCVKQP [ HQT
VJG NQCF QH / -I
O
/GCP GNGXCVKQP [O
6JG NGPIVJ QH VJG DGCO NNO DGVYGGP VJG MPKHGGFIGU KU OGCUWTGF 6JG DTGCFVJ
NDO CPF VJG VJKEMPGUU FO QH VJG DGCO CTG OGCUWTGF YKVJ C XGTPKGT ECNKRGTU CPF
UETGY ICWIG TGURGEVKXGN[
6JG;QWPIOUOQFWNWUQHVJGOCVGTKCNQHVJGDGCOKUECNEWNCVGFWUKPIVJGHQTOWNC
;
/IN
DF [
Properties of Matter
1.41
,6+$3(' *,5'(56
&GHKPKVKQP
6JG IKTFGTU YKVJ WRRGT CPF NQYGT UGEVKQP DTQCFGPGF CPF VJG OKFFNG
UGEVKQP VCRGTGF UQ VJCV KV ECP DG YKVJUVCPF JGCX[ NQCFU QXGT KV 6JG
ETQUU UGEVKQP QH VJG IKTFGTU VCMGU VJG UJCRG QH VJG ECRKVCN NGVVGT + KU ECNNGF
CU + UJCRG IKTFGTU
# IKTFGT YKNN DGPF WPFGT KVU QYP YGKIJV CPF VJG NQCF RNCEGF QXGT KV #
UOCNN FGRTGUUKQP KU RTQFWEGF CV VJG OKFFNG RQTVKQP +V KU UKOKNCT VQ PQPWPKHQTO
DGPFKPI 6JG FGRTGUUKQP RTQFWEGF KP VJG ECUG QH VJG PQPWPKHQTO DGPFKPI HQT C
TGEVCPIWNCT DGCO KU IKXGP D[
[
YJGTG
9N
DF;
9 KU VJG NQCF CRRNKGF
N
KU VJG NGPIVJ QH VJG IKTFGT
[
KU VJG FGRTGUUKQP RTQFWEGF
D
KU VJG DTGCFVJ
F
KU VJG VJKEMPGUU QH VJG IKTFGT
Fig. 1.21 I - shaped girders
1.42
Engineering Physics
'SWCVKQP UJQYUVJCVVJGFGRTGUUKQPKUFKTGEVN[RTQRQTVKQPCNVQNKPXGTUGN[
RTQRQTVKQPCN VQ VJG DTGCFVJ D CPF KPXGTUGN[ RTQRQTVKQPCN VQ VJG EWDKE RQYGT QH
VJG VJKEMPGUU F 6JKU UJQYU VJCV VJG FGRTGUUKQP ECP DG OKPKOK\GF D[ TGFWEKPI
VJG NGPIVJ N CPF D[ KPETGCUKPI VJG VJKEMPGUU CPF VJG DTGCFVJ QH VJG IKTFGT
&WG VQ FGRTGUUKQP QH VJG IKTFGT VJG WRRGT RCTVU CDQXG VJG PGWVTCN CZKU
EQPVTCEVU CPF VJG NQYGT RCTVU DGNQY VJG PGWVTCN CZKU GZVGPFU *GPEG VJG UVTGUUGU
JCXG OCZKOWO XCNWG CV VJG VQR CPF DQVVQO UWTHCEGU QH VJG IKTFGTU 6JGUG UVTGUUGU
RTQITGUUKXGN[ FGETGCUGU CU YG OQXG HTQO VJG VQR CPF DQVVQO VQYCTFU VJG OKFFNG
NC[GT 6JG UVTGUUGU CTG PGINKIKDN[ UOCNN CV VJG OKFFNG NC[GT 6JGTGHQTG VJG WRRGT
CPF NQYGT RCTVU QH VJG IKTFGT OWUV DG UVTQPIGT YKVJ DTQCFGT UWTHCEGU
+V KU UWHHKEKGPV KH VJG IKTFGTU KU FGUKIPGF YKVJ NGUU COQWPV QH OCVGTKCN KP
VJG OKFFNG NC[GT VJGP VJQUG KP VJG WRRGT CPF NQYGT NC[GTU 6JKU FGUKIP VJWU IKXGU
+ NGVVGT UJCRG QH VJG IKTFGT HKI &WG VQ VJKU UJCRG VJG OCVGTKCN YKNN DG
UCXGF YKVJQWV TGFWEKPI VJG UVTGPIVJ QH VJG OCVGTKCN
#FXCPVCIGU QH + UJCRGF IKTFGTU
6JG OCVGTKCN ECP DG UCXGF
6JG UVTGPIVJ QH VJG IKTFGT KU PQV CHHGEVGF
6JG UVCDKNKV[ CPF VJG GHHKEKGPE[ QH VJG IKTFGTU CTG OCKPVCKPGF
#RRNKECVKQPU QH + UJCRG IKTFGTU
6JG[ CTG WUGF CU UWRRQTVKPI DGCOU HQT VJG EGKNKPIU KP VJG EQPUVTWEVKQP
QH DWKNFKPIU
6JG[ CTG WUGF KP VJG EQPUVTWEVKQP QH DTKFIGU QXGT VJG TKXGTU
5KPEG KV OQTG UVCDNG UVTQPIGT CPF FWTCDNG KV KU CRRNKGF KP VJG
EQPUVTWEVKQP QH KTQP DGCOU VQ UWRRQTV VJG DTKFIGU HQT VJG JGCX[ XGJKENGU
CPF CNUQ KP VJG EQPUVTWEVKQP QH FCOU
6JG[ CTG XGT[ OWEJ WUGHWN KP VJG RTQFWEVKQP QH KTQP TCKNU YJKEJ CTG
GORNQ[GF KP TCKNYC[ VTCEMU
7:,67,1* &283/( 2) $ :,5( 25 &</,1'(5
.GV WU EQPUKFGT C E[NKPFTKECN YKTG #$ QH NGPIVJ NNO CPF TCFKWU NCO DG HKZGF CV
KVU WRRGT GPF 9JGP C EQWRNG KU CRRNKGF CV KVU NQYGT GPF KP C RNCPG RGTRGPFKEWNCT
VQ VJG NGPIVJ QH VJG E[NKPFGT KVU GCEJ EKTEWNCT ETQUUUGEVKQP IGVU TQVCVGF CDQWV VJG
CZKU QH VJG YKTG 6JKU KU ECNNGF VYKUVKPI QH VJG E[NKPFGT CPF VJG CPING VJTQWIJ
YJKEJ VJG TCFKWU QH GCEJ ETQUU UGEVKQP TQVCVGU KU ECNNGF VJG CPING QH VYKUV 6JKU
CPING QH VYKUV NO KU RTQRQTVKQPCN VQ FKUVCPEG QH ETQUUUGEVKQP HTQO VJG ENCORGF GPF
Properties of Matter
1.43
KG  KU OCZKOWO CV VJG HTGG GPF CPF FGETGCUGU VQYCTFU VJG ENCORGF GPF CU
UJQYP KP HKI 9G UJGNN HKPF VJG EQWRNG TGSWKTGF KP VYKUVKPI VJG YKTG VJTQWIJ CPING 
Fig. 1.22 Twisting couple of a cylinder
6JG YKTG ECP DG CUUWOGF VQ DG OCFG WR QH CP KPHKPKVG PWODGT QH EQCZKCN
E[NKPFGTU .GV VJG KPPGT CPF QWVGT TCFKK QH QPG UWEJ E[NKPFGT T CPF TFT
9JGP VJG E[NKPFGT KU VYKUVGF VJTQWIJ CP CPING  VJG TCFKWU $3 YKNN TQVCVG
VJTQWIJ VJG UCOG CPING CPF OQXG VQ VJG RQUKVKQP $3 6JG NQPIKVWFKPCN NKPG
23 OQXGU VQ C PGY RQUKVKQP 23 UWEJ VJCV VJG CPING 323 KU  #V GSWKNKDTKWO
RQUKVKQP VJG VYKUVKPI EQWRNG KU GSWCN CPF QRRQUKVG VQ VJG TGUVQTKPI EQWRNG
(TQO HKIWTG

33
33
CPF
T
N
33NT
#PING QH UJGCT 
 4KIKFKV[ OQFWNWU 
T
N
5JGCTKPIUVTGUU
#PINGQHUJGCT
 5JGCTKPI UVTGUU 0
1.44
Engineering Physics
5WDUVKVWVKPI HQT  HTQO GSP KP GSP YG JCXG
5JGCTKPI UVTGUU 
0T
N
9G MPQY UJGCTKPI UVTGUU 
5JGCTKPIHQTEG
#TGC
5JGCTKPI #TGCQPYJKEJVJGUJGCTKPI


HQTEGKUCEVKPI
UVTGUU
5JGCTKPI HQTEG
KG (
0T
TFT
N
(
0
TFT
N
6JKU HQTEG CEVU CV GXGT[ RQKPV QP VJG ETQUU UGEVKQP QH VJG E[NKPFTKECN YKTG
VCPIGPVKCN VQ VJG EKTEWOHGTGPEG PQTOCN VQ VJG TCFKWU CV VJG RQKPV
 /QOGPV QH VJG HQTEG CDQWV VJG #$ CZKU QH VJG E[NKPFGT
 5JGCT HQTEG  2GTRGPFKEWNCT FKUVCPEG
/QOGPV QH VJG HQTEG 

0
TFTT
N
0 T FT
N
6QVCN OQOGPV QH VJG HQTEG CEVKPI QP VJG GPVKTG E[NKPFTKECN YKTG QH TCFKWU
NCO KU FGVGTOKPGF D[ KPVGITCVKPI HTQO VQ C
6QVCN OQOGPV QH VJG HQTEG CEVKPI QP VJG YKTG
C
0
%
TFT
N
C
0  T 
 

N
 

%
0 C

N
0C
N
YJGP CPIWNCT VYKUV  TCFKCP VJGP
Properties of Matter
1.45
6JG VYKUVKPI EQWRNG RGT WPKV CPIWNCT VYKUV KU
0C
%
N
'SWCVKQP IKXGU VJG EQWRNG RGT WPKV VYKUV QH YKTG
7256,21$/ 3(1'8/80
# DQF[ UWURGPFGF D[ C YKTG YJKEJ VYKUVU HKTUV KP QPG FKTGEVKQP CPF
VJGP KP VJG TGXGTUG FKTGEVKQP KP VJG JQTK\QPVCN RNCPG KU ECNNGF VQTUKQPCN
RGPFWNWO 6JG HKTUV VQTUKQP RGPFWNWO YCU FGXGNQRGF D[ 4QDGTV .GUNKG KP 6QTUKQPCN RGPFWNWO KU C RGPFWNWO YJKEJ RGTHQTOU VQTUKQPCN
QUEKNNCVKQPU +V KU WUGF VQ FGVGTOKPG VJG OQOGPV QH KPGTVKC CPF VQTUKQPCN
TKIKFKV[ QH C IKXGP DQF[
2TKPEKRNG
9JGP VJG FKUE KU VWTPGF KP VJG JQTK\QPVCN RNCPG UQ CU VQ VYKUV VJG
YKTG CPF VJGP TGNGCUGF KV GZGEWVGU VQTUKQPCN XKDTCVKQPU CDQWV VJG YKTG
CU CP CZKU 6JG VKOG RGTKQF NVO QH VJG VQTUKQPCN XKDTCVKQP KU IKXGP D[


V
+
%
YJGTG N+O KU VJG OQOGPV QH KPGTVKC QH VJG YKTG CPF N%O KU VJG EQWRNG RGT
WPKV CPIWNCT VYKUV QH VJG YKTG
&GUETKRVKQP
+V EQPUKUV QH C WPKHQTO EKTEWNCT FKUE YKVJ C EJWEM CV VJG EGPVTG (KI 6JG GZRGTKOGPVCN YKTG YJQUG TKIKFKV[ OQFWNWU KU VQ DG FGVGTOKPGF ECP DG HKTON[
Fig. 1.23 Torsional Pendulam
1.46
Engineering Physics
ITKRRGF D[ VJKU EJWEM 6JG QVJGT GPF QH VJG YKTG ECP DG ITKRRGF QP CP CPQVJGT
EJWEM YJKEJ KU HKZGF QP C TKIKF UVCPF 6JG NGPIVJ QH VJG YKTG DGVYGGP VJG EJWEMU
ECP DG CFLWUVGF
9QTMKPI
9JGP VJG FKUE KU TQVCVGF YKVJ UOCNN CORNKVWFG VJG VQTUKQPCN RGPFWNWO YKNN
GZGEWVG VQTUKQPCN QUEKNNCVKQPU KP C JQTK\QPVCN RNCPG 9JGP VJG FKUE KU TQVCVKPI VJG
YKTG IGV VYKUVGF .GV NO DG VJG CPING QH VYKUVKPI # TGUVQTKPI HQTEG KU CNUQ CEVKPI
QP VJG YKTG 6JG TGUVQTKPI HQTEG VTKGU VQ DTKPI VJG YKTG KPVQ KVU QTKIKPCN RQUKVKQP
6JGTGHQTG VJG YKTG CPF VJG FKUE YKNN TQVCVG KP VJG QRRQUKVG FKTGEVKQP FWG VQ VJG
TGUVQTKPI HQTEG CPF JGPEG VJG YKTG YKNN GZRGTKGPEG C TGUVQTKPI EQWRNG 6JG YKTG
KU VYKUVGF CICKP CPF VJG UCOG VTCPUHQTOCVKQP KU TGRGCVGF 6JWU VJG FKUE UVCTVU
VQ QUEKNNCVGU KP VJG JQTK\QPVCN RNCPG CDQWV VJG YKTG CU CP CZKU 6JGUG
QUEKNNCVKQPU CTG ECNNGF VQTUKQPCN QUEKNNCVKQPU
 6JG TGUVQTKPI EQWRNG %
YJGTG % KU TGUVQTKPI EQWRNG RGT WPKV VYKUV 6JG OKPWU UKIP KU RWV CU VJG EQWRNG
KU FKTGEVGF QRRQUKVG VQ VJG VYKUV  NO KU VJG CPIWNCT FKURNCEGOGPV QT VYKUV CV CP[
KPUVCPV NVO
6JG CPIWNCT CEEGNGTCVKQP KP VJG YKTG KU FWG VQ VJG TGUVQTKPI EQWRNG
C
F
FV
+H + DG VJG OQOGPV QH KPGTVKC QH VJG FKUE CDQWV VJG YKTG CU CP CZKU VJGP
F
VJG KPVGTPCN VQTSWG CRRNKGF QP VJG YKTG + FV
6JGP VJG GSWCVKQP QH VJG FKUE KU
+
+
QT QT F
FV
F
FV
%
%
F
%
 
+
FV
F
FV

Properties of Matter
YJGTG
1.47
%
 C EQPUVCPV
+
6JWU VJG CPIWNCT CEEGNGTCVKQP KU RTQRQTVKQPCN VQ VJG VYKUV QT VJG CPIWNCT
FKURNCEGOGPV *GPEG VJG OQVKQP KU UKORNG JCTOQPKE UQ VJCV KVU VKOG RGTKQF KU
IKXGP D[


6
&KURNCEGOGPV
#EEGNGTCVKQP
5WDUVKVWVKPI VJG XCNWGU QH CEEGNGTCVKQP CPF FKURNCEGOGPV KP VJG CDQXG
GSWCVKQP YG IGV


%
 + 
 
6

%
+
6
'SWCVKQP IKXGU VJG RGTKQF QH VQTUKQPCN RGPFWNWO
+H NTO KU VJG TCFKWU QH VJG YKTG CPF NNO KU VJG NGPIVJ QH VJG YKTG UWURGPFGF
VJGP YG MPQY
6JG VYKUVKPI EQWRNG RGT WPKV CPIWNCT VYKUV %
0T
N
5WDUVKVWVKPI GSP KP GSP YG IGV


6
6
+N
0T
N+
0T
QT 4KIKFKV[ OQFWNWU QH VJG YKTG 0
+N
6T
0O
1.48
Engineering Physics
1.12.1 Determination of Rigidity modulus of the wire using Torsional
Pendulum
%QPUKFGT C VQTUKQPCN RGPFWNWO EQPUKUVKPI QH C EKTEWNCT FKUE +V KU UWURGPFGF
WUKPI C YKTG QH NGPIVJ N 1PG ECP FGVGTOKPG VJG VQTUKQPCN TKIKFKV[ OQFWNWU QH VJG
IKXGP YKTG D[ OGCUWTKPI VJG VKOG RGTKQF QH VJG FKUE WPFGT VJTGG EQPFKVKQPU
YKVJQWV CP[ CFFKVKQPCN OCUUGU
VYQ GSWCN OCUUGU CV C FKUVCPEG F HTQO VJG EGPVTG QH VJG FKUE CPF
VYQ GSWCN OCUUGU CV C FKUVCPEG F HTQO VJG EGPVTG QH VJG FKUE
6JG UEJGOCVKE TGRTGUGPVCVKQP QH VJG RGPFWNWO KU UJQYP KP HKI 6JG VKOG RGTKQF 6 QH VJG FKUE YKVJ CFFKVKQPCN OCUU KU


6
YJGTG
+
%
+ KU VJG OQOGPV QH KPGTVKC QH VJG FKUE
% KU VJG EQWRNG RGT CPIWNCT VYKUV
.GV 6 DG VJG VKOG RGTKQF QH VJG FKUE D[ RNCEKPI VJG VYQ GSWCN OCUUGU QP
GKVJGT UKFG QH VJG FKUE CV C FKUVCPEG F HTQO VJG EGPVTG QH VJG FKUE
Fig. 1.24 Torsional pendulum-rigidity modulus
Properties of Matter
1.49
 6JG VKOG RGTKQF QH FKUE YKVJ VYQ GSWCN OCUU CV VJG RQUKVKQP F


+OF
6
%
YJGTG O KU VJG OCUU QH VJG QDLGEV RNCEGF CV C FKUVCPEG F
5KOKNCTN[ .GV 6 DG VJG VKOG RGTKQF QH FKUE D[ RNCEKPI VJG VYQ GSWCN OCUU
QP GKVJGT UKFG CV C FKUVCPEG F HTQO VJG EGPVTG QH VJG FKUE
 6JG VKOG RGTKQF QH VJG FKUE YKVJ VYQ GSWCN OCUU CV FKUVCPEG F


+OF
6
%
5SWCTKPI GSWCVKQPU CPF YG IGV
6
CPF
6
+OF
%
+OF
%
5WDVTCEVKPI VJG CDQXG GSWCVKQPU YG IGV
66

OFF
%
5WDUVKVWVKPI VJG XCNWG QH %QWRNG %
0T
KP GSWCVKQP YG IGV
N
ON 66
FF
0T

ON
0T
FF
ON FF
 6QTUKQPCN TKIKFKV[ 0
0T 66
'SWCVKQP IKXGU VJG TKIKFKV[ OQFWNWU QH VJG IKXGP YKTG WUKPI VQTUKQPCN
RGPFWNWO
1.50
Engineering Physics
EXAMPLE
1
# UVGGN YKTG QH NGPIVJ O CPF JCXKPI FKCOGVGT O KU CVVCEJGF
VQ C DGCO KP KVU WRRGT TQF +H C NQCF QH MI KU UWURGPFGF HTQO VJG NQYGT
CPF JQY OWEJ YKNN VJG YKTG DG GZVGPFGF! ;QWPIOU OQFWNWU
0O Given Data
;QWPIOU OQFWNWU QH VJG YKTG ;0O

.GPIVJ QH VJG YKTG
.O
&KCOGVGT QH VJG YKTG
FOQTTO
#RRNKGF NQCF
OMI
5QNWVKQP
9G MPQY ;QWPIOU OQFWNWU ;
6JG (QTEG
(.
#N
(OI
0
 6JG GZVGPFGF YKTG NGPIVJ N

(.
#;


NO
6JG KPETGCUG KP NGPIVJ QH YKTG NO
EXAMPLE
2
# EQRRGT YKTG QH O NGPIVJ CPF OO FKCOGVGT KU UWDLGEVGF VQ C VGPUKQP
QH 0 %CNEWNCVG VJG GNQPICVKQP RTQFWEGF KP VJG YKTG KH VJG ;QWPIOU
OQFWNWU QH GNCUVKEKV[ QH EQRRGT KU )2C
0QX Given Data
;QWPIOU OQFWNWU ;2C
.GPIVJ QH VJG YKTG .O
Properties of Matter
1.51
4CFKWU QH VJG YKTG TO
(QTEG (0

5QNWVKQP
(QTOWNC ;QWPIOU /QFWNWU ;
(.
#N
N
(.
#;



#T
 
 
O
O
 6JG GNQPICVKQP RTQFWEGF KP VJG YKTG  OO
EXAMPLE
3
# EKTEWNCT CPF C USWCTG ECPVKNGXGT CTG OCFG QH UCOG OCVGTKCN CPF JCXG
GSWCN CTGC QH ETQUUUGEVKQP CPF NGPIVJ (KPF VJG TCVKQ QH VJGKT FGRTGUUKQPU
HQT C IKXGP NQCF
&GE 
5QNWVKQP
(QTOWNC &GRTGUUKQP HQT C ECPVKNGXGT [
/IN
;+I
[E
/IN
;+I
&GRTGUUKQP HQT C EKTEWNCT ECPVKNGXGT
E
&GRTGUUKQP HQT C USWCTG ECPVKNGXGT
[U
/IN
;+I
U
+I HQT EKTEWNCT ETQUU UGEVKQP 
T
+I HQT USWCTG ETQUU UGEVKQP 
DF C

1.52
Engineering Physics
5KPEG DFC HQT USWCTG
*GPEG
[E
[U

+I
E
+I
U

C
C
 
T
T
5KPEG VJG CTGC QH ETQUUUGEVKQPU CTG GSWCN YG ECP YTKVG TC
C
QT 
T

 
 C 
    
[U   T 
 [E
 4CVKQ QH FGRTGUUKQPU 
EXAMPLE

4
# ECPVKNGXGT QH NGPIVJ EO HKZGF CV QPG GPF KU FGRTGUUGF D[ OO CV
VJG NQCFGF GPF %CNEWNCVG VJG FGRTGUUKQP CV C FKUVCPEG QH EO HTQO VJG
HKZGF GPF
0QX Given Data
&GRTGUUKQP [ CV C FKUVCPEG QH EO O
&GRTGUUKQP [ CV C FKUVCPEG QH EO !

5QNWVKQP
&GRTGUUKQP CV C FKUVCPEG QH EO [


9 N
;+I  9  

;+I
9  Z 
N Z N 
&GRTGUUKQP CV C FKUVCPEG QH EO [
;+I  Z


;
 
9 


;+I 

9 
;+I
Properties of Matter
1.53
&KXKFKPI GSWCKVQP D[ GSWCVKQP YG IGV
 



[

[



[O
[OO
 &GRTGUUKQP QH VJG ECPVKNGXGT CV C FKUVCPEG QH EO
HTQO VJG HKZGF GPF KU OO
EXAMPLE
5
# TGEVCPIWNCT UQNKF JCU FKOGPUKQPU EO # HQTEG QH 0 CRRNKGF
VCPIGPVKCNN[ VQ VJG WRRGT UWTHCEG 6JGTG KU C FKURNCEGOGPVQHOOTGNCVKXG
VQ VJG NQYGT UWTHCEG %CNEWNCVG VJG UJGCTKPI UVTGUU UVTCKP CPF VJG TKIKFKV[
OQFWNWU
Given Data
(0
(QTEG
NO
&KURNCEGOGPV

.GPIVJ QH UQNKF
.O
$TGCFVJ QH UQNKF
DO
*GKIJV QH UQNKF
JO
5QNWVKQP
5JGCT UVTGUU 

(QTEG
#TGC

0O
1.54
Engineering Physics
5JGCT UVTCKP 

&KURNCEGOGPVQHVJGWRRGTNC[GT
*GKIJV



6JG TKIKFKV[ OQFWNWU )

5JGCTUVTGUU
5JGCTUVTCKP
)0O
5JGCTKPI UVTGUU  0O
5JGCT UVTCKP

4KIKFKV[ OQFWNWU 00O
EXAMPLE
6
&GVGTOKPG VJG ;QWPIOU OQFWNWU QH VJG OCVGTKCN QH C TQF KH KV KU DGPF
WPKHQTON[ QXGT VYQ MPKHG GFIGU UGRCTCVGF D[ C FKUVCPEG QH O CPF NQCFU
QH MI CTG JWPI CV O CYC[ HTQO VJG MPKHG GFIGU 6JG DTGCFVJ CPF
VJKEMPGUU QH VJG TQF CTG O CPF O TGURGEVKXGN[ 6JG GNGXCVKQP
CV VJG OKFFNG QH VJG TQF KU O
Given Data
/CUU
6JG FKUVCPEG
/MI
CO
&KUVCPEG DGVYGGP VYQ MPKHG GFIGU NO
$TGCFVJ QH VJG TQF
DO
6JKEMPGUU QH VJG TQF
FO
6JG GNGXCVKQP
[O
Properties of Matter

1.55
5QNWVKQP
;QWPIOU OQFWNWU KP WPKHQTO DGPFKPI KU IKXGP CU
;

/ICN
DF[


;0O
;QWPIOU OQFWNWU ;0O
EXAMPLE
7
%CNEWNCVG VJG UJGCTKPI HQTEG TGSWKTGF VQ FKUVQTV C DNQEM QH CNWOKPKWO D[
EO KH KVU NGPIVJ KU EO CPF KVU UWTHCEG CTGC KU EO 6JG UJGCT
OQFWNWU QH CNWOKPKWO KU 0O

Given Data
4KIKFKV[ OQFWNWU
00O
.GPIVJ
.O
5WTHCEG CTGC
#O
&KURNCEGOGPV
NO
5QNWVKQP
4KIKFKV[ OQFWNWU
0
(.
#N
QT 5JGCT HQTEG
(
0#N
.



(0

6JG UJGCTKPI HQTEG (0
1.56
Engineering Physics
EXAMPLE
8
# NCVJG QH YKFVJ EO CPF VJKEMPGUU OO UWRRQTVGF JQTK\QPVCNN[ QP
MPKHG GFIGU EO CRCTV KU NQCFGF YKVJ YGKIJV QH IO JWPI HTQO KVU
GPFU YJKEJ RTQLGEV EO DG[QPF VJG MPKHG GFIGU +H VJG EGPVTG QH VJG NCVJG
KU VJGTGD[ GNGXCVGF D[ OO ECNEWNCVG VJG ;QWPIOU OQFWNWU QH KVU OCVGTKCN
Given data:
/IOMI
NEOO
CEO
JOOO
DEOO
FOOOO

5QNWVKQP
;
;QWPIOU OQFWNWU
#-
+P VJKU ECUG
*GPEG
;
;


;
/INC
J#-
DF
/INC
JDF





)2C
;QWPIOU OQFWNWU ;)2C
Properties of Matter
EXAMPLE
1.57
9
6JG DGPFKPI OQOGPV QH C DGCO HQWPF VQ DG 0O YJGP KV KU DGPV KP
VJG HQTO QH CP CTE QH C EKTENG QH TCFKWU O 'UVKOCVG VJG IGQOGVTKE
OQOGPV QH KPGTVKC QH VJG DGCO ;QWPIOU OQFWNWU QH OCVGTKCN QH VJG DGCO
 )2C Given data:
6JG DGPFKPI OQOGPV 0O
4CFKWU 4O
;QWPIOU OQFWNWU QH VJG OCVGTKCN QH VJG DGCO ;)2C

5QNWVKQP
$GPFKPI OQOGPV 
;#-
4
QT )GQOGVTKE OQOGPV QH KPGTVKC
#-


#-O
EXAMPLE 10
%CNEWNCVG VJG 2QKUUQPOU TCVKQ HQT VJG OCVGTKCN IKXGP ; 0O
CPF 00O

5QNWVKQP

;

0
  
 
   


 2QKUUKQPOU TCVKQ 
1.58
Engineering Physics
EXAMPLE 11
9JCV EQWRNG OWUV DG CRRNKGF VQ C O NQPI YKTG YKVJ OO FKCOGVGT KP
QTFGT VQ VYKUV QPG GPF QH KV VJTQWIJ  VJG QVJGT GPF TGOCKPKPI HKZGF!
6JG TKIKFKV[ OQFWNWU KU 0O
Given data
NO
.GPIVJ QH VJG YKTG
O
&KCOGVGT QH VJG YKTG
00O
4KIKFKV[ OQFWNWU QH VJG YKTG
#PING QH VYKUVKPI

TCFKCP
5QNWVKQP
6JG EQWRNG TGSWKTGF VQ VYKUV VJG YKTG VJTQWIJ 
%
0T

N
0T 

N %
0T
N
5WDUVKVWVKPI 0TCPFN XCNWGU KP VJG CDQXG GSWCVKQP
YG IGV
%


%
 6JG EQWRNG VQ DG CRRNKGF KU 0O
Properties of Matter
1.59
EXAMPLE 12
%CNEWNCVG VJG EQWRNG VQ DG CRRNKGF VQ VYKUV C YKTG QH NGPIVJ EO HKZGF
CV QPG GPF HTGG CV VJG QVJGT VJTQWIJ  4CFKWU QH YKTG  EO CPF
TKIKFKV[ OQFWNWU QH OCVGTKCN QH YKTG  2C 
Given data
.GPIVJ QH YKTG
EOO
4CFKWU QH VJG YKTG
EOO
#PING QH VYKUVKPI
 

TCFKCVKQP
K
 
4KIKFKV[ OQFWNWU
02C
5QNWVKQP
%QWRNG TGSWKTGF VQ VYKUV VJG YKTG
%

0T

N
 



 

0O
 %QWRNGF TGSWKTGF %0O
1.60
Engineering Physics
&GHKPG RGTHGEVN[ GNCUVKE CPF RGTHGEVN[ RNCUVKE DQF[
'NCUVKE OCVGTKCNU
/CVGTKCN YJKEJ TGICKP VJGKT QTKIKPCN NGPIVJ QT UJCRG QT UK\G CHVGT VJG TGOQXCN
QH VJG FGHQTOKPI HQTEG CTG ECNNGF GNCUVKE OCVGTKCNU
2NCUVKE OCVGTKCNU
/CVGTKCNU YJKEJ FQ PQV TGICKP VJGKT QTKIKPCN NGPIVJ QT UJCRG QT UK\G CHVGT
VJG TGOQXCN QH FGHQTOKPI HQTEG CTG ECNNGF RNCUVKE OCVGTKCN
&GHKPG UVTGUU CPF UVTCKP CPF YTKVG FQYP VJGKT WPKVU
#7 0QX ,CP 5VTGUU
Q6JG HQTEG CEVKPI RGT WPKV CTGC QH ETQUUUGEVKQP QH VJG YKTG KU ECNNGF CU VJG
UVTGUUR
 5VTGUU 
(
(QTEGU

#
#TGCQHVJGETQUUUGEVKQP
6JG WPKV HQT UVTGUU KU 0O
6JG VGTO QUVTGUU KU CNUQ FGHKPGF CU VJG TGUVQTKPI HQTEG CEVKPI RGT WPKV CTGC
QH ETQUUUGEVKQPR
5VTCKP
6JGTCVKQQHVJGEJCPIGKPFKOGPUKQPQHVJGOCVGTKCNVQVJGQTKIKPCNFKOGPUKQP
KU ECNNGF UVTCKP
5VTCKP
%JCPIGKPFKOGPUKQP
1TKIKPCNFKOGPUKQP
5VTCKP KU C FKOGPUKQPNGUU SWCPVKV[ CPF KV JCU PQ WPKV
5VCVG *QQMGOU NCY
*QQMGOU NCY UVCVGU VJCV YKVJKP VJG GNCUVKE NKOKV VJG UVTGUU FGXGNQRGF KP VJG
DQF[ KU FKTGEVN[ RTQRQTVKQPCN VQ UVTCKP RTQFWEGF KP KV
KG 5VTGUU5VTCKP
5VTGUU%QPUVCPV5VTCKP
Properties of Matter
1.61
&GHKPG OQFWNWU QH GNCUVKEKV[
5VTGUU
%QPUVCPVQTOQFWNWUQHGNCUVKEKV[
5VTCKP
+P QVJGT YQTFU VJG TCVKQ DGVYGGP UVTGUU CPF UVTCKP KU C EQPUVCPV 6JKU
EQPUVCPV QH RTQRQTVKQPCNKV[ KU ECNNGF OQFWNWU QH GNCUVKEKV[
+VU WPKV KU 0O QT 2C
9JCV FQ [QW KPHGT HTQO UVTGUUUVTCKP FKCITCO!
#7 &GE z
6JG UVTGUU KU FKTGEVN[ RTQRQTVKQPCN VQ VJG UVTCKP YKVJKP GNCUVKE NKOKV
z
+V FGVGTOKPGU VJG WNVKOCVG UVTGPIVJ QH VJG OCVGTKCN
z
+V FKUVKPIWKUJGU VJG GNCUVKE CPF RNCUVKE NKOKV QH C OCVGTKCN
z
6JKU FKCITCO CNUQ JGNRU WU VQ FKUVKPIWKUJ VJG OCVGTKCN DCUGF QP VJG
RTQRGTVKGU UWEJ CU FWEVKNKV[ CPF DTKVVNGPGUU
9JCV CTG VJG VJTGG V[RGU QH GNCUVKE OQFWNKK CPF YJCV FQ VJG[ TGHGT!
6JG VJTGG V[RGU QH GNCUVKE OQFWNKK CTG
;QWPIOU OQFWNWU ;
$WNM OQFWNWU - QT 8QNWOG OQFWNWU CPF
5JGCT OQFWNWU QT 4KIKFKV[ OQFWNWU 0
6JG ;QWPIOU OQFWNWU TGHGTU VQ C UVTGVEJKPI QT EQORTGUUKQP VJG DWNM OQFWNWU
VQ C EJCPIG QH XQNWOG YKVJ RTGUUWTG CPF UJGCT OQFWNWU VQ C VYKUVKPI
&GHKPG ;QWPIOU OQFWNWU
+V KU FGHKPGF CU VJG TCVKQ DGVYGGP NQPIKVWFKPCN UVTGUU VQ NQPIKVWFKPCN UVTCKP
YKVJKP VJG GNCUVKE NKOKVU
KG ;QWPIOUOQFWNWU
.QPIKVWFKPCNUVTGUU
.QPIKVWFKPCNUVTCKP
+VU WPKV KU 0O QT 2CUECN
&GHKPG $WNM OQFWNWU
+V KU FGHKPGF CU VJG TCVKQ DGVYGGP VJG XQNWOG UVTGUU QT DWNM UVTGUU VQ VJG
XQNWOG UVTCKP QT DWNM UVTCKP YKVJKP VJG GNCUVKE NKOKVU
KG $WNM OQFWNWU 
+VU WPKV KU 0O
8QNWOGUVTGUU
8QNWOGUVTCKP
1.62
Engineering Physics
&GHKPG 2QKUUQPOU TCVKQ
6JG TCVKQ QH NCVGTCN UVTCKP  VQ NQPIKVWFKPCN UVTCKP  KU ECNNGF 2QKUUQPOU
TCVKQ CPF KU FGPQVGF D[ 
2QKUUQPOU TCVKQ 

.CVGTCNUVTCKP

.QPIKVWFKPCNUVTCKP 
&GHKPG GNCUVKE NKOKV
6JG OCZKOWO UVTGUU WRVQ YJKEJ C DQF[ ECP TGEQXGT KVU QTKIKPCN UJCRG CPF
UK\G CHVGT VJG TGOQXKPI GZVGTPCN HQTEGU KU ECNNGF CU GNCUVKE .KOKV
9JKEJ KU OQTG GNCUVKE C UVGGN YKTG QT TWDDGT! ,WUVKH[
5KPEG GNCUVKEKV[ KU VJG RTQRGTV[ QH C OCVGTKCN VQ TGVWTP VQ KVU QTKIKPCN UJCRG
CPF UK\G YKVJQWV CP[ RGTOCPGPV FGHQTOCVKQP YJGP VJG FGHQTOKPI HQTEGU CTG
TGOQXGF UVGGN YKTG KU OQTG GNCUVKE VJCP TWDDGT
/GPVKQP VJG HCEVQTU CHHGEVKPI VJG GNCUVKEKV[ QH C OCVGTKCN
#7 &GE 0QX 1EV 6JG HQNNQYKPI HCEVQTU CHHGEV VJG GNCUVKE RTQRGTVKGU QH VJG OCVGTKCN
'HHGEV QH UVTGUU
'HHGEV QH CPPGCNKPI
'HHGEV QH JCOOGTKPI CPF TQNNKPI
'HHGEV QH VGORGTCVWTG
'HHGEV QH KORWTKVKGU
'HHGEV QH ET[UVCNNKPG PCVWTG
*QY FQ VGORGTCVWTG CPF KORWTKV[ KP C OCVGTKCN CHHGEV VJG GNCUVKEKV[
QH VJG OCVGTKCNU!
#7 &GE 'HHGEV QH 6GORGTCVWTG
#EJCPIGKPVGORGTCVWTGCHHGEVUVJGGNCUVKERTQRGTVKGUQHCOCVGTKCN0QTOCNN[
VJG GNCUVKEKV[ FGETGCUGU YKVJ VJG KPETGCUG QH VGORGTCVWTG 6JKU OC[ FWGVQKPETGCUG
QH ITCKP UK\G YKVJ TKUG QH VGORGTCVWTG
'ZCORNG
.GCF KU PQV C XGT[ IQQF GNCUVKE OCVGTKCN $WV VJG GNCUVKE RTQRGTV[ QH NGCF
KPETGCUGU YJGP VJG VGORGTCVWTG KU FGETGCUGF
Properties of Matter
1.63
'HHGEV QH +ORWTKVKGU
6JG GNCUVKE RTQRGTV[ QH C OCVGTKCN OC[ KPETGCUG QT FGETGCUG FWG VQ VJG
CFFKVKQP QH KORWTKVKGU 6JG KPETGCUG QT FGETGCUG QH GNCUVKEKV[ FGRGPFU QP VJG V[RG
QH KORWTKV[ CFFGF VQ KV
'ZCORNG
9JGP RQVCUUKWO KU CFFGF VQ IQNF VJG GNCUVKE RTQRGTV[ QH IQNF KPETGCUGU
9JCV CTG VJG GHHGEVU QH JCOOGTKPI CPF CPPGCNKPI QP GNCUVKEKV[ QH C
OCVGTKCN!
#7 /C[ 0QX 'HHGEV QH CPPGCNKPI
#PPGCNKPI KU VJG RTQEGUU QH JGCVKPI VJG OCVGTKCN CV C RCTVKEWNCT VGORGTCVWTG
CPF ITCFWCNN[ EQQNKPI
9JGP UQNKF OCVGTKCN CTG UWDLGEVGF VQ CPPGCNKPI VJG ET[UVCN ITCKP VGPF VQ
QTKGPV KPVQ QPG RCTVKEWNCT FKTGEVKQP 6JKU HQTO NCTIG ET[UVCN *GPEG VJG GNCUVKE
RTQRGTVKGU QH VJG OCVGTKCNU CTG TGFWEGF
'HHGEV QH JCOOGTKPI CPF TQNNKPI
9JGP ET[UVCNU CTG JCOOGTGF QT TQNNGF VJG ITCKPU CTG TGFWEGF VQ UOCNNGT
WPKVU YKVJ VJG TGUWNV VJCV VJGTG KU CP KPETGCUG KP GNCUVKE RTQRGTVKGU
&GHKPG DGCO
# DGCO KU FGHKPGF CU C TQF QT DCT QH WPKHQTO ETQUU UGEVKQP QH JQOQIGPGQWU
KUQVTQRKE GNCUVKE OCVGTKCNU GKVJGT EKTEWNCT QT TGEVCPIWNCT YJQUG NGPIVJ KU XGT[ NCTIG
EQORCTGF VQ KVU DTGCVJ CPF VJKEMPGUU UQ VJCV VJG UJGCTKPI UVTGUUGU CV CP[ RQKPV
QH VJG TQF CTG XGT[ UOCNN CPF PGINKIKDNG
&GHKPG DGPFKPI OQOGPV QH C DGCO
#7 0QX 9JGP C DGCF KU DGPF VJG FGHQTOCVKQP RTQFWEGF D[ VJG NQCF DTKPIU CDQWV
TGUVQTKPI HQTEGU FWG VQ GNCUVKEKV[ 6JG OQOGPV QH VJG EQWRNG FWG VQ VJG GNCUVKE
TGCEVKQP TGUVQTKPI EQWRNG YJKEJ DCNCPEGU VJG GZVGTPCN EQWRNG FWG VQ VJG CRRNKGF
NQCF KU ECNNGF VJG DGPFKPI OQOGPV
&GHKPG PGWVTCN CZKU
#7 &GE 6JG OKFFNG NCTIGT QT HKNCOGPV QH C C DGCO YJKEJ TGOCKPU WPCNVGTGF GXGP
YKVJ VJG RTGUGPEG QH NQCF QP VJG DGCO KU ECNNGF PGWVTCN CZKU (KNCOGPVU YJKEJ
CTG N[KPI CDQXG KV CTG GNQPICVGF CPF VJQUG CTG N[KPI DGNQY KV CTG EQORTGUUGF
1.64
Engineering Physics
9JCV KU WPKHQTO DGPFKPI CPF YJ[ KU KV UCKF VQ DG WPKHQTO!
9JGP C DGCO KU UWRRQTVGF U[OOGVTKECNN[ QP VYQ MPKHG GFIGU CPF NQCFGF
YKVJ GSWCN YGKIJVU CV GCEJ GPFU VJG DGPF DGCOU HQTO CP CTE QH C EKTENG 6JG
GNGXCVKQP KP VJG DGCO KU RTQFWEGF 6JKU DGPFKPI KU ECNNGF WPKHQTO DGPFKPI 5KPEG
VJG DGCO DGPFU KPVQ CP CTE QH C EKTENG VJG DGPFKPI KU UCKF VQ DG WPKHQTO
9JCV KU PQPWPKHQTO DGPFKPI!
+H VJG DGCO KU NQCF CV KVU OKFRQKPV VJG FGRTGUUKQP RTQFWEGF YKNN PQV HQTO
CP CTE QH C EKTENG 6JKU V[RG QH DGPFKPI KU ECNNGF PQPWPKHQTO DGPFKPI
&KUVKPIWKUJ DGVYGGP WPKHQTO CPF PQPWPKHQTO DGPFKPI
+P WPKHQTO DGPFKPI GXGT[ GNGOGPV QH VJG DGCO KU DGPV YKVJ VJG UCOG TCFKWU
QH EWTXCVWTG +P PQPWPKHQTO DGPFKPI VJG TCFKWU QH EWTXCVWTG KU PQV C UCOG HQT
CNN VJG GNGOGPVU KP VJG DGCO
9JCV KU +UJCRGF IKTFGTU!
6JG IKTFGTU YKVJ WRRGT CPF NQYGT UGEVKQP DTQCFGPGF CPF VJG OKFFNG UGEVKQP
VCRGTGF UQ VJCV KV ECP DG YKVJUVCPF JGCX[ NQCFU QXGT KV 6JG ETQUU UGEVKQP QH VJG
IKTFGTU VCMGU VJG UJCRG QH VJG ECRKVCN NGVVGT + KU ECNNGF CU + UJCRG IKTFGTU
'ZRNCKP VJG CFXCPVCIG QH +UJCRG IKTFHCT
#7 0QX /C[ 6JG OCVGTKCN ECP DG UCXGF
6JG UVTGPIVJ QH VJG IKTFGT KU PQV CHHGEVGF
6JG UVCDKNKV[ CPF VJG GHHKEKGPE[ QH VJG IKTFGTU CTG OCKPVCKPGF
9JCV KU 6QTUKQPCN 2GPFWNWO
# DQF[ UWURGPFGF D[ C YKTG YJKEJ VYKUVU HKTUV KP QPG FKTGEVKQP CPF VJGP KP
VJG TGXGTUG FKTGEVKQP KP VJG JQTK\QPVCN RNCPG KU ECNNGF VQTUKQPCN RGPFWNWO
6QTUKQPCN RGPFWNWO KU C RGPFWNWO YJKEJ RGTHQTOU VQTUKQPCN QUEKNNCVKQPU +V
KU WUGF VQ FGVGTOKPG VJG OQOGPV QH KPGTVKC CPF VQTUKQPCN TKIKFKV[ QH C IKXGP DQF[
Properties of Matter
1.65
&GTKXG CP GZRTGUUKQP HQT VJG KPVGTPCN DGPFKPI OQOGPV QH C DGCO KPVGTOU QH
TCFKWU QH EWTXCVWTG
0QX
¾ 9JCV KU ECPVKNGXGT! 1DVCKP CP GZRTGUUKQP HQT VJG FGRTGUUKQP CV VJG NQCFGF
GPF QH C ECPVKNGXGT YJQUG QVJGT GPF KU HKZGF CUUWOKPI VJCV KVU QYP YGKIJV
KU PQV GHHGEVKXG KP DGPFKPI
&GE &GE #RT &GE
¾ 9JCV KU ECPVKNGXGT! &GTKXG CP GZRTGUUKQP HQT VJG FGRTGUUKQP CV VJG HTGG
GPF QH C ECPVKNGXGT YJGP VJG QVJGT GPF KU TKIKFN[ HKZGF CUUWOG VJG YGKIJV
QH VJG ECPVKNGXGT KU PGINKIKDNG /C[ ¾ &GTKXG CP GZRTGUUKQP HQT FGRTGUUKQP CV VJG HTGG GPF QH C ECPVKNGXGT FWG
VQ NQCF
0QX ¾ &GUETKDGCPGZRGTKOGPVVQFGVGTOKPGVJG;QWPIOUOQFWNWUQHVJGECPVKNGXGT
OCVGTKCN WUKPI VJKU GZRTGUUKQP
0QX ¾ &GUETKDG YKVJ PGEGUUCT[ VJGQT[ VJG OGVJQF VQ FGVGTOKPG VJG ;QWPIOU
OQFWNWU QH VJG OCVGTKCN QH C TGEVCPIWNCT DCT D[ WPKHQTO DGPFKPI
,WP ¾ &GUETKDGCPGZRGTKOGPVVQFGVGTOKPGVJG;QWPIOUOQFWNWUQHCDGCOWUKPI
DGPFKPI QH DGCOU!
&GE ¾
K &GTKXG CP GZRGTKOGPV HQT VJG GNGXCVKQP CV VJG EGPVTG QH C ECPVKNGXGT
YJKEJ KU NQCFGF CV DQVJ GPFU
KK &GUETKDG CP GZRGTKOGPV VQ FGVGTOKPG ;QWPIOU OQFWNWU QH C DGCO D[
WPKHQTO DGPFKPI
0QX ¾ 'ZRNCKP YKVJ PGEGUUCT[ VJGQT[ VJG FGVGTOKPCVKQP QH ;QWPIOU OQFWNWU QH
GNCUVKEKV[ QH VJG OCVGTKCN QH C DGCO UWRRQTVGF CV KVU GPFU CPF NQCFGF KP
VJG OKFFNG
&GE ¾ *QY YKNN [QW FGVGTOKPG VJG ;QWPIOU OQFWNWU QH C OCVGTKCN QH C DCT D[
PQPWPKHQTO DGPFKPI OGVJQF! 'ZRNCKP KP DTKGH VJG VJGQT[ DGJKPF VJG
FGVGTOKPCVKQP QH ;QWPIOU OQFWNWU
/C[ ¾ 9TKVG C PQVG +UJCRG IKTFGT 9JCV CTG KVU CFXCPVCIGU!
#7 ,CP ¾ )KXG CP CEEQWPV QH +UJCRG IKTFGTU
¾ &GTKXG CP GZRTGUUKQP HQT VJG VQTUKQPCN EQWRNG RGT WPKV CPIWNCT VYKUV YJGP
C E[NKPFGT KU VYKUVGF
&GE ,WN[ ¾ &GTKXG CP GZRTGUUKQP HQT EQWRNG RGT WPKV VYKUV QP C VJKP E[NKPFGT
&GE &GE 1.66
Engineering Physics
¾ &GTKXG CP GZRTGUUKQP HQT VJG RGTKQF QH QUEKNNCVKQP QH C VQTUKQP RGPFWNWO
*QY ECP KV DG WUGF VQ FGVGTOKPG VJG VQTUKQPCN TKIKFKV[ QH C YKTG!
/C[ /C[ 0QX &GE ¾ 9JCV KU VQTUKQPCN RGPFWNWO! 'ZRNCKP JQY KV KU WUGF VQ FGVGTOKPG VJG
TKIKFKV[ OQFWNWU QH VJG OCVGTKCN QH C VJKP YKTG
/C[ # UVGGN TQF QH NGPIVJ O KU HKZGF TKIKFN[ DGVYGGP VYQ UWRRQTVU 6JG EQGHHKEKGPV
QH NKPGCT GZRCPUKQP QH UVGGN % %CNEWNCVG VJG UVTGUU KP VJG TQF
HQT CP KPETGCUG KP VGORGTCVWTG QH % 6JG ;QWPIOU OQFWNWU QH GNCUVKEKV[ QH
UVGGN 0O
#PU 5VTGUU 0O
%CNEWNCVGVJG2QKUUQPOUTCVKQHQTVJGOCVGTKCNIKXGP;0OCPF
00O
#PU %CNEWNCVG VJG ;QWPIOU OQFWNWU QH C OCVGTKCN YJQUG DWNM OQFWNWU
-0O CPF TKIKFKV[ OQFWNWU 00O
#PU ;0O
+P CP GZRGTKOGPV C DCT QH NGPIVJ O KU ENCORGF JQTK\QPVCNN[ CV QPG GPF
CPF C NQCF QH MI CVVCEJGF CV KVU HTGG GPF %CNEWNCVG VJG FGRTGUUKQP CV
VJG NQCFGF GPF KH ;0O CPF VJG DCT KU QH DTGCFVJ O
CPF VJKEMPGUU O
#PU O
# ECPVKNGXGT QH UVGGN HKZGF JQTK\QPVCNN[ KU UWDLGEVGF VQ C NQCF QH IO CV
KVU HTGG GPF VJG IGQOGVTKE OQOGPV QH KPGTVKC QH VJG ECPVKNGXGT KU
O +H VJG NGPIVJ QH ECPVKNGXGT CPF ;QWPIOU OQFWNWU QH UVGGN CTG
O CPF 2C TGURGEVKXGN[ ECNEWNCVG VJG FGRTGUUKQP CV VJG NQCFGF GPF
#PU EO
# UVGGN TQF QH NGPIVJ EO KU HKZGF CV QPG GPF CPF -I NQCF KU UWURGPFGF
CV VJG QVJGT GPF %CNEWNCVG VJG FGRTGUUKQP KH VJG ;QWPIOU OQFWNWU KU
 2CUECNU &KCOGVGT KU EO
#PU EO
# E[NKPFTKECN YKTG QH NGPIVJ O CPF TCFKWU OO KU TKIKFKV[ ENCORGF CV QPG
GPF %CNEWNCVG VJG EQWRNG TGSWKTG VQ VYKUV VJG HTGG CPF VJTQWIJ CP CPING
 6JG TKIKFKV[ OQFWNWU QH VJG OCVGTKCN QH VJG YKTG KU  2C
#PU %0O