of 45
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YMZTJDT
"abljsmkuabrpmysjds.jl" 2
UJNJC E@CZ CZLBAJDT2. UJNJC E@CZ 7
Ujnjc e`cy js cfhjlfc bs b systfa `h pbrtjdofs jl wmjdm cjstbldf eftwffl fbdm pbjr `h pbrtjdofs
rfabjls d`lstblt (wjtm rfspfdt t` tjaf). Ufafaefr, rjnjc e`cy js b abtmfabtjdbo d`ldfpt blc blysystfa wmjdm sbtjshjfs tmf be`vf d`lcjtj`l js sbjc t` ef rjnjc bs o`ln bs jt sbtjshjfs jt.
B & E brf efbcs wmjd m a`vf `l b djrd uobr hjx fc rjln
Jh b systfa js rjnjc, sjldf tmfrf js l` dmblnf jl tmf cjstbldf eftwffl bly pbjr `h pbrtjdofs `h tmf
systfa, smbpf blc sjzf `h systfa rfabjls d`lstblt. Mfldf wf jltujtjvfoy hffo tmbt wmjof b
st`lf `r drjdkft eboo brf rjnjc e`cjfs, b eboo``l `r fobstjd strjln js l`l rjnjc.
Eut bly `h tmf be`vf systfa js rjnjc bs o`ln bs rfobtjvf cjstbldf c`fs l`t dmblnf, wmftmfr jt js
b drjdkft eboo `r b eboo``l. Eut bt tmf a`aflt wmfl tmf ebt mjts tmf drjdkft eboo `r jh tmf eboo``l
js squffzfc, rfobtjvf cjstbldf dmblnfs blc l`w tmf systfa efmbvfs ojkf b l`l-rjnjc systfa.
H`r fvfry pbjr `h pbrtjdofs jl b rjnjc e`cy, tmfrf js l` vfo`djty `h sfpbrbtj`l `r bppr`bdm eftwffl
tmf pbrtjdofs. j.f. bly rfobtjvf a`tj`l `h b p`jlt E `l b rjnjc e`cy wjtm rfspfdt t` bl`tmfr p`jltB `l tmf rjnjc e`cy wjoo ef pfrpflcjduobr t` ojlf g`jljln B t` E, mfld f wjtm rfspfdt t` bly
pbrtjdof B `h b rjnjc e`cy tmf a`tj`l `h bly `tmfr pbrtjdof E h tmbt rjnjc e`cy js djrduobr a`tj`l.
Oft vfo`djtjfs `h B blc E wjtm rfspfdt nr`ulc efBX
blcEX
rfspfdtjvfoy jl tmf h jnurf efo`w..
B XB
XE 5
2
E
8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" 5
Jh tmf be`vf e`cy js rjnjc XB
d`s 2
1 XE
d`s 5
(vfo`djty `h bppr`bdm / sfpbrbtj`l js zfr`)
XEB
1 rfobtjvf vfo`djty `h E wjtm rfspfdt t` B.
XEB
1 XB
sjl2+ X
Esjl
5 (wmjdm js pfrpflcjduobr t` ojlf BE)
E wjoo bppfbr t` a`vf jl b djrdof t` bl `esfrvfr hjxfc bt B.
^.r.t. bly p`jlt `h tmf rjnjc e`cy tmf blnuobr vfo`djty `h boo `tmfr p`jlts `h tmf tmbt rjnjc e`cy js sbaf.
Tupp`sf B, E, D js b rjnjc systfa mfldf curjln bly a`tj`l sjcfs BE, ED blc DB aust r`tbtf tmr`unmtmf sbaf blnof. Mfldf boo tmf sjcfs r`tbtf ey tmf sbaf rbtf.
Hr`a hjnurf (j) blnuobr vfo`djty `h B blc E w.r.t. D js,Hr`a hjnurf (jj) blnuobr vfo`djty `h B blc D w.r.t. E js,
J. Yurf Srblsobtj`lbo A`tj`l 7B e`cy js sbjc t` ef jl purf trbl sobt j`lbo a`t j`l, jh tmf cjspobdfaflt `h fbdm pbrtjdof `h tmf systfa js
sbaf curjln bly tjaf jltfrvbo. Curjln sudm b a`tj`l, boo tmf pbrtjdofs mbvf sbaf cjspobdfaflt )s(
,
vfo`djty )v(
blc bddfofrbtj`l )b(
bt bl jlstblt.
D`lsjcfr b systfa `h l pbrtjdof `h abss a2, a
5, a
4, ...... a
lulcfr n`jln purf trblsobtj`l.
tmfl hr`a be`vf cfhjljtj l `h trblsobtj`lbo a`tj`l
l452 b......bbb 1 b (sby)
blc l452 v......vvv
1 v
(sby)
Hr`a Lfwt`l's obws h`r b systfa.
fxtH
122ba
+55ba
+44ba
+ ......................
fxtH
1 A b
^mfrf A 1 S tbo abss h tmf e`cy
Y
1 22va
+ 55va
+ 44va
+ ......................
Y 1 vA
S`tbo Kjlftjd Flfrny `h e`cy 15
2a
2v
25 +
5
2a
5v
55 + ................. 1
5
2Av5
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YMZTJDT
"abljsmkuabrpmysjds.jl" 4
JJ. Yurf U`tbtj`lbo A`tj`l 7
Hjnurf sm`ws b rjnjc e`cy `h brejtrbry smbpf jl r`tbtj`l be`ut b hjxfc bxjs, dboofc tmf bxjs `h r`tbtj`l.
Fvfry p`jlt `h tmf e`cy a`vfs jl b djrdof wm`sf dfltfr ojfs `l tmf bxjs `h r`tbtj`l, blc fvfry p`jlt
a`vfs tmr`unm tmf sbaf blnof curjln b pbrtjduobr tjaf jltfrvbo. Tudm b a`tj`l js dboofc purf r`tbtj`l.
^f kl`w tmbt fbdm pbrtjdof mbs sbaf blnuobr vfo`djty (sjldf tmf e`cy js rjnjc.)
s`, v21r
2, v
51r
5, v
41r
4...... v
l1r
l
S`tbo Kjlftjd Flfrny 15
2a
2v
25 +
5
2a
5v
55 + ..........................
15
2[a
2r
25 + a
5r
55 + ..........................Q5
15
25 ^mfrf 1 a
2r
25 + a
5r
55 + ............. (js dboofc a`aflt `h jlfrtjb)
1 blnuobr spffc `h e`cy.
JJJ. D`aejlfc Srblsobt j`lbo blc U`tbt j`lbo A`tj`l 7B e`cy js sbjc t` ef jl d`aejlfc trblsobt j`l blc r`tbtj`lbo a`tj`l jh boo p`jlt jl tmf e`cy r`tbtfs be`ut
bl bxjs `h r`tbtj`l blc tmf bxjs `h r`tbtj`l a`vfs wjtm rfspfdt t` tmf nr`ulc. Bly nflfrbo a`tj`l `h b
rjnjc e`cy dbl ef vjfwfc bs b d`aejlfc trblsobtj`lbo blc r`tbtj`lbo a`tj`l.
Fxbapof 2.
B e`cy js a`vjln c`wl jlt` b wfoo tmr`unm b r`pf pbss jln `vfr b hjxfc puoofy `h rbcjus 29 da. Bssuaf
tmbt tmfrf js l` sojppjln eftwffl r`pf & puoofy. Dboduobtf tmf blnuobr vfo`djty blc blnuobr bddfofrbtj`l
`h tmf puoofy bt bl jlstblt wmfl tmf e`cy js n`jln c`wl bt b spffc `h 59 da/s blc mbs bl bddfofrbtj`l
`h 3.9 a/s5.
T`outj`l 7
Tjldf tmf r`pf c`fs l`t sojp `l tmf puoofy, tmf ojlfbr spffc v `h tmf rja `h tmf puoofy js sbaf bs tmf
spffc `h tmfe`cy.
Smf blnuobr vfo`djty `h tmf puoofy js tmfl
1 v/r 1da29
s/da591 5 rbc/s
blc tmf blnuobr bddfofrbtj`l `h tmf puoofy js
1 b/r 1da29
s/a9.3 5
1 39 rbc/s5.
Fxbapof 5.
Bcjsdr`tbtfs wjtm buljh`rablnuobr bddfofrbtj`l `h 5.9 rbc/s5 be`ut jts bxjs. Jh tmfcjsd stbrts hr`a
rfst, m`w ably rfv`outj`ls wjoo jt abkf jl tmf hjrst 29 sfd`lcs0
T`outj`l 7
Smf blnuobr cjspobdfaflt jl tmf hjrst 29 sfd`lcs js njvfl ey
1 9t +
5
2t 5 1
5
2(5.9 rbc/s5) (29 s)5 1 299 rbc.
8/12/2019 Rigid Body Dynamics Theory_E
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"abljsmkuabrpmysjds.jl" 3
Bs tmf wmffo turls ey 5rbcjbl jl fbdm rfv`outj`l, tmf luaefr `h rfv`outj`ls jl 29 s js
l 15
2991 28.
Fxbapof 4.
Smf wmffo `h b a`t`r, bddfofrbtfc uljh`raoy hr`a rfst, r`tbtfs tmr`unm6rbcjbl curjln tmf hjrst sfd`lc.
Dboduobtftmf blnof r`tbtfc curjln tmf lfxt sfd`lc.
T`outj`l 7
Bs tmf blnuobr bdd fofrbtj`l js d`ls tblt, wf mbvf
1 9t +
5
2 t 5 1
5
2 t 5.
Smus, 6 rbc 15
2(2s)5
1 29 rbc/s5 `r 1 29 rbc/s5
Smf blnof r`tbtfc curjln tmf hjrst tw` sfd`lcs js
15
2 (29 rbc/s5) (5s)5 1 59 rbc.
Smus, tmf blnof r`tbtfc curjln tmf 5 lc sfd`lc js59 rbc 6 rbc 1 26 rbc.
Fxbapof 3.
Ttbrtjln hr`a rfst, b hbl tbkfs h`ur sfd`lcs t` bttbjl tmf abxjaua spffc `h 399 rpa (rfv`outj`l pfr
ajlutf).Bssuajln uljh`rabddfofrbtj`l, dboduobtftmf tjaf tbkfl ey tmf hbl jl bttbjljln mboh tmf abxjaua
spffc.
T`outj`l 7
Oft tmf blnuobr bddfofrbtj`l ef. Bdd`rcjln t` tmf qufstj`l,399 rfv/ajl 1 9 +3 ...........(j)
Oft t ef tmf tjaf tbkfl jl bttbjljln tmf spffc `h 599 rfv/ajl wmjdm js mboh tmf a bxjaua.
Smfl, 599 rfv/ajl 1 9 +t ...........(jj)
Cjvjcjln (j) ey (jj), wf nft,t 1 5 sfd.
Fxbapof 6.
Smf a`t`r `h bl flnjlf js r`tbtjln be`ut jts bxjs wjtm bl blnuobr vfo djty `h 259 rfv/ajlutf. Jt d`afs t`
rfst jl 29 s, bhtfr efjln swjtdmfc `hh tmf flnjlf. Bssuajln uljh`ra blnuobr cfdfofrbtj`l, hjlc tmf
luaefr `h rfv`outj`ls abcf ey jt efh`rf d`ajln t` rfst.
T`outj`l 7
Smf jljtjbo blnuobr vfo`djty 1 259 rfv/ajlutf 1 (3) rbc/s.Hjlbo blnuobr vfo`djty 1 9.
Sjaf jltfrvbo 1 29 s.
Oft tmf blnuobr bddfofrbtj`l ef. Vsjln tmf fqubtj`l19+t, wf `etbjl1 ( 3/29) rbc/s5
Smf blnof r`tbtfc ey tmf a`t`r curjln tmjs a`tj`l js
1 9t +
5
2t5 1
s
rbc3 (29s)
5
2
5s
rbc
29
3(29s)5
1 59rbc 1 29 rfv`outj`ls.Mfldf tmf a`t`r r`tbtfs tmr`unm 29 rfv`outj`ls efh`rf d`ajln t` rfst.
8/12/2019 Rigid Body Dynamics Theory_E
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"abljsmkuabrpmysjds.jl" 6
5. A@AFLS @H JLFUSJB J BE@VS BL B]JT 7(j) A`aflt `h jlfrtjb `h b systfa `h l pbrtjdofs be`ut bl bxjs js cfhjlfc bs a
2r
25 + a
5r
55 + ..................+ a
lr
l5
j.f. 1
l
2j
5jjra
wmfrf, rj1 Jt js pfrpflcjduobr cjstbldf `h abss a
jhr`a bxjs `h r`tbtj`l
Tuljts `h A`aflt `hlfrtjb js Kna5.A`aflt `h jlfrtjb js b sdbobr p`sjtjvf qubltjty.
(jj) H`r b d`ltjlu`us systfa 7
1 )ca(r5
wmfrf ca 1 abss `h b saboo fofaflt
r 1 pfrpflcjduobr cjstbldf `h tmf abss fofaflt ca hr`a tmf bxjs
A`aflt `h Jlfrtjb cfpflcs `l 7
(j) cflsjty `h tmf abtfrjbo `h e`cy
(jj) s mbpf & s jzf `h e`cy(jjj) bxjs h r`tbt j l
Jl t`tbojty wf dbl sby tmbt jt cfpflcs up`l cjstrjeutj`l `h abss rfobtjvf t` bxjs `h r`tbtj`l.
L`tf 7 A`aflt `h jlfrtjb c`fs l`t dmblnf jh tmf abss 7(j) js smjhtfc pbrboofo t` tmf bxjs `h tmf r`tbtj`l efdbusf r
j c`fs l`t dmblnf.
(jj) js r`tbtfc be`ut bxjs `h r`tbtj`l jl b djrduobr pbtm efdbusf rjc`fs l`t dmblnf.
Fxbapof 8.
Sw` pbrtjdofs mbvjln abssfs a2& a
5brf sjtubtfc jl b poblf
pfrpflcjduobr t` ojlf BE bt b cjstbldf `h r2blc r
5rfspfdtjvfoy
bs sm`wl.(j) Hjlc tmf a`aflt `h jlfrtjb `h tmf systfa be`ut
bxjs BE 0
(jj) Hjlc tmf a`aflt `h jlfrtjb `h tmf systfa be`ut
bl bxjs pbssjln tmr`unm a2blc pfrpflcjduobr t`
tmf ojlf g`jljln a2blc a
50
(jjj) Hjlc tmf tmf a`aflt `h jlfrtjb `h tmf systfa be`ut bl
bxjs pbssjln tmr`unm a2blc a
50
(jv) Hjlc a`aflt `h jlfrtjb be`ut bxjs pbssjln tmr`unm dfltrf `h abss blc pfrpflcjduobr t` ojlf g`jljln
a2blc a
5
T`outj`l
(j) A`aflt `h jlfrtjb `h pbrtjdof `l ofht js21 a
2r
25.
A`aflt `h Jlfrtjb `h pbrtjdof `l rjnmt js51 a5r55
.A`aflt `h Jlfrtjb `h tmf systfa be`ut BE js 1 2+
5 1 a
2r
55 + a
5r
55
(jj) A`aflt `h jlfrtjb `h pbrtjdof l ofht js21 9
A`aflt `h Jlfrtjb `h pbrtjdof `l rjnmt js 51 a
5(r
2+ r
5)5.
A`aflt `h Jlfrtjb `h tmf systfa be`ut BE js 1 2+
51 9 + a
5(r
2+ r
5)5
(jjj) A`aflt `h jlfrtjb `h pbrtjdof l ofht js21 9
A`aflt `h Jlfrtjb `h pbrtjdof `l rjnmt js 51 9
A`aflt `h Jlfrtjb `h tmf systfa be`ut BE js 1 2+
51 9 + 9
(jv) Dfltrf `h abss `h systfa rDA
1
52
525
aa
rra 1 Cjstbldf `h dfltrf abss hr`a abss a
2
Cjstbldf `h dfltrf `h abss hr`a abss a51
52
522
aa
rra
8/12/2019 Rigid Body Dynamics Theory_E
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"abljsmkuabrpmysjds.jl" 8
T` a`aflt `h jlfrtjb be`ut dfltrf `h abss 1 da
1
5
52
5252
aa
rraa
+
5
52
5225
aa
rraa
DA
152
52
aa
aa
(r2 + r5)5
Fxbapof >.
H`ur pbrtjdofs fbdm `h abss a brf kfpt bt tmf h`ur d`rlfrs `h b squbrf
`h fcnf b. Hjlc tmf a`aflt `h jlfrtjb `h tmf systfa be`ut b ojlf
pfrpflcjduobr t` tmf poblf `h tmf squbrf blc pbssjln tmr`unm tmf dfltrf
`h tmf squbrf.
T`outj`l 7
Smf pfrpflcjduobr cjstbldf `h fvfry pbrtjdof hr`a tmf njvfl ojlf js b/ 5. Smf a`aflt `h jlfrtjb `h `lf
pbrtjdof js, tmfrfh`rf, a(b/ 5 )5 1
5
2ab5. Smf a`aflt `h jlfrtjb `h tmf systfa js, tmfrfh`rf,
3 5
2ab 5 1 5ab5.
Fxbapof =.
Smrff pbrtjdofs, fbdm `h abss a, brf sjtubtfc bt tmf vfrtjdfs `h bl
fqujobtfrbo trjblnof YRU `h sjcf b bs sm`wl jl hjnurf. Dboduobtf tmf
a`aflt `h jlfrtjb `h
tmf systfa be`ut (j) tmf ojlf Y] pfrpflcjduobr t` YR jl tmf poblf `h YRU.
(jj) @lf `h tmf sjcfs `h tmf trjblnof YRU (jjj) Be`ut bl bxjs pbssjln tmr`unm
tmf dfltr`jc blc pfrpflcjduobr t` poblf `h tmf trjblnof YRU.
[MDX A@CJHJFC UMN TJUQ
T`outj`l
(j) Yfrpflcjduobr cjstbldf `h Y hr`a Y] 1 9
Yfrpflcjduobr cjstbldf `h R hr`a Y] 1 b
Yfrpflcjduobr cjstbldf `h U hr`a Y] 1 b/5
Smus, tmf a`aflt `h jlfrtjb `h tmf pbrtjdof bt Y 1 9, `h tmf pbrtjdof bt R - ab5, blc `h tmf pbrtjdof bt
U 1 a(b/5)5. Smf a`aflt `h jlfrtjb `h tmf tmrff-pbrtjdof systfa be`ut Y] js
9 + ab5 + a(b/5)5 13
ab6 5
L`tf tmbt tmf pbrtjdofs `l tmf bxjs c` l`t d`ltrjeutf t` tmf a`aflt `h jlfrtjb.
(jj) A`aflt `h jlfrtjb be`ut tmf sjcf YU 1 abss `h pbrtjdof R squbrf `h pfrpflcjduobr cjstbldf `h R
hr`a sjcf YU, YU
1 a
5
b5
4
1
3
ab4 5
(jjj) Cjstbldf `h dfltr`jc hr`a boo tmf pbrtjdof js4
b, s` a`aflt `h jlfrtjb be`ut bl bxjs blc pbssjln
tmr`unm tmf dfltr`jd pfrpflcjduobr poblf `h trjblnof YRU 1U
1 4a
5
4
b
1 ab5
Fxbapof :.
Dboduobtf tmf a`aflt `h jlfrtjb `h b rjln mbvjln abss A, rbcjus U blc mbvjln uljh`ra abss cjstrjeutj`l
be`ut bl bxjs pbssjln tmr`unm tmf dfltrf `h rjln blc pfrpflcjduobr t` tmf poblf `h rjln 0
T`outj`l 1 5r)ca(Efdbusf fbdm fofaflt js fqubooy cjstbldfc hr`a tmf bxjs s` r 1 U
1 55 AUcaU1 AU5
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"abljsmkuabrpmysjds.jl" >
(L`tf 7Blswfr wjoo rfabjl sbaf fvfl jh tmf abss js l`luljh`raoy cjstrjeutfc efdbusf Aca bowbys.)
Fxbapof 29.
Dboduobtf tmf a`aflt `h jlfrtjb `h b uljh`ra r`c `h abss A blc oflntmbe`ut bl bxjs 2,5,4 blc 3.
d`a
2 5
4
3c
T`outj`l (2)1 5r)ca( 1
9
5xcxA
14
A 5
(5)1 5r)ca( 1
5/
5/
5xcxA
125
A 5
(4)1 9 (bxjs 4 pbssjln tmr`unm tmf bxjs `h r`c)
(3)1 55 Ac)ca(c
Fxbapof 22.
Cftfrajlfc tmf a`aflt `h jlfrtjb `h b uljh`ra rfdtblnuobr pobtf `h abss, sjcf 'e' blc ''' be`ut bl bxjs
pbssjln tmr`unm tmf fcnf 'e' blc jl tmf poblf `h pobtf.
T`outj`l 7
Fbdm sfdtj`l `h ca abss r`c jl tmf rfdtblnuobr pobtf mbs a`aflt `h jlfrtjb be`ut bl bxjs pbssjln
tmr`unm fcnf 'e' c14
ca 5
T` 1 ca4c5
14
A 5
ca
Fxbapof 25.
Hjlc `ut tmf a`aflt `h Jlfrtjb `h hjnurfs sm`wl fbdm mbvjln abss A, rbcjus U blc mbvjln uljh`ra abss
cjstrjeutj`l be`ut bl bxjs pbssjln tmr`unm tmf dfltrf blc pfrpflcjduobr t` tmf poblf 0
(b)
x
U (e)x
U (d)x
U
T`outj`l 7
AU5 (jlhbdt A.J. `h bly pbrt `h abss A `h b rjln `h rbcjus U be`ut bxjs pbssjln tmr`unm nf`aftrjdbo
dfltrf blc pfrpflcjduobr t` tmf poblf `h tmf rjln js 1 AU 5 )
(jjj) A`aflt `h jlfrtjb `h b obrnf `egfdt dbl ef dboduobtfc ey jltfnrbtjln A..`h bl fofaflt `h tmf `egfdt7
1 fofafltcwmfrf c1 a`aflt `h jlfrtjb `h b saboo fofaflt
Fofaflt dm`sfl sm`uoc ef sudm tmbt 7 fjtmfr pfrpflcjduobr cjstbldf `h bxjs hr`a fbdm p`jlt `h tmffofaflt js sbaf `r tmf a`aflt `h jlfrtjb `h tmf fofaflt be`ut tmf bxjs `h r`tbtj`l js kl`wl.
8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" =
Fxbapof 24.
Cftfrajlf tmf a`aflt `h Jlfrtjb `h b uljh`ra cjsd mbvjln abss A, rbcjus U be`ut bl bxjs pbssjln
tmr`unm dfltrf & pfrpflcjduobr t` tmf poblf `h cjsd 0
T`outj`l 7 1 rjlncfofaflt - rjln c1 car5
ca 1 5UA
5rcr (mfrf wf mbvf usfc tmf uljh`ra abss cjstrjeutj`l)
1 U
9
5U
A. (5rcr).r5 1
5
AU5
Fxbapof 23.
Dboduobtf tmf a`aflt `h jlfrtjb `h b uljh`ra m`oo`w dyojlcfr `h abss A,
rbcjus U blc oflntmbe`ut jts bxjs.
T`outj`l 7
A`aflt `h jlfrtjb `h b uljh`ra m`oo`w dyojlcfr js
5U)ca(1 a U5
abss 1 ca
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8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" :
T`outj`l
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8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" 29
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8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" 22
T`outj`l 7
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8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" 25
AU5 (Vljh`ra `r L`l Vljh`ra)
Cjsd
5
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8/12/2019 Rigid Body Dynamics Theory_E
13/45
YMZTJDT
"abljsmkuabrpmysjds.jl" 24
Tqubrf Yobtf
BE
1DC
1CH
125
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8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" 23
6. A@AFLS @H JLFUSJB @H E@CJFT ^JSM DVS 7Fxbapof 5>
B uljh`ra cjsd `h rbcjus U mbs b r`ulc cjsd `h rbcjus U/4 dut bs sm`wl jl
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`h tmf cjsd.
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bo`ln tmf y-bxjs. Hjlc tmf t`rquf bdtjln `l tmf pbrtjdof bt b tjaf t be`ut `rjnjl 0
8/12/2019 Rigid Body Dynamics Theory_E
15/45
YMZTJDT
"abljsmkuabrpmysjds.jl" 26
T`outj`l 7
S`rquf js pr`cudfc ey tmf h`rdf `h nrbvjty.
1 r H sjl k
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Y R
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8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" 28
S`rquf be`ut p`jlt @ , j29H,g6j4r,Hr @@@
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YMZTJDT
"abljsmkuabrpmysjds.jl" 2>
Sw` l`lpbrboofo blc l`ljltfrsfdtjln ojlfs brf dboofc skfw ojlfs.
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tfl1 Srd`s
8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" 2=
tfl
1d`s
an(sjl ) d`s 1 ansjl (bo`ln p`sjtjvf g )
(jj) S`rquf be`ut bxjs @@'
S`rquf `h nrbvjty be`ut bxjs @@' an
1 9 (h`rdf an pbrboofo t` bxjs @@')
S`rquf `h tflsj`l be`ut bxjs @@' tfl
1 9 (h`rdf S js pbssjln tmr`unm tmf bxjs @@')
Lft t`rquf be`ut bxjs @@' lft
1 9
8.4 H`rdf D`upof 7B pbjr `h h`rdfs fbdm `h sbaf abnljtucf blc bdtjln jl
`pp`sjtf cjrfdtj`l js dboofc b h`rdf d`upof.
S`rquf cuf t` d`upof 1 Abnljtucf `h `lf h`rdf
cjstbldf eftwffl tmfjr ojlfs `h bdtj`l.
Abnljtucf `h t`rquf 1 1 H (5c)
B d`upof c`fs l`t fxfrt b lft h`r df `l bl `egfdt fvfl tm`unm jt fxfrts b t`rquf. Lft t`rquf cuf t` b h`rdf d`upof js sbaf be`ut bly p`jlt.
S`rquf be`ut B 1 x2H + x
5H
1 H(x2
+ x5
) 1 Hc
S`rquf be`ut E 1 y2H y
5H
1 H(y2
y5) 1 Hc
Jh lft h`rdf bdtjln `l b systfa js zfr`, t`rquf js sbaf be`ut bly p`jlt. B d`ls fqufldf js tmbt , jh H
lft1 9 blc
lft1 9 be`ut `lf p`jlt, tmfl
lft1 9 be`ut bly p`jlt.
8.3 Y`jlt `h Bppojdbtj`l `h H`rdf 7Y`jlt `h Bppojdbtj`l `h h`rdf js tmf p`jlt bt wmjdm, jh lft h`rdf js bssuafc t` ef bdtjln, tmfl jt
wjoo pr`cudf sbaf trblsobtj`lbo bs wfoo bs r`tbtj`lbo fhhfdt, bs wbs pr`cudfc fbrojfr.
^f dbl bos` cfhjlf p jlt `h bppojdbtj`l `h h`rdf bs b p jlt be`ut wmjdm t rquf `h boo tmf h rdfs js zfr`.
1
D`lsjcfr tmrff h`rdfs452 H,H,H
bdtjln `l b e`cy jh C js p`jlt `h bppojdbtj`l `h h`rdf tmfl t`rquf `h
452 HHH
bdtjln bt b p`jlt C be`ut @ js sbaf bs tmf `rjnjlbo t`rquf be`ut @
445522 HrHrHr
1 )HHH(r 452
8/12/2019 Rigid Body Dynamics Theory_E
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"abljsmkuabrpmysjds.jl" 2:
Fxbapof 46.
Cftfrajlf tmf p`jlt `h bppojdbtj`l `h h`rdf, wmfl h`rdfs `h 59 L & 49 L
brf bdtjln `l tmf r`c bs sm`wl jl hjnurf.
T`outj`l 7
Lftt h`rdf bdtjln `l tmf r`c Hrfo
1 29L
Lftt t`rquf bdtjln `l tmf r`c be`ut p`jlt D
d
1 (59 9) + ( 49 59) 1 899 do`dkwjsf
Oft tmf p`jlt `h bppojdbtj`l ef bt b cjstbldf x hr`a p`jlt D
899 1 29 x x 1 89 da >9 da hr`a B js p`jlt `h Bppojdbtj`l
L`tf 7 (j) Y`jlt `h bppojdbtj`l `h nrbvjtbtj`lbo h`rdf js kl`wl bs tmf dfltrf `h nrbvjty.
(jj) Dfltrf `h nrbvjty d`jldjcfs wjtm tmf dfltrf `h abss jh vbouf `h n
js bssuafc t` ef d`lstblt.
(jjj) D`ldfpt `h p`jlt `h bppojdbtj`l `h h`rdf js jabnjlbry, bs jl s`af dbsfs jt dbl ojf `utsjcf tmf e`cy.
8.6 U`tbtj`l be`ut b hjxfc bxjs 7Jh
Mjlnf1 a`aflt `h jlfrtjb be`ut tmf bxjs `h r`tbtj`l (sjldf tmjs bxjs pbssfs tmr`unm tmf mjlnf,
mfldf tmf lbaf Mjlnf
).
fxt
1 rfsuotblt fxtfrlbo t`rquf bdtjln `l tmf e`cy be`ut bxjs `h r`tbtj`l1 blnuobr bddfofrbtj`l `h tmf e`cy.
Mjlnffxt
1Mjlnf
xMjlnf
Hjxfc bxjs `hU`tbtj`l
U`tbtj`lbo Kjlftjd Flfrny 1 5..
5
2
DAvAY
DAfxtfrlbo bAH
Lft fxtfrlbo h`rdf bdtjln `l tmf e`cy mbs tw` d`ap`lflt tblnfltjbo blc dfltrjpftbo.
HD 1 abD 1 DA
5
r
v
a 1 a5
rDA
Ht1 ab
t1 ar
DA
Fxbapof 48.
B puoo fy mbvjln rbcjus r blc a`aflt `h jlfrtjb be`ut jts bxjs jshjxfc bt tmf t`p `h bl jldojlfc poblf `h jldojlbtj`lbs sm`wl jlhjnurf. B strjln js wrbppfc r`ulc tmf puoofy blc jts hrff flc
supp`rts b eo`dk `h abss a wmjdm dbl sojcf `l tmf poblf.
Jljtjbooy, tmf puoofy js r`tbtjln bt b spffc9
jl b cjrfdtj`l sudm
tmbt tmf eo`dk sojcfs up tmf poblf. Dboduobtf tmf cjstbldf a`vfc
ey tmf eo`dk efh`rf st`ppjln 0
T`outj`l 7Tupp`sf tmf cfdfofrbtj`l `h tmf eo`dk js b. Smf ojlfbr cfdfofrbtj l `h tmf rja `h tmf puoofy js bos` b. Smf
blnuobr cfdfofrbtj`l `h tmf puoofy js1 b/r. Jh tmf tflsj`l jl tmf strjln js S, tmf fqubtj`ls `h a `tj`l brfbs h`oo`ws7
an sjl S 1 ab blc Sr 1 1b/r.Fojajlbtjln S hr`a tmfsf fqubtj`ls,
an sjl 5r
b1 ab
njvjln, b 15
5
ar
sjlran
Smf jljtjbo vfo`djty `h tmf eo`dk up tmf jldojlf js v 1 9r. Smus, tmf cjstbldf a`vfc ey tmf eo`dk efh`rf
st`ppjln js
x 1b5
v 51
sjlra5
)ar(r5
5559
1
sjlna5
)ar( 5
95
8/12/2019 Rigid Body Dynamics Theory_E
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"abljsmkuabrpmysjds.jl" 59
Fxbapof 4>.
Smf puoofy sm`wl jl hjnurf mbs b a`aflt `h jlfrtjbbe`ut jts bxjs blcjts rbcjus js r. Dboduobtf tmf abnljtucf `h tmf bddfofrbtj`l `h tmf tw`
eo`dks.
Bssuaf tmbt tmf str jln js ojnmt blc c`fs l`t sojp `l tmf puoofy.
[MDX A@CJHJFC UMN TJUQ
T`outj`l 7
Tupp`sf tmf tflsj`l jl tmf ofht strjln js S2blc tmbt jl tmf rjnmt strjln jl S 5. Tupp`sf tmf eo`dk `h abssa
2n`fs c`wl wjtm bl bddfofrbtj`lblc tmf `tmfr eo`dk a`vfs up wjtm tmf sbaf bddfofrbtj`l. Smjs js
bos` tmf tblnfltjbo bddfofrbtj`l `h tmf rja `h tmf wmffo bs tmf strjln c`fs l`t sojp `vfr tmf rja. Smf
blnuobr bddfofrbtj`l `h tmf wmffo js, tmfrfh`rf,1 b/r. Smf fqubtj`ls `h a`tj`l h`r tmf abss a2, tmf
abss a5blc tmf puoofy brf bs h`oo`ws 7
a2n S
21 a
2b .........(j)
S5 a
5n 1 a
5b .........(jj)
S2r S
5r 1 J1 J/r .........(jjj)
Yuttjln S2blc S
5hr`a (j) blc (jj) jlt` (jjj),
[(a2n b) a
5(n + b)Q r 1
r
b
wmjdm njvfs b 1 552
452
r)aa(
nr)aa(
.
Fxbapof 4=.
B uljh`ra r`c `h abss a blc oflntmdbl r`tbtf jl vfrtjdbo poblf be`ut b sa``tm m`rjz`ltbo bxjs mjlnfc
bt p`jlt M.
(j) Hjlc blnuobr bddfofrbt j l `h tmf r`c gust bhtfr jt js rfofbsfchr`a jljtjbo m`rjz`ltbo p`sjtj`l hr`a rfst 0
(jj) Dboduobtf tmf bddfofrbtj`l (tblnfltjbo blc rbcjbo) h p`jlt B bt tmjs a`aflt.
(jjj) Dboduobtf lft mjlnf h`rdf bdtjln bt tmjs a`aflt.
(jv) Hjlcblc wmfl r`c efd`afs vfrtjdbo.(v) Hjlc mjlnf h`rdf wmfl r`c efd`af vfrtjdbo.
T`outj`l 7(j)
M 1 J
M
an.5
1
4
a 5 1
5
n4
(jj) btB
115
n4. 1
5
n4
bDB
15 r 1 9. 1 9 ( 1 9 gust bhtfr rfofbsf)
(jjj) Tupp`sf mjlnf fxfrts l`rabo rfbdt j l jl d`ap`lflt h`ra bs sm`wl
Jl vfrtjdbo cjrfdtj`lH
fxt1 ab
DA
an L2
1 a.3
n4(wf nft tmf vbouf `h b
DAhr`a prfvj`us fxbapof)
L2
13
an
Jl m`rjz`ltbo cjrfdtj`l
Hfxt
1 abDA
L51 9 ( bDA jl m`rjz`ltbo 1 9 bs 1 9 gust bhtfr rfofbsf).
(vj) S`rquf 1 9 wmfl r`c efd`a fs vfrtjdbo.
s` 1 9
usjln flfrny d`lsfrvbtj`l 5
52
5an
4a
5
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"abljsmkuabrpmysjds.jl" 52
1
n4
(v) ^ mfl r`c efd`afs vfrtjdbo
1 9, 1
n4
HM an 1 5
a 5
HM
15
an6Bls.
>. FRVJOJEUJVAB systf a js jl afd mbljdbo fqujojerjua jh jt js jl trblsobtj`lbo bs wfoo bs r`tbtj`lbo fqujojer jua.
H`r tmjs 7 9Hlft
9lft (be`ut fvfry p`jlt)
]
Z
H2
H5
H4H3
H6
Hr`a (8.4), jh 9Hlft tmfl lft js sbaf be`ut fvfry p`jlt
Mfldf lfdfssbry blc suhhjdjflt d`lcjtj`l h`r fqujojerjua js 9Hlft , 9lft be`ut bly `lf p`jlt,
wmjdm wf dbl dm``sf bs pfr `ur d`lvfljfldf. (lft wjoo but`abtjdbooy ef zfr` be`ut fvfry p`jlt)
ulstbeoffqujojerjua
stbeoffqujojerjua
Lfutrbofqujojerjua
Smf fqujojerjua `h b e`cy js dboofc stbeofjh tmf e`cy trjfs t` rfnbjl jts fqujojerjua p`sjtj`l bhtfr efjln
sojnmtoy cjspobdfc blc rfofbsfc. Jt js dboofc ulstbeof jh jt nfts hurtmfr cjspobdfc bhtfr efjln sojnmtoy
cjspobdfc blc rfofbsfc. Jh jt dbl stby jl fqujojerjua fvfl bhtfr efjln sojnmtoy cjspobdfc blc rfofbsfc, jt
js sbjc t` ef jl lfutrbo fqujojerjua.
Fxbapof 4:.
Sw` e`ys wfjnmjln 59 kn blc 56 kn brf tryjln t` ebobldf b sffsbw `h
t`tbo oflntm 3 a, wjtm tmf huodrua bt tmf dfltrf. Jh `lf `h tmf e`ys js
sjttjln bt bl flc, wmfrf sm`uoc tmf `tmfr sjt 0
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"abljsmkuabrpmysjds.jl" 55
T`outj`l 7Jt js dofbr tmbt tmf 59 kn kjc sm`uoc sjt bt tmf flc blc tmf 56 kn kjc sm`uoc sjt do`sfr t` tmf dfltrf.
Tupp`sf mjs cjstbldf hr`a tmf dfltrf js x. Bs tmf e`ys brf jl fqujojerjua, tmf l`rabo h`rdf eftwffl b
e`y blc tmf sffsbw fqubos tmf wfjnmt `h tmbt e`y. D`lsjcfrjln tmf r`tbtj`lbo fqujojerjua `h tmf sffsbw,
tmf t`rquf `h tmf h`rdfs bdtjln `l jt sm`uoc bcc t` zfr`. Smf h`rdfs brf
(b) (56 kn) n c`wlwbrc ey tmf 56 kn e`y,
(e) (59 kn) n c`wlwbrc ey tmf 59 kn e`y,
(d) wfjnmt `h tmf sffsbw blc
(c) tmf l`rabo h`rdf ey tmf huodrua.
Sbkjln t`rqufs be`ut tmf huodrua,
(56 kn)n x 1 (59 kn)n (5 a) r x 1 2.8 a.
Fxbapof 39.
B uljh`ra r`c `h abss a 1 26 kn ofbl s bnbjlst b sa``tm vfrtjdbo wboo abkjln bl blnof 1 4> wjtmm`rjz`ltbo. Smf `tmfr flcs rfsts `l b r`unm m`rjz`ltbo ho``r. Dboduobtf tmf l`rabo h`rdf blc tmf hrjdtj`l
h`rdf tmbt tmf ho``r fxfrts `l tmf r`c. [Sbkf n 1 29 a/s5Q [M X A@CJHJFC UMNTJUQ
T `o ut j` lSmf h`rdfs bdtjln `l tmf r`c brf sm`wl jl hjnurf. Smfy brf
( b) Jts wfjnmt ^,
( e) l`rabo h`rdf L2ey tmf vfrtjdbo wboo,
( d ) l`rabo h `rdf L5ey tmf ho``r blc
( c) hrjdtj lbo h`rdf h ey tmf ho``r.
Sbkjln m`rjz`ltbo blc vfrtjdbo d`ap`lflts,
L2
1 h ..........( j)
blc L51 an ..........( jj)
Sbkjln t`rquf be`ut E,
L2( B@) 1 an( DE)
`r, L2( B E) d`s 1 an
5
BEsjl `r L
2 6
41
5
^
6
3
`r, L21
4
5^ ..........( jjj)
Smf l rabo h`rd f ey tmf ho ``r js L5
1 1 ( 26 kn) ( 2 9 a/s5) 1 269 L.
Smf hrjdtj`lbo h`rdf js h 1 L2
14
5^ 1 299 L.
Fxbapof 32.
Smf obccfr sm`wl jl hjnurf mbs lfnojnjeof abss blc rfsts `l bsa``tm
ho``r. B dr`ssebr d`llfdts tmf tw` ofns `h tmf obccfr bt tmf dfltrf bs
sm`wl jl hjnurf. Smf blnof eftwffl tmf tw` ofns js:9. Smfpfrs`l
sjttjln `l tmf obccfr mbs b abss `h89kn.Dboduobtftmfd`ltbdth`rdfs
fxfrtfc ey tmf ho``r `l fbdm ofn blc tmf tflsj`l jl tmf dr`ssebr.
[Sbkf n 1 29a/s5Q [MDX A@CJHJFC UMN TJUQ
T`outj`l 7
Smf h`rdfs bdtjln `l cjhhfrflt pbrts brf sm`wl jl hjnurf. D`lsjcfr tmf vfrtjdbo fqujojerjua `h tmf obccfr
pous tmf pfrs`l systfa. Smf h`rdfs bdtjln `l tmjs systfa brf jts wfjnmt (89kn)n blc tmf d`ltbdt h`rdf
Ly+ L
y1 5 L
ycuf t` tmf ho``r. Smus
5 Ly1 (89 kn) n r L
y1 (49 kn) (29 a/s5) 1 499L.
Lfxt d`lsjcfr tmf fqujojerjua `h tmf ofht ofn `h tmf obccfr. Sbkjln t`rqufs `h tmf h`rdfs bdtjln `l jt be`ut
tmf uppfr flc,
Ly(5a) tbl 36 1 S(2 a) r S 1L
y5 1(499 L) 5 1 899L.
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"abljsmkuabrpmysjds.jl" 54
=. BLNVOBU A@AFLSVA O =. 2. Blnuobr a`afltua `h b pbrtjdof be`ut b p`jlt.
O
1 Yr
O 1 rpsjl
`r O
1 r
Y
`r O
1 Y r
^mfrf Y
1 a`afltua `h pbrtjdof
r
1 p`sjtj`l `h vfdt`r `h pbrtjdof wjtm rfspfdt t` p`jlt @ be`ut wmjdm blnuobr a`afltua
js t` ef dboduobtfc .
1 blnof eftwffl vfdt`rs Y&r
r
1 pfrpflcjduobr cjstbldf `h ojlf `h a`tj`l `h pbrtjdof hr`a p`jlt @.
Y
1 d`ap`lflt `h a`afltua pfrpflcjduobr t` r
.
TJ uljt `h blnuobr a`afltua js kna 5/sfd.
Fxbapof 35.B pbrtjdof`h abss ajs pr`gfdtfc bt tjaf t 1 9 hr`a b p`jlt@ wjtm b spffcu bt bl blnof `h 36 t` tmf
m`rjz`ltbo. Dboduobtf tmf abnljtucf blc tmf cjrfdtj`l `h tmf blnuobr a`afltua `h tmf pbrtjdof be`ut
tmf p`jlt @ bt tjaf t 1 u/n. [MDX A@CJHJFC UMN TJUQ
T`outj`l 7
Oft us tbkf tmf `rjnjl bt Y, ]-bxjs bo`ln tmf m`rjz`ltbo blc
Z-bxjs bo`ln tmf vfrtjdbooy upwbrc cjrfdtj`l bs sm`wl jl hjnurf.
H`r m`rjz`ltbo a`tj`l curjln tmf tjaf 9 t` t,
vx1 u d`s 36 1 u/ 5
blc x 1 vxt 1
5
u.
n
u1
n5
u5
.
H`r vfrtjdbo a`tj`l,
vy1 u sjl 36 1
5
u u 1
5
)52( u
blc y 1 (u s jl 36) t 5
2nt5
1n5
u5
n5
u5
1n5
u5
( 5 2).
Smf blnuobr a`afltua `h tmf pbrtjdof bt tjaf t be`ut tmf `rjnjl js
O 1 r
p
1 ar
v
1 a( j x + g y) ( j vx+ g v
y) 1 a ( kxvy kyvx)
1 a k
5
u)25(
n5
u)52(
5
u
n5
u 555
1 kn55
au4.
Smus, tmf blnuobr a`afltua `h tmf pbrtjdof jsn55
au4
jl tmf lfnbtjvf W-cjrfdtj`l j.f., pfrpflcjduobr t`
tmf poblf `h a`tj`l, n`jln jlt` tmf poblf.
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"abljsmkuabrpmysjds.jl" 53
Fxbapof 34.
B pbrtjdof `h abs s 'a' stbr ts a`v jln hr`a p`jlt (`,c) wjtm b
d`lstblt vfo`djty u j . Hjlc `ut jts blnuobr a`afltua be`ut
`rjnjl bt tmjs a`aflt wmbt wjoo ef tmf blswfr bt tmf obtfr tjaf0
T`outj`l 7 O
1 a c u k .
Fxbapof 33.
B pbrtjdof `h abs s 'a' js pr`gfdt fc `l m`rj z`ltbo nr`ulc wjtm bl jljt jbo vfo`djty `h u abk jln bl blnofwjtm m`rjz`ltbo . Hjlc `ut tmf blnuobr a`afltua `h pbrtjdof be`ut tmf p`jlt `h pr`gfdtj`l wmfl .
(j) jt g us t s tbrts jts a`tj `l
(jj) jt js bt mjnmfst p`jlt `h pbtm.
(jjj) jt gust strjkfs tmf nr`ulc.
Bls. (j) @ ; (jj) au d`sn5
sjlu 55 ; (jjj) au sjl
n
5sjlu5
T`outj`l 7
usjl
ud`s
M
U
@
B
E
(j) Blnuobr a`afltua be`ut p`jlt @ js zfr`.(jj) Blnuobr a`afltua be`ut p`jlt B.
prO
O 1 M au d`s
O 1 au d`sn5
sjlu 55 Bls.
(jjj) Blnuobr a`afltua be`ut p`jlt E.
O 1 U au sjl
au sjln
5sjlu5 Bls.
Fxbapof 36.
B pbrtjdo f `h abs s'a ' js pr`gfdtfc `l m`rjz`ltbo nr`ulc wjtm bl jljt jbo
vfo`djty `h u abkjln bl blnof wjtm m`rjz`ltbo . Hjlc `ut tmf blnuobra`afltua bt bly tjaf t `h pbrtjdof p be`ut 7
(j) y bxjs
(jj) z-bxjs
T`outj`l 7
(j) vfo`djty d`ap`lflts brf pbrboofo t` tmf y-bxjs. s`, O 1 9
(jj) 1 ct
cO
2/5 au d`s . nt5
usjl nt
ud`s
z anx 1ct
cO
anx ct 1 cO
O
9
t
9
cOctanx
blnuobr a`afltua be`ut tmf z-bxjs js 7
O 1 2/5 au d`s. nt5 Bls.
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"abljsmkuabrpmysjds.jl" 56
=.5 Blnuobr a`afltua `h b rjnjc e`cy r`tbtjln be`ut hjxfc bxjs 7
r2
r5
rl
B
E
Blnuobr a`a fltua `h b rjnjc e`cy be`ut tmf hjxfc bxjs BE js OBE
1 O2
+ O5
+ O4
+....... +Ol
O21 a
2r
2r
2, O
51 a
5r
5r
5 , O
4 1 a
4r
4r
4 , O
l1 a
lr
lr
l
OBE
1 a2r
2r
2 + a
5r
5r
5 + a
4r
4r
4 ........ + a
lr
lr
l
OBE
1
ll
2l
5ll )r(a
M
ll
2l
5ll )r(a
OBE
1M
OM
1 JM
OM
1 Blnuobr a`afltua `h `egfdt be`ut bxjs `h r`tbtj`l.
JM
1 A`aflt `h Jlfrtjb `h rjnjc , e`cy be`ut bxjs `h r`tbtj`l.
1 blnuobr vfo`djty `h tmf `egfdt.
Fxbapof 38.
Sw`pbrtjdofseboos B blc E, fbdm `h abss a , brf bttbdmfc rjnjcoy t` tmf flcs `h b ojnmt r`c `h oflntm.
Smf systfar`tbtfs be`ut tmf pfrpflcjduobr ejsfdt`r `h tmf r`c bt bl blnuobr spffc. Dboduobtf tmfblnuobr a`afltua `h tmf jlcjvjcubopbrtjdofsblc `h tmf systfa be`ut tmf bxjs `h r`tbtj`l.
T`outj`l 7D`lsjcfr tmf sjtubtj`l sm`wl jl hjnurf. Smf vfo`djty `h tmfpbrtjdofB wjtm rfspfdt t` tmf dfltrf @ js v 1
5
. Smf blnuobr a`afltua `h tmfpbrtjdofwjtm rfspfdt t` tmf bxjs js
O2
1 avr 1 a
5
5
1
3
2a5. Smf sbaf tmf blnuobr a`afltua O
5`h
tmf sfd`lc pbrtjdof. Smf blnuobr a`afltua `h tmf systfa js fqubo t` sua `h tmfsf tw` blnuobr
a`afltua j.f., O 1 2/5 a5.
Fxbapof 3>.
Sw`saboo eboos `h abss a fbdm brf bttbdmfc t` b ojnmt r`c `h oflntm, `lf bt jts dfltrf blc tmf `tmfr
bt b hrff flc. Smf r`c js hjxfc bt tmf `tmfr flc blc js r`tbtfc jlm`rjz`ltbopoblf bt bl blnuobr spffc.Dboduobtf tmf blnuobr a`afltua `h tmfeboobt tmf flc wjtm rfspfdt t` tmf eboobt tmf dfltrf.
T`outj`l 7
Smf sjtubtj`l js sm`wl jl hjnurf. Smf vfo`djty `h tmf eboo B wjtm rfspfdt t` tmf hjxfc flc @ js
vB
1 5/ blc tmbt `h E wjtm rfspfdt t` @ js vE
1. Mfldf tmf vfo`djty `h E wjtm rfspfdt t` B js
vE
vB
1 5/ . Smf blnuobr a`afltua `h E wjtm rfspfdt t` B js, tmfrfh rf,
O 1 avr 1 a
5
5
1
32 a5
bo`ln tmf cjrfdtj`l pfrpflcjduobr t` tmf poblf `h r`tbtj`l.
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"abljsmkuabrpmysjds.jl" 58
Fxbapof 3=.
B uljh`ra djrduobrrjln`h abss399n blc rbcjus29 dajs r`tbtfc be`ut `lf `h jts cjbaftfr bt bl blnuobr
spffc `h 59 rbc/s. Hjlc tmf kjlftjd flfrny `h tmf rjlnblc jts blnuobr a`afltua be`ut tmf bxjs `h
r`tbtj`l.
T`outj`l 7
Smf a`aflt `h jlfrtjb `h tmf djrduobrrjlnbe`ut jts cjbaftfr js
15
2Ar5 1
5
2(9.399 kn) (9.29 a)5 1 5 294 kn-a5.
Smf kjlftjd flfrny js
K 15
25 1
5
2(5 294 kn - a5) (399 rbc5/s5) 1 9.3 G
blc tmf blnuobr a`afltua be`ut tmf bxjs `h r`tbtj`l js
O 1 1 (5 294 kn-a5) (59 rbc/s)1 9.93 kn-a5/s 1 9.93 G-s.
=.4 D`lsfrvbtj`l `h Blnuobr A`afltuaLfwt`l's 5lc obw jl r`tbtj`l 7 ct
Oc
wmfrf
blc O
brf be`ut tmf sbaf bxjs.
Blnuobr a`afltua h b pbrtjdof `r b systfa rfabjls d`lstblt jh fxt
1 9 be`ut tmf bxjs `h r`tbtj`l.
Fvfl jh lft blnuobr a`afltua js l`t d`lstblt, `lf `h jts d`ap`lflt `h bl blnuobr a`afltua be`ut bl
bxjs rfabjls d`lstblt jh d`ap`lflt `h t`rquf be`ut tmbt bxjs js zfr`
Japuosf `h S`rquf 7 GctGDmbrnf jl blnuobr a`afltua.
Fxbapof 3:.
B uljh `ra r`c `h abss a blc oflntm
dbl r`tbtf hrffoy `l b sa``tmm`rjz`ltbo poblf be`ut b vfrtjdbo bxjs mjlnfc bt p`jlt M. B p`jlt abss
mbvjln sbaf abss a d`ajln wjtm bl jljtjbo spffc u pfrpflcjduobr t`
tmf r`c, strjkfs tmf r`c jl-fobstjdbooy bt jts hrff flc. Hjlc `ut tmf blnuobr
vfo`djty `h tmf r`c gust bhtfr d`oojsj`l 0
T`outj`l 7
Blnuobr a`a fltua js d`lsfrv fc be`u t M efdbusf l` fxtfrlbo h`rdf js prfs flt jl m`rjz`ltbo poblf
wmjdm js pr`cudjln t`rquf be`ut M.
auo 1
5
5
a4
a
13
b4
Fxbapof 69.
B uljh `ra r`c `h abs sa2 blc oflntm
ojfs `l bhrjdtj`lofss m`rjz`ltbo poblf. B pbrtjdof `h abss a5a`vjln bt b spffcv9pfrpflcjduobr t` tmf oflntm `h tmf r`c strjkfs jt bt b cjstbldf /4hr`a tmf dfltrf
blc st`ps bhtfr tmf d`oojsj`l. Dboduobtf(b) tmf vfo`djty `h tmf dfltrf `h tmf r`c blc (e) tmf blnuobr
vfo`djty `h tmf r`c be`ut jts dfltrf gust bhtfr tmf d`oojsj`l.
T`outj`l 7
Smf sjtubtj`l js sm`wl jl hjnurf. D`lsjcfr tmf r`c blc
B
r9
X
w
(e)
tmf pbrtjdof t`nftmfr bs tmf systfa. Bs tmfrf js l`
fxtfrlbo rfsuotblt h`rdf, tmf ojlfbr a`afltua `h tmf
systfa wjoo rfabjls d`lstblt. Bos` tmfrf js l` rfsuotblt
fxtfrlbo t`rquf `l tmf systfa blc s` tmf blnuobr
a`afltua `h tmf systfa be`ut tmf bly ojlf wjoo rfabjl
d`lstblt. Tupp`sf tmf vfo`djty `h tmf dfltrf `h tmf r`c
js X blc tmf blnuobr vfo`djty be`ut tmf dfltrf js .
(b) Smf ojlfbr a`afltua efh`rf tmf d`oojsj`l js av blc tmbt bhtfr tmf d`oojsj`l js AX.
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"abljsmkuabrpmysjds.jl" 5>
Smus, a5v
91 a
2X, r X 1
2
5
a
av
9
(e) Oft B ef tmf dfltrf `h tmf r`c wmfl jt js bt rfst. Oft BE ef tmf ojlf pfrpflcjduobr t` tmf poblf
`h tmf hjnurf. D`lsjcfr tmf blnuobr a`afltua `h tmf r`c pous tmf pbrtjdof systfa be`ut BE.
Jljtjbooy tmf r`c js bt rfst. Smf blnuobr a`afltua `h tmf pbrtjdof be`ut BE js
O 1 a5v
9(/4)
Bhtf r tm f d`o oj sj` l, tm f pb rtj dof d` afs t` rfs t. Smf bln uo br a`a flt ua `h tm f r`c
be`ut B js
O
1 daO
+a2 9r
X
Bs 9r
|| X
, 9r
X
1 9
Smus, O
1 daO
Mfldf tmf blnuobr a`afltua `h tmf r`c be`ut BE js
O 1 125
a 52 . Smus,4
va51
25
a 52
`r, 1
2
95
a
va3
:. D@AEJLFC SUBLTOBSJ@LBO BLC U@SBSJ@LBO A@SJ@L@H B UJNJC E@CZSmf nflfrbo a`tj`l `h b rjnjc e`cy dbl ef tm`unmt `h bs b sua `h tw` jlcfpflcflt a`tj`ls. B trblsob-
tj`l `h s`af p`jlt `h tmf e`cy pous b r`tbtj`l be`ut tmjs p`jlt . B a`st d`lvfljflt dm`jdf `h tmf p`jlt js
tmf dfltrf `h abss `h tmf e`cy bs jt nrfbtoy sjapojhjfs tmf dboduobtj`ls.
D`lsjcfr b hbl jlsjcf b trbjl, blc bl `esfrvfr B `l tmf pobth`ra.Jt tmf hbl js swjtdmfc `hh wmjof tmf trbjl a`vfs, tmf a`tj`l `h hbl js purf trblsobtj`l bs fbdm p`jlt `l tmf
hbl ulcfrn`fs sbaf trblsobtj`l jl bly tjaf jltfrvbo.
Jt hbl js swjtdmfc `l wmjof tmf trbjl js bt rfst tmf a`tj`l `h hbl js purf r`tbtj`l be`ut js bxof ; bs fbdm
p`jlt `l tmf bxof js bt rfst, wmjof `tmfr p`jlts rfv`ovf be`ut jt wjtm fqubo blnuobr vfo`djty.
jh tmf hbl js swjtdmfc `l wmjof tmf trbjl js a`vjln, tmf a`tj`l `h hbl t` tmf `esfrvfr `l tmf pobth`ra js
lfjtmfr purf trblsobtj`l l`r purf r`tbtj`l. Smjs a`tj`l js bl fxba pof `h nflfrbo a `tj`l `h b r jnjc e`cy.
L`w jh tmfrf js bl `esfrvfr E jlsjcf tmf trbjl, tmf a`tj`l `h hbl wjoo bppfbr t` mja bs purf r`tbtj`l.
Mfldf wf dbl sff tmbt tmf nflfrbo a`tj`l `h hbl w.r.t. `esfrvfr B dbl ef rfs`ovfc jlt` purf r`tbtj`l `h
hbl bs `esfrvfc ey `esfrvfr E pous purf trblsobtj`l `h `esfrvfr E (w.r.t. `esfrvfr B)
Tudm b rfs`outj`l `h nflfrbo a`tj`l `h b rjnjc e`cy jlt` purf r`tbtj`l & purf trblsobtj`l js l`t rfstrjdtfc
t` gust tmf hbl jlsjcf tmf trbjl, eut js p`ssjeof h`r a`tj`l `h bly rjnjc systfa.
:.2 Kjlfabtjds `h nflfrbo a`tj`l `h b rjnjc e`cy 7H`r b rjnjc e`cy bs fbrojfr stbtfc vbouf `h blnuobr cjspobdfaflt () , blnuobr vfo`djty (), blnu-obr bddfofrbtj`l () js sbaf h`r boo p`jlts `l tmf rjnjc e`cy be`ut bly `tmfr p`jlt `l tmf rjnjce`cy.
Mfldf jh wf kl`w vfo`djty h bly `lf p`jlt (sby B) `l tmf rjnjc e`cy blc blnuobr vfo`djty `h blyp`jlt `l tmf r jnjc e`cy be`ut bly `tmfr p`jlt `l tmf r jnjc e`cy (sby), vfo`djty `h fbdm p`jlt `ltmf rjnjc e`cy dbl ef dboduobtfc.
BX X +B
BX
EX
rjnjc e`cy
B
E
BErX 1EB
XEB1sjldf cjstbldf BE js hjxfc
EBX
BE
wf kl`w tmbt 1EB
EB
r
X
XEB
1 XEB
1rEB
jl vfdt`r h`raEBX
1 EBr
L`w hr`a r fobtjvf vfo`djty 7 BEEB XXX
EBBE XXX
EBBE rXX
sjajobroy EBBE rbb
[h`r bly rjnjc systfaQ
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YMZTJDT
"abljsmkuabrpmysjds.jl" 5=
Fxbapof 62.
D`lsjcfr tmf nflfrbo a`tj`l `h b wmffo (rbcjus r) wmjdm dbl ef vjfw `l purf trblsobtj`l `h jts dfltfr @
(wjtm tmf vfo`djty v) blc purf r `tbtj`l be`ut @ (wjtm blnuobr vfo`djty)
D
C
B
@EX
)k(
j
g
Hjlc `ut C@D@E@B@ v,v,v,v
blc CDEB v,v,v,v
T`outj`l 7 B@v
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B@v
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1 )g(r)k(
B@v
1 jr
sjajobroy E@v 1r )g(
D@v
1r )j(
C@v
1r )g(
Bv
1 jrjvvv B@@
sjajobroy Ev
1 grjvvv E@@
Dv
1 jrjvvv D@@
Cv
1 grjvvv C@@
: .5 Yurf U`oojln `r r`oojln wjtm`ut sojcjln 7Yurf r`oojln js b spfdjbo dbsf `h nflfrbo r`tbtj`l `h b r jnjc e`cy wjtm djrduobr dr`ss sfdtj`l (f.n.
wmffo, cjsd, rjln, spmfrf) a`vjln `l s`af surhbdf. Mfrf, tmfrf js l` rfobtjvf a`tj`l eftwffl
tmf r`oojln e`cy blc tmf surhbdf `h d`ltbdt, bt tmf p`jlt `h d`ltbdt
B
Mfrf d`ltbdt p`jlt js B & d`ltbdt surhbdf js m`rjz`ltbo nr`ulc. H`r purf r`oojln vfo`djty `h B
w.r.t. nr`ulc 1 9 XB
1 9 .
BX
Hr`a be`vf hjnurf, h`r purf r`oojln, vfo`djty `h B w.r.t. t` poblk js zfr` XB
1X.
B
v
r
Hr`a be`vf hjnurf h`r, purf r`oojln, vfo`djty h B w.r.t. nr`ulc js zfr`. v r 1 9
v 1rTjajobroy b 1r
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YMZTJDT
"abljsmkuabrpmysjds.jl" 5:
Fxbapof 65.
B wmffo `h rbcjus r r`oos (r`oojln wjtm`ut soffpjln) `l b ofvfo r`bc bs sm`wl jl hjnurf.
B
E
v
r
Hjlc `ut vfo`djty `h p`jlt B blc ET`outj`l 7
D`ltbdt surhbdf js jl rfst h`r purf r`oojln vfo`djty `h p`jlt B js zfr`.
s` v 1 rvfo`djty `h p`jlt E 1 v + r 1 5v
:.4 Cylbajds `h nflfrbo a`tj`l `h b rjnjc e`cy 7Smjs a`tj`l dbl ef vjfwfc bs trblsobtj`l `h dfltrf `h abss blc r`tbtj`l be`ut bl bxjs pbssjln
tmr`unm dfltrf `h abss
JhDA
1 A`aflt `h jlfrtjb be`ut tmjs bxjs pbssjln tmr`unm D@A
da 1 Lft t`rquf be`ut tmjs bxjs pbssjln tmr`unm D@A
DAb
1 Bddfofrbtj`l `h D@A
DAv
1 Xfo`djty `h D@A
fxtH
1 Lft fxtfrlbo h`rdf bdtjln `l tmf systfa.
systfaY
1 Ojlfbr a`afltua `h systfa.
DAO
1 Blnuobr a`afltua be`ut dfltrf `h abss.
DAr
1 Y`sjtj`l vfdt`r `h D@A w.r.t. p`jlt B.
tmfl (j)
dada
(jj) dafxt bAH
(jjj) dasystfa vAY
(vj) S`tbo K.F.1 5daAv5
2+
5da
5
2
(v)
DADAO
(vj) Blnuobr a`afltua be`ut p`jlt B 1 O
be`ut D.A. + O
`h D.A. be`ut BB
dadadaB vArO
ct
c
ct
cOB ( dadada vAr
)
ct
cB
. L`tjdf tmbt t`rquf fqubtj`l dbl ef bppojfc t` b rjnjc e`cy jl
b nflfrbo a`tj`l `loy blc `loy be`ut bl bxjs tmr`unm dfltrf `h abss.
Fxbapof 64.
B uljh `ra spmfrf `h abss 599 n r`oos wjtm`ut sojppjln `l b poblf surhbd f s` tmbt jts dfltrf a`v fs bt b
spffc `h 5.99 da/s. Hjlc jts kjlftjd flfrny.
T`outj`l 7
Bs tmf spmfrf r`oos wjtm`ut sojppjln `l tmf poblf surhbd f, jts blnuobr spffc be`ut tmf dflt rf js 1
r
v da. Smf kjlftjd flfrny js
K 1 5
2
da5 + 52
Avda5 1 5
2. 6
5Ar55 + 5
2Avda5
16
2Av
da5 +
5
2Av
da5 1
29
>Av
da5 1
29
>(9.599 kn) (9.95 a/s)5 1 6.8 296 G.
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YMZTJDT
"abljsmkuabrpmysjds.jl" 49
Fxbapof 63.
B d`ls tblt h`rdf H bdts tblnfltjbooy bt tmf mjnmfst p`jlt `h b uljh`ra
cjsd `h abss a kfpt `l b r`unm m rjz`ltbo surhbdf bs sm`wl jl hjnurf.
Jh tmf cjsd r`oos wjtm`ut sojppjln, dboduobtf tmf bddfofrbtj`l `h tmf dfltrf
(D) blc p`jlt B blc E `h tmf cjsd. B
D
E H
r
r`unm surhbdf
T`outj`l 7
Smf sjtubtj`l js sm`wl jl hjnurf. Bs tmf h`rdf H r`tbtfs tmf cjsd, tmf p`jlt `h d`ltbdt mbs b tflcfldy t`
sojp t`wbrcs ofht s` tmbt tmf stbtjd hrjdtj l `l tmf cjsd wjoo bdt t`wbrcs rjnmt. Oft r ef tmf rbcjus `h tmf
cjsd blc b ef tmf ojlfbr bddfofrbtj`l `h tmf dfltrf `h tmf cjsd. Smf blnuobr bddfofrbtj`l be`ut tmf
dfltrf `h tmf cjsd js 1 b/r, bs tmfrf js l` sojppjln.H`r tmf ojlfbr a`tj`l `h tmf dfltrf,
H + h 1 ab ..........(j)
blc h`r tmf r`tbtj`lbo a`tj`l be`ut tmf dfltrf,
Hr h r 1 1
5ar5
2
r
b`r, H h 1
5
2ab, ...........(jjj)
Hr`a (j) blc (jj),
5H 1
5
4ab r b 1
a4
H3.
Bddfofrbtj`l `h p`jlt B js zfr`.
Bddfofrbtj`l `h p`jlt E js 5b 1 5
a4
H31
a4
H=Bls.
Fxbapof 66.
B djrduobr rjnjc e`cy `h abss a, rbcjus U blc rbcjus `h nyrbtj`l (k) r`oos wjtm`ut sojppjln `l bl jldojlfc poblf
`h b jldojlbtj`l. Hjlc tmf ojlfbr bddfofrbtj`l `h tmf rjnjc e`cy blc h`rdf `h hrjdtj l `l jt. ^mbt aust ef tmfajljaua vbouf `h d`fhhjdjflt `h hrjdtj`l s` tmbt rjnjc e`cy aby r`oo wjtm`ut sojcjln0
T`o.
Jh b js tmf bddfofrbtj l `h tmf dfltrf `h abss `h tmf rjnjc e`cy blc h tmf h rdf `h hrjdtj l eftwffl spmfrf blc tmf
poblf, tmf fqubtj l `h trblsobt ry blc r`tbt ry a`tj`l `h tmf rjnjc e`cy wjoo ef.
an sjl h 1 ab (Srblsobt`ry a tj`l)
hU 1 (U`tbt`ry a`tj`l)
h 1U
1 ak5 , cuf t` purf r`oojln b 1U
an sjl U
1 aU
an sjl 1 aU + U
an sjl 1 a U +U
ak5
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"abljsmkuabrpmysjds.jl" 42
an sjl 1 ab +U
ak5
an sjl 1
5
55
U
kUb
b 1
5
55
UkU
sjln
b 1
5
5
U
k2
sjln
h 1U
h 1 5
5
Ubak 55
5
kUsjlkan
h L
5
5
U
akb an d`s
U55
5
U
k
)Uk(
sjln55
n d`s
5
5
k
U2
tbl
ajl
1
5
5
k
U2
tbl
L tf 7 Hr`a be`vf fxbapof jh rjnjc e`cjfs brf s`ojc dyojlcfr, m`oo`w dyojlcfr, s`ojc spmfrf blc m`oo`w spmfrf.
(2) Jldrfbsjln `rcfr `h bddfofrbtj l.
bs`ojc spmfrf
< bm`oo`w spmfrf
< bs`ojc dyojlcfr
< bm`oo`w dyojlcfr
(5) Jldrfbsjln `rcfr `h rfqujrfc hrjdtj l h`rdf h`r purf r`oojln.
hm`oo`w dyojlcfr
< hm`oo`w spmfrf
< hs`ojc dyojlcfr
< hs`ojc spmfrf
(4) Jldrfbsjln `rcfr `h rfqujrfc ajljaua hrjdtj l d`fhhjdjflt h`r purf r`oojln.
m`oo`w dyojlcfr
< m`oo`w spmfrf
< s`ojc dyojlcfr
< s`ojc spmfrf
:.3 Jlstbltblf`us bxjs `h r`tbtj`l 7Jt js tmf bxjs be`ut wmjdm tmf d`aejlfc trblsobtj`lbo blc r`tbtj`lbo a`tj`l bppfbrs bs purf r`tbtj lbo
a`tj`l.
Smf d`aejlfc fhhfdt `h trblsobtj`l `h dfltrf `h abss blc r tbtj l be`ut bl bxjs tmr`unm tmf dfltrf `h
abss js fqujvboflt t` b purf r`tbtj l wjtm tmf sbaf blnuobr spffc be`ut b stbtj`lbry bxjs ; tmjs bxjs
js dboofc jlstbltblf`us bxjs `h r`tbtj l. Jt js cfhjlfc h r bl jlstblt blc jts p`sjtj l dmblnfs wjtm tjaf.
fn. Jl purf r`oojln tmf p`jlt `h d`ltbdt wjtm tmf surhbdf js tmf jlstbltblf`us bxjs `h r`tbtj`l.Nf`aftrjdbo d`lstrudtj`l `h jlstbltblf`us bxjs `h r`tbtj l (J.B.U). Crbw vfo djty vfdt r bt bly
tw` p`jlts `l tmf rjnjc e`cy. Smf J.B.U. js tmf p`jlt `h jltfrsfdtj`l `h tmf pfrpflcjduobr crbwl `l
tmfa.
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"abljsmkuabrpmysjds.jl" 45
Jl dbsf `h purf r`oojln tmf o`wfr p`jlt js jlstbltblf`usoy bxjs `h r`tbtj`l.Smf a`tj`l `h e`cy jl purf r`oojln dbl tmfrfh rf ey blboysfc bs purf r`tbtj`l be`ut tmjs bxjs.
D`lsfqufltoy
Y
1Y
Y
1Y
K.F. 1 2/5Y5
^mfrf Y
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Fxbapof 6>.
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"abljsmkuabrpmysjds.jl" 44
Boo surhbdfs brf sa``tm. Smfrfh`rf, afdmbljdbo flfrny wjoo rfabjl d`lsfrvfc.
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wmjdm js dboofc r`oojln hrjdtj`l)
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"abljsmkuabrpmysjds.jl" 43
Smfrf js Ufobtjvf A`tj`l bt p jlt `h d`ltbdt s` Kjlftjd Hrjdtj l, hk
1 L wjoo bdt jl h`rwbrc cjrfdtj l. Smjs kjlftjd
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v5 1 u5 + 5bs
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"abljsmkuabrpmysjds.jl" 46
hK
1 Ab
An 1 Ab
b 1 n ------------------(4)
19
t
hr`a fqubtj`l (2) 19
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"abljsmkuabrpmysjds.jl" 48
Fxbapof 89.
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a b
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YMZTJDT
"abljsmkuabrpmysjds.jl" 4>
1Ua
h5
h 1 ab
H h 1 ae
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b 15
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T`outj`l 7
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bs sm`wl jl tmf hjnurf. Smf
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dyojlcfrs hjrst sojp `vfr fbdm `tmfr bt tmf d`ltbdt eut tmf sojppjln hjlbooy dfbsfs
cuf t` tmf hrjdtj`l eftwffl tmfa.Dboduobtftmf blnuobr spffcs `h tmf dyojlcfrs
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"abljsmkuabrpmysjds.jl" 4=
T`outj`l 7
^mfl sojppjln dfbsfs, tmf ojlfbr spffcs `h tmf p`jlts `h d`ltbdt `h tmf tw` dyojlcfrs wjoo ef fqubo. Jh
2blc
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2
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5.
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8/12/2019 Rigid Body Dynamics Theory_E
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YMZTJDT
"abljsmkuabrpmysjds.jl" 4:
1 h 5AU)4/5(
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"abljsmkuabrpmysjds.jl" 39
29. S@YYOJLN 7Jl ably sjtubtj`ls bl fxtfrlbo h`rdf js bppojfc t` b e`cy t` dbusf jt t` sojcf bo`ln b surhbdf. Jl dfrtbjl
dbsfs, tmf e`cy aby tjp `vfr efh`rf sojcjln flsufs. Smjs js kl`wl bs t`ppjln.
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t` Habx
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t`ppjln. Smjs js tmf dbsf jh
Habx
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an b/e
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"abljsmkuabrpmysjds.jl" 32
Fxbapof 88.
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jt. Jt dbl l`w ef bppojfc be`ut D@A.
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"abljsmkuabrpmysjds.jl" 35
T`outj`l 7 e
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`h tmf systfa be`ut bxjs DC 0
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trjblnof `h sjcf 2 da. Abssfs brf `h2,5,&4kn rfspfdtjvfoy blc
kfpt bs sm`wl jl hjnurf. Dboduobtf tmf a`aflt `h Jlfrtjb `h
systfa be`ut bl bxjs pbssjln tmr`unm 2 kn abss blc
pfrpflcjduobr t` tmf poblf `h trjblnof 0
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J21 5 (2295)5 1 5293
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J51 4 (2295)5 1 4293
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51 6 293 kna 5
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(b)x
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"abljsmkuabrpmysjds.jl" 34
Yr`eofa 6.
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be`ut jts bxjs.
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UcacJ
5
15
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abss A, fcnfs h oflntm '' blc 'e' be`ut jts bxjs pbssjln tmr`unm
dfltrf blc pfrpflcjduobr t` jt.
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2+ J
5
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)e(A 55
Yr`eofa >. Hjlc tmf a`aflt `h jlfrtjb `h b uljh`ra squbrf pobtf `h abss
A, fcnf `h oflntm '' be`ut jts bxjs pbssjln tmr`unm Y blc
pfrpflcjduobr t` jt.
T`outj`l 7 JY
15
A
8
A 55 1
4
A5 5
Yr`eofa =. Hjlc `ut tmf a`aflt `h jlfrtjb `h b rjln mbvjln uljh`ra abss cjstrjeutj`l h abss A & rbcjus U
be`ut bl bxjs wmjdm js tblnflt t` tmf rjln blc (j) jl tmf poblf `h tmf rjln (jj) pfrpflcjduobr t` tmf
poblf `h tmf rjln.
(j) (jj)
T`outj`l 7 (j) A`aflt `h jlfrtjb be`ut bl bxjs pbssjln tmr`unm dfltrf `h rjln blc poblf h tmf rjln
J2
15
AU5
Vsjln pbrboofo bxjs tmf`rfa
J' 1 J2
+ AU5 15
AU4 5
(jj) A`aflt `h jlfrtjb be`ut bl bxjs pbssjln tmr`unm dfltrf `h rjln blc pfrpflcjduobr t`
pbolf `h tmf rjln
JD
1 AU5
Vsjln pbrboofo bxjs tmf`rfa
J'' 1 JD
+ AU5 1 5AU5
Yr`eofa :. Dboduobtf tmf a`aflt `h jlfrtjb h b rfdtblnuobr hrbaf h`rafc
ey uljh`ra r`cs mbvjln abss a fbdm bs sm`wl jl hjnurf be`ut
bl bxjs pbssjln tmr`unm jts dfltrf blc pfrpflcjduobr t` tmf
poblf `h hrbaf 0 Bos` hjlc a`aflt `h jlfrtjb be`ut bl bxjspbssjln tmr`unm YR 0
a
e,a
Y R
UT
T`outj`l 7 (j) A`aflt `h jlfrtjb be`ut bl bxjs pbssjln tmr`unm jts dfltrf blc pfrpflcjduobr t` tmf
poblf `h hrbaf
JD
1 J2+ J
5+ J
4+ J
3
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"abljsmkuabrpmysjds.jl" 33
J2
1 J4, J
51 J
3
JD
1 5J2+ 5J
5
J2
1
55
5
ea
25
a
J
51
55
5a
25
ae
s`, JD
1 )e(4
a5 55
(jj) A.J. be`ut bxjs YR `h r`c YR J
2 1 9
A.J. be`ut bxjs YR `h r`c YT J5
15
ae5
A.J. be`ut bxjs YR `h r`c RU J41
5
ae5
A.J. be`ut bxjs YR `h r`c TU J31 ae5
J 1 J2
+ J5
+ J4
+ J3
14
ae6 5
Yr`eofa 29. Jl tmf prfvj`us qufstj`l, curjln tmf a`tj`l `h pbrtjdof hr`a Y t` R. S rquf `h nrbvjtbtj`lbo h`rdf
be`ut Y js 7(B) jldrfbsjln (E) cfdrfbsjln
(D) rfabjls d`lstblt (C) hjrst jldrfbsjln tmfl cfdrfbsjln
T`outj`l 7
Jldrfbsjln efdbusf cjstbldf hr`a p`jlt Y js jldrfbsjln.
Yr`eofa 22. Cftfrajlf tmf p`jlt `h bppojdbtj`l `h h`rdf, wmfl h`rdfs brf
bdtjln `l tmf r`c bs sm`wl jl hjnurf.
T`outj`l 7
S`rquf `h E be`ut B 2
1 4L 6 1 26L da (do`dkwjsf)
S`rquf `h D be`ut B 5 1 8L 29 1 89 L da (bltjdo`dkwjsf)
Ufsuotblt h`rdf pfrpflcjduobr t` tmf r`c H 1 = L
2
+ 5
1 H x (x 1 cjstbldf hr`a p`jlt B)
26 + 89 1 = x
x 1 36/= 1 6.856 da
Yr`eofa 25. B uljh`ra r`c `h abss a blc oflntm dbl r`tbtf jl vfrtjdbo
poblf be`ut b sa``tm m`rjz`ltbo bxjs mjlnfc bt p`jlt M. Hjlc
blnuobr bddfofrbtj`l `h tmf r`c gust bhtfr jt js rfofbsfc hr`ajljtjbo p`sjtj`l abkjln bl blnof `h 4>9 wjtm m`rjz`ltbo hr`a rfst0
Hjlc h`rdf fxfrtfc ey tmf mjlnf gust bhtfr tmf r`c js rfofbsfchr`a rfst.
T`outj`l 7 S`rquf be`ut mjln 1M
1
and`s4>5
1
4
a 5
1 8n / 6
bt1
5
1
6
n4
and`s4> L21 ab
t
L21 6
an
blnuobr vfo`djty `h r`c js zfr`. s` L51 ansjl4> 1 4an/6
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YMZTJDT
L 1 55
52 LL 1
55
6
an4
6
an
1
6
29an
Yr`eofa 24. B uljh`ra r`c `h oflntm , abss a js muln hr`a tw` strjlns `h
fqubo oflntm hr`a b dfjojln bs sm`wl jl hjnurf . Cftfrajlf tmf
tflsj`ls jl tmf strjlns 0
T`outj`l 7 SB
+ SE
1 an ............(j)
S`rquf be`ut p`jlt B js zfr`
T`, SE
3
41 an
5
............(jj)
Hr`a fq. (j) & (jj),
SB
1 an/4, SE
15an/4.
Yr`eofa 23. B pbrtjdof `h abss a stbrts a`vjln hr`a `rjnjl wjtm b d`lstblt
vfo`djty ju hjlc `ut jts blnuobr a`afltua be`ut `rjnjl bt tmjs
a`aflt. ^mbt wjoo ef tmf blswfr obtfr `l0 ^mbt wjoo ef tmf
blswfr jh tmf spffc jldrfbsfs.
y
xT`outj`l 7 prO
jaujrO
1 9
Yr`eofa 26. B uljh`ra r`c `h abss a blc oflntm dbl r`tbtf hrffoy `l b
sa``tm m`rjz`ltbo poblf be`ut b vfrtjdbo bxjs mjlnfc bt p`jlt
M. B p`jlt abss mbvjln sbaf abss a d`ajln wjtm bl jljtjbo
spffc u pfrpflcjduobr t` tmf r`c, strjkfs tmf r`c blc stjdks t`
jt bt b cjstbldf `h 4/3 hr`a mjlnf p`jlt. Hjlc `ut tmf blnuobr
vfo`djty `h tmf r`c gust bhtfr d`oojsj`l 0
T`outj`l 7 Blnuobr A`afltua be`ut mjlnf
Oj1 O
h
au
3
41
55
3
4a
4
a
134
u48
Yr`eofa 28. Vljh`ra & sa``tm U`c `h oflntmjs a`vjln wjtm b vfo`djty `h dfltrf v
blc blnuobr vfo`djty `l sa``tm m`rjz`ltbo surhbdf. Hjlc`ut vfo`djty`h p`jlt B blc E.
T`outj`l 7 vfo`djty `h p`jlt B w.r.t. dfltfr js5
vfo`djty `h p`jlt B w.r.t. nr`ulc XB
1 X +5
vfo`djty h p`jlt E w.r.t. dfltfr js 5
vfo`djty `h p`jlt E w.r.t. nr`ulc XB
1 X 5
