Chapter 10 Conservation of Angular Momentum

Transcription

1 Chpte Consevton o Angul Momentum Coneptul Polems * ( Tue. The oss podut o the vetos A nd B s dened to e A B ABsnφ nˆ. A nd B e pllel, snφ. ( Tue. B denton, s long the s. ( Tue. The deton o toque eeted oe s detemned the denton o the oss podut. Detemne the Conept The oss podut o the vetos A nd B s dened to e A B ABsnφ nˆ. Hene, the oss podut s mmum when snφ. Ths ondton s stsed povded A nd B e pependul. ( s oet. Detemne the Conept nd p e elted odng to p. Fom ths denton o the oss podut, nd p e pependul;.e., the ngle etween them s 9. 4 Detemne the Conept nd p e elted odng to p. Beuse the moton s long lne tht psses though pont P, nd so s. ( s oet. *5 Detemne the Conept nd p e elted odng to p. ( Beuse s detl popotonl to p Doulng p doules. : ( Beuse s detl popotonl to Doulng doules. : 75

2 76 Chpte 6 Detemne the Conept The gue shows ptle movng wth onstnt speed n stght lne (.e., wth onstnt velot nd onstnt lne momentum. The mgntude o s gven psnφ mv(snφ. Reeng to the dgm, note tht the dstne snφ om P to the lne long whh the ptle s movng s onstnt. Hene, mv(snφ s onstnt nd so s onstnt. 7 Flse. The net toque tng on ottng sstem equls the hnge n the sstem s ngul momentum;.e., τ net d dt, whee. Hene, τ net s eo, ll we n s o sue s tht the ngul momentum (the podut o nd s onstnt. hnges, so must. *8 Detemne the Conept Yes, ou n. mgne ottng the top hl o ou od wth ms lt t sdes though (oughl 9 ngle. Beuse the net ngul momentum o the sstem s, the ottom hl o ou od ottes n the opposte deton. Now etend ou ms out nd otte the top hl o ou od k. Beuse the moment o net o the top hl o ou od s lge thn t ws pevousl, the ngle whh the ottom hl o ou od ottes though wll e smlle, ledng to net otton. You n epet ths poess s neess to otte though n t ngle. 9 Detemne the Conept s onstnt, we know tht the net toque tng on the sstem s eo. Thee m e multple onstnt o tme-dependent toques tng on the sstem s long s the net toque s eo. (e s oet. Detemne the Conept No. n ode to do wok, oe must t ove some dstne. n eh nelst ollson the oe o stt ton does not t though n dstne. Detemne the Conept t s ese to wl dll outwd. n t, dll nwd oe s equed just to pevent ou om sldng outwd. * Detemne the Conept The pull tht the student eets on the lok s t ght ngles to ts moton nd eets no toque (ell tht F nd F snθ. Theeoe, we τ τ

3 Consevton o Angul Momentum 77 n onlude tht the ngul momentum o the lok s onseved. The student does, howeve, do wok n dsplng the lok n the deton o the dl oe nd so the lok s eneg neses. ( s oet. * Detemne the Conept The hdoled egg s sold nsde, so evethng ottes wth unom velot. B ontst, t s dult to get the vsous lud nsde w egg to stt ottng; howeve, one t s ottng, stoppng the shell wll not stop the moton o the nteo lud, nd the egg m stt ottng gn te momentl stoppng o ths eson. 4 Flse. The eltonshp whethe t s spnnng. τ d dt deses the moton o gosope ndependentl o 5 Ptue the Polem We n dvde the epesson o the knet eneg o the ojet the epesson o ts ngul momentum to otn n epesson o K s unton o nd. Epess the ottonl knet eneg o the ojet: K Relte the ngul momentum o the ojet to ts moment o net nd ngul velot: Dvde the st o these equtons the seond nd solve o K to otn: K nd so ( s oet. 6 Detemne the Conept The pupose o the seond smlle oto s to pevent the od o the helopte om ottng. the e oto ls, the od o the helopte wll tend to otte on the mn s due to ngul momentum eng onseved. 7 Detemne the Conept One n use ght-hnd ule to detemne the deton o the toque equed to tun the ngul momentum veto om est to south. ettng the nges o ou ght hnd pont est, otte ou wst untl ou nges pont south. Note tht ou thum ponts downwd. ( s oet.

4 78 Chpte 8 Detemne the Conept n tunng est, the mn edets the ngul momentum veto om noth to est eetng lokwse toque (vewed om ove on the gosope. As onsequene o ths toque, the ont end o the sutse wll dp downwd. (d s oet. 9 ( The ltng o the nose o the plne ottes the ngul momentum veto upwd. t vees to the ght n esponse to the toque ssoted wth the ltng o the nose. ( The ngul momentum veto s otted to the ght when the plne tuns to the ght. n tunng to the ght, the toque ponts down. The nose wll move downwd. Detemne the Conept ponts up nd the tvels ove hll o though vlle, the oe on the wheels on one sde (o the othe wll nese nd wll tend to tp. ponts owd nd tuns let o ght, the ont (o e o the wll tend to lt. These polems n e veted hvng two dentl lwheels tht otte on the sme sht n opposte detons. Detemne the Conept The ottonl knet eneg o the womn-plus-stool sstem s gven K ot. Beuse s onstnt (ngul momentum s onseved nd he moment o net s gete wth he ms etended, ( s oet. * Detemne the Conept Consde the ovehed vew o tethe pole nd ll shown n the djonng gue. The ll ottes ountelokwse. The toque out the ente o the pole s lokwse nd o mgntude RT, whee R s the pole s dus nd T s the tenson. So must deese nd (e s oet. Detemne the Conept The ente o mss o the od-nd-putt sstem moves n stght lne, nd the sstem ottes out ts ente o mss.

5 Consevton o Angul Momentum 79 4 ( Tue. The net etenl toque tng sstem equls the te o hnge o the ngul d momentum o the sstem;.e., τ, et. dt ( Flse. the net toque on od s eo, ts ngul momentum s onstnt ut not neessl eo. Estmton nd Appomton *5 Ptue the Polem Beuse we hve no nomton egdng the mss o the skte, we ll ssume tht he od mss (not nludng he ms s 5 kg nd tht eh m hs mss o 4 kg. et s lso ssume tht he ms e m long nd tht he od s lndl wth dus o m. Beuse the net etenl toque tng on he s eo, he ngul momentum wll emn onstnt dung he pouette. Epess the onsevton o he ngul momentum dung he pouette: Epess he totl moment o net wth he ms out: Tetng he od s though t s lndl, lulte ts moment o net o he od, mnus he ms: Modelng he ms s though the e ods, lulte the moment o net when she hs them out: o ( msout msout msout + od ms m od. kg m msn ms [ ( 4kg( m ].67 kg m msn ( 5kg(.m Susttute to detemne he totl ms out. kg m +.67 kg m moment o net wth he ms out:.67 kg m Epess he totl moment o net msn [ ] wth he ms n:. kg m + ( 4kg(.m + od ms. kg m

6 74 Chpte Solve equton ( o msn nd susttute to otn: msn ms out msn.67 kg m. kg m 4.7 ev/s ms out (.5 ev/s 6 Ptue the Polem We n epess the peod o the eth s otton n tems o ts ngul velot o otton nd elte ts ngul velot to ts ngul momentum nd moment o net wth espet to n s though ts ente. We n deentte ths epesson wth espet to nd then use deentls to ppomte the hnges n nd T. Epess the peod o the eth s otton n tems o ts ngul velot o otton: T π Relte the eth s ngul velot o otton to ts ngul momentum nd moment o net: Susttute to otn: T π Fnd dt/d: Solve o dt/t nd ppomte T: Susttute o nd to otn: Susttute numel vlues nd evlute T: dt d π T dt d o T T T T 5 5 T m M R E E 5m T T M 9 (. kg ( ( d 4 6 kg s 6 6 d E 4h 6s d d h

7 Consevton o Angul Momentum 74 7 Ptue the Polem We n use mv to nd the ngul momentum o the ptle. n ( we n solve the equton l( l + h l l + nd the ppomte vlue o l. ( Use the denton o ngul o ( mv momentum to otn: ( kg( m/s( 4 m.4 8 kg m /s ( Solve the equton l( l + h o l ( l + : l h ( l + Susttute numel vlues nd l + evlute l ( : l( l.4 kg m /s.5 J s Beuse l >>, ppomte ts vlue wth the sque oot o l l + : ( l.9 6 ( The quntton o ngul momentum s not noted n mosop phss euse no epement n deentte etween l 6 +. l 6 nd *8 Ptue the Polem We n use onsevton o ngul momentum n pt ( to elte the eoe-nd-te ollpse otton tes o the sun. n pt (, we n epess the tonl hnge n the ottonl knet eneg o the sun s t ollpses nto neuton st to dede whethe ts ottonl knet eneg s gete ntll o te the ollpse. ( Use onsevton o ngul momentum to elte the ngul moment o the sun eoe nd te ts ollpse: ( Usng the gven omul, ppomte the moment o net 5 o the sun eoe ollpse:.59(.99 kg( 6.96 km.59mr 5.69 sun 46 kg m

8 74 Chpte Fnd the moment o net o the sun when t hs ollpsed nto sphel neuton st o dus km nd unom mss dstuton: Susttute n equton ( nd solve o to otn: MR 5 5 (.99 kg( km kg m kg m kg m Gven tht ev/5 d, evlute : 8 ev 7.5 5d.86 7 ev/d The ddtonl ottonl knet eneg omes t the epense o gvttonl potentl eneg, whh deeses s the sun gets smlle. Note tht the ottonl peod deeses the sme to o / nd eomes: T π 7.86 π ev π d d h d ev 4h 6s. s ( Epess the tonl hnge n the sun s ottonl knet eneg s onsequene o ts ollpse nd smpl to otn: K K K K K K K Susttute numel vlues nd evlute K/K : 7 K.86 ev/d ev/5d K eneg neses to o ppomtel 7 8. (.e., the ottonl knet 9 Ptue the Polem We n solve CMR o C nd susttute numel vlues n ode to detemne n epementl vlue o C o the eth. We n then ompe ths vlue to those o sphel shell nd sphee n whh the mss s unoml dstuted to dede whethe the eth s mss denst s getest ne ts oe o ne ts ust.

9 Consevton o Angul Momentum 74 ( Epess the moment o net o the eth n tems o the onstnt C: Solve o C to otn: Susttute numel vlues nd evlute C: ( ll o the mss wee n the ust, the moment o net o the eth would e tht o thn sphel shell: the mss o the eth wee unoml dstuted thoughout ts volume, ts moment o net would e: CMR C MR C 8. 4 ( 5.98 kg( 67 km. sphel shell sold sphee 5 MR MR 7 kg m Beuse epementll C < /5 ne the ente o the eth..4, the mss denst must e gete * Ptue the Polem et s estmte tht the dve wth ms etended ove hed s out.5 m long nd hs mss M 8 kg. We ll lso ssume tht t s esonle to model the dve s unom stk ottng out ts ente o mss. Fom the photo, t ppes tht he spng out m n the, nd tht the dvng od ws out m hgh. We n use these ssumptons nd estmted qunttes, togethe wth the dentons, to estmte nd. Epess the dve s ngul velot nd ngul momentum : Usng onstnt-eleton equton, epess hs tme n the : θ ( t nd ( t t se m + t ll 6 m up + g g down Susttute numel vlues nd evlute t: t ( m ( 6m 9.8m/s + 9.8m/s.89s Estmte the ngle though whh he otted n.89 s: θ.5ev π d

10 744 Chpte Susttute n equton ( nd evlute π d :.66 d/s.89s Use the stk ottng out n s though ts ente o mss model to ppomte the moment o net o the dve: Susttute n equton ( to otn: Susttute numel vlues nd evlute : M M ( 8kg(.5m (.66 d/s 69. kg m /s 7kg m /s Remks: We n hek the esonleness o ths estmton n nothe w. Beuse he ose out m n the, the ntl mpulse tng on hm must e out 6 kg m/s (.e., p Mv. we estmte tht the leve m o the oe s oughl l.5 m, nd the ngle etween the oe eeted the od nd lne unnng om hs eet to the ente o mss s out 5, we otn lsn5 78 kg m /s, whh s not too d onsdeng the ppomtons mde hee. Ptue the Polem Fst we ssume sphel dve whose mss M 8 kg nd whose dmete, when uled nto ll, s m. We n estmte hs ngul velot when he hs uled hmsel nto ll om the to o hs ngul momentum to hs moment o net. To estmte hs ngul momentum, we ll guess tht the leve m l o the oe tht lunhes hm om the dvng od s out.5 m nd tht the ngle etween the oe eeted the od nd lne unnng om hs eet to the ente o mss s out 5. Epess the dve s ngul velot when he uls hmsel nto ll n md-dve: Usng onstnt-eleton equton, elte the speed wth whh he let the dvng od v to hs mmum heght nd ou estmte o hs ngle wth the vetl deton: ( v + whee os5 v v Solve o v : v g os 5 Susttute numel vlues nd evlute v ( 9.8m/s ( m : v os5 7.7 m/s

11 Consevton o Angul Momentum 745 Appomte the mpulse tng on the dve to lunh hm wth the speed v : ettng l epesent the leve m o the oe tng on the dve s he leves the dvng od, epess hs ngul momentum: Use the unom sphee model to ppomte the moment o net o the dve: Susttute n equton ( to otn: Susttute numel vlues nd evlute : p Mv lsn 5 Mvl sn 5 MR 5 5 Mv lsn 5 5vl sn 5 MR R 5 ( 7.7 m/s(.5m (.5m.d/s sn 5 * Ptue the Polem We ll ssume tht he lunhes hmsel t n ngle o 45 wth the hoontl wth hs ms sped wde, nd then pulls them n to nese hs ottonl speed dung the jump. We ll lso ssume tht we n model hm s -m long lnde wth n vege dus o.5 m nd mss o 6 kg. We n then nd hs tke-o speed nd tme usng onstnt-eleton equtons, nd use the ltte, togethe wth the denton o ottonl velot, to nd hs ntl ottonl velot. Fnll, we n ppl onsevton o ngul momentum to nd hs ntl ngul momentum. Usng onstnt-eleton equton, elte hs tkeo speed v to hs mmum elevton : v v + o, euse v v sn45, v, nd g, v sn 45 g Solve o v to otn: v g sn 45 g sn 45 Susttute numel vlues nd evlute v ( 9.8m/s (.6m : v sn m/s Use ts denton to epess Goeel s ngul velot: θ t Use onstnt-eleton equton to epess Goeel s tme t: t tse.6 m g

12 746 Chpte Susttute numel vlues nd evlute t: (.6m t.699s 9.8m/s Susttute numel vlues nd evlute : Use onsevton o ngul momentum to elte hs tke-o ngul velot to hs vege ngul velot s he peoms quduple ut: 4ev.699s π d ev 6.d/s Assumng tht he n hnge hs ngul momentum to o pullng hs ms n, solve o nd evlute : Epess hs tke-o ngul momentum: Assumng tht we n model hm s sold lnde o length l wth n vege dus nd mss m, epess hs moment o net wth ms dwn n (hs tke-o onguton: Susttute to otn: Susttute numel vlues nd evlute : ( 6 d/s 8. d/s ( m m whee the to o epesents ou ssumpton tht he n doule hs moment o net etendng hs ms. m ( 6kg(.5m ( 8d/s 4.kg m /s Veto Ntue o Rotton Ptue the Polem We n epess F nd n tems o the unt vetos î nd ĵ nd then use the denton o the oss podut to nd τ. Epess F n tems o F nd the unt veto î : F Fˆ Epess n tems o R nd the unt veto : ĵ Rˆj

13 Consevton o Angul Momentum 747 Clulte the oss podut o nd ( F τ F FR ˆj ˆ : FR ( ˆ ˆj FRkˆ 4 Ptue the Polem We n nd the toque s the oss podut o nd F. Compute the oss podut o nd : F τ F ( ˆ + ˆj ( mgˆj mg( ˆ ˆj mg( ˆj ˆj mgkˆ 5 Ptue the Polem The oss podut o the vetos A A ˆ + nd B B ˆ ˆ + B j s gven A B A ( ˆ ˆ + ( ˆ ˆj + ( ˆj ˆ + ( ˆj ˆ B A B A B A B j A ( ( ˆ ( ˆ B + A B k + A B k + A B ( A B kˆ + A B kˆ ( ( ( Fnd A B o A 4 î nd B 6 î + 6 ĵ : A B 4ˆ 4 4 A ˆj ( 6ˆ + 6 ˆj ( ˆ ˆ + 4( ˆ ˆj ( + 4kˆ 4kˆ ( Fnd A B o A 4 î nd B 6 î + 6 kˆ : ( Fnd A B o A î + nd B î + ĵ : ĵ A B 4ˆ A B 4 4 ( 6ˆ + 6kˆ ( ˆ ˆ + 4( ˆ kˆ ( + 4( ˆj 4 ˆj ( ˆ + ˆj ( ˆ + ˆj 6( ˆ ˆ + 4( ˆ ˆj + 9( ˆj ˆ + 6( ˆj ˆj 6( + 4( kˆ + 9( kˆ + 6( 5kˆ

14 748 Chpte *6 Ptue the Polem The mgntude o A B s gven AB snθ. Equte the mgntudes o A B nd A B : AB snθ AB osθ sn θ osθ o tn θ ± Solve o θ to otn: θ tn ± ± 45 o ± 5 7 Ptue the Polem et e n the plne. Then ponts n the postve deton. We n estlsh the esults lled o n ths polem omng the ppopte oss poduts nd deenttng v. ( Epess usng unt vetos: kˆ Epess usng unt vetos: ˆ Fom the oss podut o nd : kˆ ˆ ( kˆ ˆ ˆj vˆj v ( Deentte v wth espet to t to epess : dv d ( dt dt d d + dt dt d + v dt t + ( t + whee ( nd t nd e the tngentl nd

15 Consevton o Angul Momentum 749 entpetl eletons, espetvel. 8 Ptue the Polem Beuse B, we n epess B s B B ˆ + oss podut wth A to detemne B nd B. Epess B n tems o ts omponents: B B ˆ + B ˆj B ˆj nd om ts ( Epess A B : A B 4 ( B + B j 4B k k Solve o B : B ˆ ˆ ˆ ˆ ˆ Relte B to B nd B : B B + B Solve o nd evlute B : B B B 5 4 Susttute n equton (: B 4 ˆ + ˆj 9 Ptue the Polem We n wte B n the om B B ˆ + B ˆj + B kˆ nd use the dot podut o A nd B to nd B nd the oss podut to nd B nd B. Epess B n tems o ts omponents: B B ˆ + B ˆj + B kˆ Evlute A B : A B B Evlute A : nd B 4 B A B ˆj ( B ˆ + 4 ˆj + B kˆ B Beuse A B 9 î : B nd B. kˆ + B ˆ ( Susttute n equton ( to otn: B 4 ˆj + kˆ

16 Chpte 75 4 Ptue the Polem The dot podut o A wth the oss podut o B nd C s sl quntt nd n e epessed n detemnnt om s. We n epnd ths detemnnt mnos to show tht t s equvlent to ( C B A, ( B A C, nd ( A C B. The dot podut o A wth the oss podut o B nd C s sl quntt nd n e epessed n detemnnt om s: ( C B A Epnd the detemnnt mnos to otn: + + ( Evlute the oss podut o B nd C to otn: ( ( ( k j C B ˆ ˆ ˆ + + Fom the dot podut o A wth B C to otn: ( + + C B A ( Beuse ( nd ( e the sme, we n onlude tht: ( C B A Poeed s ove to estlsh tht: ( B A C nd

17 Consevton o Angul Momentum 75 B ( C A 4 Ptue the Polem et, wthout loss o genelt, the veto C le long the s nd the veto B le n the plne s shown elow to the let. The dgm to the ght shows the pllelepped spnned the thee vetos. We n ppl the dentons o A B C s the volume o the pllelepped. the oss- nd dot-poduts to show tht ( Epess the oss-podut o B nd : Fom the dot-podut o A wth the oss-podut o B nd C to otn: *4 Ptue the Polem Dw the tngle usng the thee vetos s shown elow. Note tht A + B C. We n nd the mgntude o the oss podut o A nd B nd o A nd C nd then use the oss podut o A nd C, usng A + B C, to show tht AC sn ABsn o B/sn C/ sn. Poeedng smll, we n etend the lw o snes to the thd sde o the tngle nd the ngle opposte t. C B C ( BC snθ ( kˆ nd B C Bsnθ C A ( e o the pllelogm ( B C A( Bsnθ ( BC snθ ( Aosφ ( e o se( heght V pllelepped C osφ

18 75 Chpte Epess the mgntude o the oss podut o A nd B : Epess the mgntude o the oss podut o A nd C : A B A C ABsn AC sn ( A + B Fom the oss podut o A wth C A C A to otn: A A + A B A B euse A A. Beuse A C A B : A C A B nd AC sn ABsn Smpl nd ewte ths epesson to otn: B C sn sn Poeed smll to etend ths esult to the lw o snes: A sn B sn C sn Angul Momentum 4 Ptue the Polem nd p e elted odng to p., then emnton o the mgntude o p wll llow us to onlude thtsn φ nd tht the ptle s movng ethe detl towd the pont, detl w om the pont, o though the pont. Beuse : p m v m v o v Epess the mgntude o v : v vsn φ Beuse nethe no v s eo: sn φ whee φ s the ngle etween nd v. Solve o φ: φ sn o 8

19 Consevton o Angul Momentum Ptue the Polem We n use the dentons to lulte the ngul momentum nd moment o net o the ptle nd the eltonshp etween,, nd to detemne ts ngul speed. ( Epess nd evlute the mgntude o : mv ( kg(.5m/s( 4m 8. kg m /s ( Epess the moment o net o the ptle wth espet to n s though the ente o the le n whh t s movng: m ( kg( 4m kg m ( Relte the ngul speed o the ptle to ts ngul momentum nd solve o nd evlute : 8. kg m /s kg m.875 d/s 45 Ptue the Polem We n use the denton o ngul momentum to lulte the ngul momentum o ths ptle nd the eltonshp etween ts ngul momentum nd ngul speed to dese the vton n ts ngul speed wth tme. ( Epess the ngul momentum o the ptle s unton o ts mss, speed, nd dstne o ts pth om the eeene pont: mvsnθ ( 6m( kg( 4.5m/s 54. kg m /s sn9 ( Beuse m : nd neses s the ptle ppohes the pont nd deeses s t eedes. *46 Ptue the Polem We n use the omul o the e o tngle to nd the e swept out t t t, dd ths e to the e swept out n tme dt, nd then deentte ths epesson wth espet to tme to otn the gven epesson o da/dt. Epess the e swept out t t t : A osθ whee θ l s the ngle etween nd v nd

20 754 Chpte s the omponent o n the deton o v. Epess the e swept out t t t + dt: Deentte wth espet to t: A A da dt + da ( + d ( + vdt d v dt onstnt Beuse snθ : v ( snθ v ( psnθ m m 47 Ptue the Polem We n nd the totl ngul momentum o the on om the sum o ts spn nd otl ngul moment. ( Epess the spn ngul spn m spn momentum o the on: Fom Polem 9-44: Susttute o to otn: MR spn 4 4 MR spn Susttute numel vlues nd evlute spn : spn 4 (.5kg(.75m ev π d s ev. 5 kg m /s ( Epess nd evlute the totl ngul momentum o the on: ot +. spn 5 + kg m spn /s ( Fom Polem -4: ot nd. 5 kg m /s (d Epess the totl ngul momentum o the on: ot + spn

21 Fnd the otl momentum o the Consevton o Angul Momentum 755 on: ± (.5kg(.5m/s(.m ot ± MvR ± kg m whee the ± s onsequene o the t tht the on s deton s not speed. /s Susttute to otn: ± kg m 5 /s kg m /s The possle vlues o e: kg m /s o kg m /s 48 Ptue the Polem Both the oes tng on the ptles eet toques wth espet to n s pependul to the pge nd though pont O nd the net toque out ths s s the veto sum. Epess the net toque out n s pependul to the pge nd τ net τ F F euse F F though pont O: ( Beuse ponts long F Toque nd Angul Momentum : ( F + F 49 Ptue the Polem The ngul momentum o the ptle hnges euse net toque ts on t. Beuse we know how the ngul momentum depends on tme, we n nd the net toque tng on the ptle deenttng ts ngul momentum. We n use onstnt-eleton equton nd Newton s nd lw to elte the ngul speed o the ptle to ts ngul eleton. ( Relte the mgntude o the toque tng on the ptle to the te t whh ts ngul momentum hnges: τ net d d t dt dt 4. N m [( 4 N m ]

22 756 Chpte ( Usng onstnt-eleton equton, elte the ngul speed o the ptle to ts eleton nd tme-n-moton: Use Newton s nd lw to elte the ngul eleton o the ptle to the net toque tng on t: + α t whee τ net α τ m net Susttute to otn: Susttute numel vlues nd evlute : τ net m t ( 4 N m t (.8kg(.4m (.9 d/s povded t s n seonds. t 5 Ptue the Polem The ngul momentum o the lnde hnges euse net toque ts on t. We n nd the ngul momentum t t 5 s om ts denton nd the net toque tng on the lnde om the te t whh the ngul momentum s hngng. The mgntude o the tonl oe tng on the m n e ound usng the denton o toque. ( Use ts denton to epess the ngul momentum o the lnde: m Susttute numel vlues nd evlute : ( 9kg(.4m 5 ev mn 77 kg m π d mn ev 6s /s ( Epess nd evlute dt d : d ( 77 kg m /s dt 5s 5.kg m /s ( Beuse the toque tng on the unom lnde s onstnt, the te dt d τ 5.kg m /s

23 Consevton o Angul Momentum 757 o hnge o the ngul momentum s onstnt nd hene the nstntneous te o hnge o the ngul momentum t n nstnt s equl to the vege te o hnge ove the tme dung whh the toque ts: (d Usng the denton o toque tht eltes the ppled oe to ts leve m, epess the mgntude o the tonl oe tng on the m: τ 5.kg m /s l.4 m 7.7 N *5 Ptue the Polem et the sstem nlude the pulle, stng, nd the loks nd ssume tht the mss o the stng s neglgle. The ngul momentum o ths sstem hnges euse net toque ts on t. ( Epess the net toque out the ente o mss o the pulle: Rg( m snθ m τ net Rm g snθ Rm g whee we hve tken lokwse to e postve to e onsstent wth postve upwd velot o the lok whose mss s m s ndted n the gue. ( Epess the totl ngul momentum o the sstem out n s though the ente o the pulle: ( Epess τ s the tme devtve o the ngul momentum: + m vr + m vr d τ dt vr R R R d dt + m + m vr R + m + m + m + m Equte ths esult to tht o pt ( nd solve o to otn: ( m θ m g R sn + m + m

24 758 Chpte 5 Ptue the Polem The oes esultng om the elese o gs om the jets wll eet toque on the speshp tht wll slow nd eventull stop ts otton. We n elte ths net toque to the ngul momentum o the speshp nd to the tme the jets must e. Relte the ng tme o the jets to the desed hnge n ngul momentum: Epess the mgntude o the net toque eeted the jets: t τ τ net FR net τ net ettng m/ t epesent the mss o gs pe unt tme ehusted om the jets, elte the oe eeted the gs on the speshp to the te t whh the gs espes: F m v t' Susttute nd solve o t to otn: t m vr t' Susttute numel vlues nd evlute t: t ( 4 kg m ( kg/s( 8 m/s( m 6 ev mn π d mn ev 6s 5.4s 5 Ptue the Polem We n use onstnt-eleton equtons to epess the pojetle s poston nd velot oodntes s untons o tme. We n use these oodntes to epess the ptle s poston nd velot vetos nd v. Usng ts denton, we n epess the pojetle s ngul momentum s unton o tme nd then deentte ths epesson to otn d dt. Fnll, we n use the denton o the toque, eltve to n ogn loted t the lunh poston, the gvttonl oe eets on the pojetle to epess τ nd omplete the demonstton tht d dt τ. Usng ts denton, epess the ngul momentum veto o the pojetle: mv ( Usng onstnt-eleton v t ( V osθ t

25 Consevton o Angul Momentum 759 equtons, epess the poston oodntes o the pojetle s unton o tme: nd + v t + t ( V snθ t gt [( V osθ t] ˆ + ( V snθ Epess the pojetle s poston veto : [ t gt ] ˆj Usng onstnt-eleton equtons, epess the velot o the pojetle s unton o tme: v v V osθ nd v v + t V snθ gt Epess the pojetle s velot veto : θ ˆ θ v v [ V os ] + [ V sn gt] j { [ gt ] ˆj } { ˆj } Susttute n equton ( to otn: [( ] ˆ V osθ t + ( V snθ t m [ V osθ ] ˆ + [ V snθ gt] ( mgt V osθ kˆ Deentte wth espet to t to otn: d d ( mgt V osθ Usng ts denton, epess the toque tng on the pojetle: Compng equtons ( nd ( we see tht: dt dt ( mgtv osθ kˆ τ Consevton o Angul Momentum kˆ ( mg ˆj [( V osθ t] ˆ + ( V snθ ( mgj ˆ o ( mgtv osθ kˆ τ d dt τ ˆ ( [ t gt ] *54 Ptue the Polem et m epesent the mss o the plnet nd ppl the denton o toque to nd the toque podued the gvttonl oe o ttton. We n use Newton s nd lw o moton n the om τ d dt to show tht s onstnt nd ppl onsevton o ngul momentum to the moton o the plnet t ponts A nd B. ( ˆj

26 76 Chpte ( Epess the toque podued the gvttonl oe o ttton o the sun o the plnet: ( Beuse τ : Notng tht t ponts A nd B v v, epess the eltonshp etween the dstnes om the sun nd the speeds o the plnets: τ F euse F ts long the deton o. d mv onstnt dt v v o v v 55 Ptue the Polem et the sstem onsst o ou, the etended weghts, nd the pltom. Beuse the net etenl toque tng on ths sstem s eo, ts ngul momentum emns onstnt dung the pullng n o the weghts. ( Usng onsevton o ngul momentum, elte the ntl nd nl ngul speeds o the sstem to ts ntl nd nl moments o net: Solve o : Susttute numel vlues nd evlute : 6kg m.8kg m (.5ev/s 5.ev/s ( Epess the hnge n the knet eneg o the sstem: K K K Susttute numel vlues nd K (.8kg m evlute K: ( 6kg m 6J ev π d 5 s ev ev π d.5 s ev

27 Consevton o Angul Momentum 76 ( Beuse no etenl gent does wok on the sstem, the eneg omes om the ntenl eneg o the mn. *56 Ptue the Polem et the sstem onsst o the lo o putt nd the tuntle. Beuse the net etenl toque tng on ths sstem s eo, ts ngul momentum emns onstnt when the lo o putt lls onto the tuntle. ( Usng onsevton o ngul momentum, elte the ntl nd nl ngul speeds o the tuntle to ts ntl nd nl moments o net nd solve o : Epess the nl ottonl net o the tuntle-plus-lo: nd + + mr lo Susttute nd smpl to otn: + mr mr + ( the lo les o tngentll to the tuntle, ts ngul momentum doesn t hnge (wth espet to n s though the ente o tuntle. Beuse thee s no etenl toque tng on the lo-tuntle sstem, the totl ngul momentum o the sstem wll emn onstnt nd the ngul momentum o the tuntle wll not hnge. Beuse the moment o net o the tle hsn t hnged ethe, the tuntle wll ontnue to spn t '. 57 Ptue the Polem Beuse the net etenl toque tng on the Susnokoh sstem s eo, the net ngul momentum o the sstem s onstnt (equl to eo euse the Susn s ntll t est nd we n use onsevton o ngul momentum to nd the ngul velot o the Susn. The speed o the okoh eltve to the loo v s the deene etween ts speed wth espet to the Susn nd the speed o the Susn t the loton o the okoh wth espet to the loo. Relte the speed o the okoh wth espet to the loo v to the speed o the Susn t the loton o the okoh: Use onsevton o ngul momentum to otn: v v ( S C

28 76 Chpte Epess the ngul momentum o the Susn: S S MR Epess the ngul momentum o the okoh: C v CC m Susttute to otn: v MR m Solve o to otn: mv MR + m Susttute n equton (: Susttute numel vlues nd evlute v : v m v v MR + m v (.5 kg(.8m (.m/s (.5m(.5m + (.5 kg(.8m.m/s 9.67mm/s *58 Ptue the Polem The net etenl toque tng on ths sstem s eo nd so we know tht ngul momentum s onseved s these dsks e ought togethe. et the numel ee to the dsk to the let nd the numel to the dsk to the ght. et the ngul momentum o the dsk wth the lge dus e postve. Usng onsevton o ngul momentum, elte the ntl ngul speeds o the dsks to the ommon nl speed nd to the moments o net: Solve o : o + ( + m m nd m Epess nd : ( Susttute nd smpl to otn: m m m + m 5

29 Consevton o Angul Momentum Ptue the Polem We n epess the ngul momentum nd knet eneg o the lok detl om the dentons. The tenson n the stng povdes the entpetl oe equed o the unom ul moton nd n e epessed usng Newton s nd lw. Fnll, we n use the wok-knet eneg theoem to epess the wok equed to edue the dus o the le to o two. ( Epess the ntl ngul momentum o the lok: mv ( Epess the ntl knet eneg o the lok: K mv ( Usng Newton s nd lw, elte the tenson n the stng to the entpetl oe equed o the ul moton: T F v m Use the wok-knet eneg theoem to elte the equed wok to the hnge n the knet eneg o the lok: W K K ( m m m K Susttute the esult om pt ( nd smpl to otn: W mv *6 Ptue the Polem Beuse the oe eeted the ue nd s pllel to the poston veto o the pont mss, the net etenl toque tng on t s eo nd we n use the onsevton o ngul momentum to detemne the speeds o the ll t ponts B nd C. We ll use mehnl eneg onsevton to nd eltng the knet nd elst potentl eneges t A nd B. ( Use onsevton o momentum to elte the ngul moment t ponts A, B nd C: Solve o v B n tems o v A : A A A B o mv mv mv v v B A A B C B B C C

30 764 Chpte Susttute numel vlues nd evlute v B : v B.6m m ( 4 m/s.4m/s Solve o v C n tems o v A : v v C A A C Susttute numel vlues nd evlute v C : v C.6m.6m ( 4 m/s 4.m/s ( Use onsevton o mehnl eneg etween ponts A nd B to elte the knet eneg o the pont mss nd the eneg stoed n the stethed ue nd: E A E B o mv + mv + A A B B Solve o : m( v v Susttute numel vlues nd evlute : (.kg (.4 m/s ( 4m/s (.6m ( m B A A B. N/m Quntton o Angul Momentum [ ] *6 Ptue the Polem The eleton s spn ngul momentum veto s elted to ts omponent s shown n the dgm. Usng tgonomet, elte the mgntude o s to ts omponent: h θ os.75h Ptue the Polem Equton -7 deses the quntton o ottonl eneg. We n show tht the eneg deene etween gven stte nd the net hghe stte s popotonl to l + usng Equton -7 to epess the eneg deene.

31 Fom Equton -7 we hve: l ( Consevton o Angul Momentum 765 K l l + E Usng ths equton, epess the deene etween one ottonl stte nd the net hghe stte: E ( l + ( l + E l( l + ( l + E E 6 Ptue the Polem The ottonl eneges o HB moleule e elted to l nd E odng to K l l( l + E whee E h. ( Epess nd evlute the moment o net o the H tom: m p 7 9 (.67 kg(.44 m kg m ( Relte the ottonl eneges to l nd E : ( K l l l + E Evlute E : 4 h (.5 J s E mev ( kg m.46 ev J.6 9 J Evlute E o l : ( (.996 mev.99mev E + Evlute E o l : E ( + (.996 mev 5.98meV Evlute E o l : E ( + (.996 mev. mev 64 Ptue the Polem We n use the denton o the moment o net o pont ptles to lulte the ottonl net o the ntogen moleule. The ottonl eneges o ntogen moleule e elted to l nd K l l l + E whee E h. to ( E odng

32 766 Chpte ( Usng gd dumell model, epess nd evlute the moment o net o the ntogen moleule out ts ente o mss: m m N m N + m N Susttute numel vlues nd evlute : 7 ( 4(.66 kg( 5.5 m.4 46 kg m ( Relte the ottonl eneges to l nd E : ( E l l l + E Evlute E : 4 h (.5 J s E mev ( kg m.4 ev J.6 9 J Susttute to otn:.44 ( mev E l l l + *65 Ptue the Polem We n otn n epesson o the speed o the ntogen moleule equtng ts tnsltonl nd ottonl knet eneges nd solvng o v. Beuse ths epesson nludes the moment o net o the ntogen moleule, we n use the denton o the moment o net to epess o dumell model o the ntogen moleule. The ottonl eneges o ntogen moleule depend on the quntum nume l odng to E / l( l h /. l + Equte the ottonl knet eneg o the ntogen moleule n ts l quntum stte nd ts tnsltonl knet eneg: E ( m Nv Epess the ottonl eneg levels o the ntogen moleule: E l l ( l + h Fo l : ( + h h E

33 Consevton o Angul Momentum 767 Susttute n equton (: h m Nv Solve o v to otn: Usng gd dumell model, epess the moment o net o the ntogen moleule out ts ente o mss: h v ( m N m nd m N mn mn + mn mn Susttute n equton (: v h h m m N N Susttute numel vlues nd evlute v: v ( kg( 5.5 m 8.5m/s 4 J s Collson Polems 66 Ptue the Polem et the eo o gvttonl potentl eneg e t the elevton o the od. Beuse the net etenl toque tng on ths sstem s eo, we know tht ngul momentum s onseved n the ollson. We ll use the denton o ngul momentum to epess the ngul momentum just te the ollson nd onsevton o mehnl eneg to detemne the speed o the ll just eoe t mkes ts peetl nelst ollson wth the od. Use onsevton o ngul momentum to elte the ngul momentum eoe the ollson to the ngul momentum just te the peetl nelst ollson: mv Use onsevton o mehnl eneg to elte the knet eneg o the ll just eoe mpt to ts ntl potentl eneg: ettng h epesent the dstne the K K + U U o, euse K U, K U v gh

34 768 Chpte ll lls, susttute o K nd U nd solve o v to otn: Susttute o v to otn: m gh Susttute numel vlues nd evlute : (.kg(.9m ( 9.8m/s (.m 4.J s *67 Ptue the Polem Beuse thee e no etenl oes o toques tng on the sstem dened n the polem sttement, oth lne nd ngul momentum e onseved n the ollson nd the velot o the ente o mss te the ollson s the sme s eoe the ollson. et the deton the lo o putt s movng ntll e the postve deton nd towd the top o the pge n the gue e the postve deton. Usng ts denton, epess the loton o the ente o mss eltve to the ente o the : m md elow the ente o the. M + m Usng ts denton, epess the velot o the ente o mss: v m mv M + m Usng the denton o n tems o nd, epess : Epess the ngul momentum out the ente o mss: m ( m mv ( d m m md mmvd mv d M + m M + m Usng the pllel s theoem, epess the moment o net o the sstem eltve to ts ente o mss: Susttute o m nd smpl to otn: m M + M m + m d ( m

35 m md md M + M + m d M + m M + m Mm d d( M + m md M + + m M + m M + ( M + m Mm d + ( M + m ( M + m mmd M + M + m Consevton o Angul Momentum 769 mm d M + ( M + ( M + m m mmd Susttute o m nd m n equton mmvd M M + m + Mmd ( nd smpl to otn: ( Remks: You n ve the epesson o m lettng m to otn M nd lettng M to otn m. m 68 Ptue the Polem Beuse thee e no etenl oes o toques tng on the sstem dened n the sttement o Polem 67, oth lne nd ngul momentum e onseved n the ollson nd the velot o the ente o mss te the ollson s the sme s eoe the ollson. Knet eneg s lso onseved s the ollson o the hd sphee wth the s elst. et the deton the sphee s movng ntll e the postve deton nd towd the top o the pge n the gue e the postve deton nd v nd V e the nl velotes o the ojets whose msses e m nd M, espetvel. Appl onsevton o lne momentum to otn: Appl onsevton o ngul momentum to otn: Set v n equton ( nd solve o V : Use onsevton o mehnl eneg to elte the knet eneges o tnslton nd otton eoe p p o mv mv' + MV ' ( o mvd mv' d + M ( mv V' ( M K K o mv MV ' + M (4 (

36 77 Chpte nd te the elst ollson: Susttute ( nd ( n (4 nd smpl to otn: m m d + M M Solve o d: d M m m 69 Ptue the Polem et the eo o gvttonl potentl eneg e dstne elow the pvot s shown n the dgm. Beuse the net etenl toque tng on the sstem s eo, ngul momentum s onseved n ths peetl nelst ollson. We n lso use onsevton o mehnl eneg to elte the ntl knet eneg o the sstem te the ollson to ts potentl eneg t the top o ts swng. Usng onsevton o mehnl eneg, elte the ottonl knet eneg o the sstem just te the ollson to ts gvttonl potentl eneg when t hs swung though n ngle θ: Appl onsevton o momentum to the ollson: Solve o to otn: K + U o, euse K U, K + U nd d Mg + mg ( osθ ( o.8dmv [ Md + (. 8d m].8dmv ( Md + md. 64 Epess the moment o net o the sstem out the pvot: m (.8d.64md + + Md Md (

37 Consevton o Angul Momentum 77 Susttute equtons ( nd ( n equton ( nd smpl to otn: d Mg + mgd θ ( os ( dmv. Md +.64md Solve o v: v (.5M +.8m( Md +.64md g( osθ.dm Evlute v o θ 9 to otn: v (.5M +.8m( Md +.64md.dm g 7 Ptue the Polem et the eo o gvttonl potentl eneg e dstne elow the pvot s shown n the dgm. Beuse the net etenl toque tng on the sstem s eo, ngul momentum s onseved n ths peetl nelst ollson. We n lso use onsevton o mehnl eneg to elte the ntl knet eneg o the sstem te the ollson to ts potentl eneg t the top o ts swng. Usng onsevton o mehnl eneg, elte the ottonl knet eneg o the sstem just te the ollson to ts gvttonl potentl eneg when t hs swung though n ngle θ : Appl onsevton o momentum to the ollson: K K + U U o, euse K U, K + U nd d Mg + mg ( osθ ( o.8dmv [ Md + (. 8d m]

38 77 Chpte Solve o to otn:.8dmv ( Md +.64md Epess the moment o net o the sstem out the pvot: m(.8d + Md (.64m + M d [.64(.kg + (.8kg ](.m.66 kg m Susttute equton ( n equton ( nd smpl to otn: d Mg +.8dmg θ ( os. ( dmv Solve o v: g(.5m +.8m( osθ v.dm Susttute numel vlues nd evlute v o θ 6 to otn: v ( 9.8m/s [.5(.8kg +.8(.kg ](.5(.66 kg m. (.m(.kg 7.74 m/s 7 Ptue the Polem et the length o the unom stk e l. We n use the mpulsehnge n momentum theoem to epess the velot o the ente o mss o the stk. B epessng the velot V o the end o the stk n tems o the velot o the ente o mss nd pplng the ngul mpulse-hnge n ngul momentum theoem we n nd the ngul velot o the stk nd, hene, the velot o the end o the stk. ( Appl the mpulse-hnge n momentum theoem to otn: Solve o v m to otn: ( Relte the velot V o the end o the stk to the velot o the ente o mss v m : Relte the ngul mpulse to the hnge n the ngul momentum o the stk: K p p p p o, euse p nd p Mv m, K Mv m v m m K M el to o m m ( l V v + v v + ( ( l K o, euse, K l ( m m

39 Consevton o Angul Momentum 77 Ree to Tle 9- to nd the moment o net o the stk wth espet to ts ente o mss: m Ml Susttute to otn: K ( l Ml Solve o : 6K Ml Susttute n equton ( to otn: ( Relte the velot V o the othe end o the stk to the velot o the ente o mss v m : V V K M v m K M 6K l + Ml v el to o m 6K l Ml v 4K M m K M ( l (d ettng e the dstne om the ente o mss towd the end not stuk, epess the ondton tht the pont t s t est: Solve o to otn: vm K M 6K Ml Solve o to otn: K M 6K Ml 6 l Note tht o mete stk stuk t the -m mk, the stton pont would e t the.-m mk. Remks: You n esl hek ths esult plng metestk on the loo nd gvng t shp low t the -m mk. 7 Ptue the Polem Beuse the net etenl toque tng on the sstem s eo, ngul momentum s onseved n ths peetl nelst ollson. ( Use ts denton to epess the totl ngul momentum o the dsk nd pojetle just eoe mpt: p m v

40 774 Chpte ( Use onsevton o ngul momentum to elte the ngul moment just eoe nd just te the ollson: Epess the moment o net o the dsk + pojetle: nd + MR m p Susttute o n the epesson o mpv to otn: MR + m p ( Epess the knet eneg o the sstem te mpt n tems o ts ngul momentum: K ( mpv ( MR + mp ( m v MR p + m p (d Epess the deene etween the ntl nd nl knet eneges, susttute, nd smpl to otn: E K K m v p MR ( m v p + m mp m pv MR + mp p *7 Ptue the Polem Beuse the net etenl toque tng on the sstem s eo, ngul momentum s onseved n ths peetl nelst ollson. The od, on ts downwd swng, ques ottonl knet eneg. Angul momentum s onseved n the peetl nelst ollson wth the ptle nd the ottonl knet o the teollson sstem s then tnsomed nto gvttonl potentl eneg s the od-plusptle swng upwd. et the eo o gvttonl potentl eneg e t dstne elow the pvot nd use oth ngul momentum nd mehnl eneg onsevton to elte the dstnes nd nd the msses M nd m. Use onsevton o eneg to elte the ntl nd nl potentl eneg o the od to ts ottonl knet eneg just eoe t olldes wth the ptle: K K + U U o, euse K, K + U U

41 Consevton o Angul Momentum 775 Susttute o K, U, nd U to otn: ( M + Mg Mg Solve o : g ettng epesent the ngul speed o the od-nd-ptle sstem just te mpt, use onsevton o ngul momentum to elte the ngul moment eoe nd te the ollson: o M M + ( ( m ' Solve o : M ' M + m Use onsevton o eneg to elte the ottonl knet eneg o the od-plus-ptle just te the ollson to the potentl eneg when the hve swung though n ngle θ m : K o, euse K, ' + Mg K + U U ( ( osθm ( osθ + mg m ( Epess the moment o net o the sstem wth espet to the pvot: M + m Susttute o θ m, nd n equton (: ( M g M + m Mg ( + mg Smpl to otn: m m ( M M Smpl equton ( lettng α m/m nd β / to otn: Susttute o α nd smpl to otn the u equton n β: 6α β + β + αβ β + 9β + 4β Use the solve unton* o ou lulto to nd the onl el vlue β.49

42 776 Chpte o β: *Remks: Most gphng lultos hve solve etue. One n solve the u equton usng ethe the gph nd te pltes o the solve etue. The oot gven ove ws ound usng SOVER on T Ptue the Polem Beuse the net etenl toque tng on the sstem s eo, ngul momentum s onseved n ths peetl nelst ollson. The od, on ts downwd swng, ques ottonl knet eneg. Angul momentum s onseved n the peetl nelst ollson wth the ptle nd the ottonl knet eneg o the te-ollson sstem s then tnsomed nto gvttonl potentl eneg s the odplus-ptle swng upwd. et the eo o gvttonl potentl eneg e t dstne elow the pvot nd use oth ngul momentum nd mehnl eneg onsevton to elte the dstnes nd nd the mss M to m. ( Use onsevton o eneg to elte the ntl nd nl potentl eneg o the od to ts ottonl knet eneg just eoe t olldes wth the ptle: Susttute o K, U, nd U to otn: K K + U U o, euse K, K + U U ( M + Mg Mg Solve o : g ettng epesent the ngul speed o the sstem te mpt, use onsevton o ngul momentum to elte the ngul moment eoe nd te the ollson: Solve o : o M M + m ( ( ( ' M ' M + m M M + m g

43 Consevton o Angul Momentum 777 Susttute numel vlues to otn: ( kg(.m ' ( kg(.m + m(.8m ( 9.8m/s.m 4.75kg m.96 kg m kg / s.96 kg +.64m / s (.64m m Use onsevton o eneg to elte the ottonl knet eneg o the od-plus-ptle just te the ollson to the potentl eneg when the hve swung though n ngle θ m : Susttute o K, U, nd U to otn: Epess the moment o net o the sstem wth espet to the pvot: K o, euse K, K + U U K + U U ' + Mg + mg M + m ( ( osθm ( osθ m Susttute o θ m, nd n equton ( nd smpl to otn: ( 4.75kg/s.96 kg +.64m.g ( M + m Susttute o M, nd nd smpl to otn: m +.m 8.9 Solve the qudt equton o ts postve oot: m.84 kg ( The eneg dsspted n the nelst ollson s: Epess U : Epess U : E U U ( U Mg ( os g M + m m U θ

44 778 Chpte Susttute n equton ( to otn: E Mg ( osθ g M + m m Susttute numel vlues nd evlute E: U ( kg( 9.8m/s (. m 6.5J kg. m ( ( ( ( os7 9.8m/s + (.85 kg(.8m 75 Ptue the Polem et nd e the ngul velotes o the od mmedtel eoe nd mmedtel te the nelst ollson wth the mss m. et e the ntl ngul velot o the od. Choose the eo o gvttonl potentl eneg e t dstne elow the pvot. We ppl eneg onsevton to detemne nd onsevton o ngul momentum to detemne. We ll ppl eneg onsevton to detemne. Fnll, we ll nd the eneges o the sstem mmedtel eoe nd te the ollson nd the eneg dsspted. Epess the eneg dsspted n the nelst ollson: Use eneg onsevton to elte the knet eneg o the sstem mmedtel te the ollson to ts potentl eneg te 8 otton: Susttute o K ottom, U top, nd U ottom to otn: E U U ( K K + U U o, euse K K top nd K K ottom, K + U U ottom top ottom + Mg + mg Mg mg ( + ( Smpl to otn: + Mg + mg ( Epess : M + m Susttute o n equton ( nd solve o to otn: ( M + m g M + m

45 Consevton o Angul Momentum 779 Susttute numel vlues nd evlute : ( (.75kg(.m + (.4kg(.8m 9.8m/s [ ] (.75kg(.m + (.4kg(.8m 7. d/s Use onsevton o ngul momentum to elte the ngul momentum o the sstem just eoe the ollson to ts ngul momentum just te the ollson: Susttute o nd nd solve o : Susttute numel vlues nd evlute : Appl onsevton o mehnl eneg to elte the ntl ottonl knet eneg o the od to ts ottonl knet eneg just eoe ts ollson wth the ptle: o ( M ( M + m nd m + M (.4 kg +.75 kg. d/s K K + U U.8m.m ( 7. d/s Susttute to otn: ( M ( M Mg + Mg Solve o : g Susttute numel vlues nd evlute : ( d/s.9 d/s ( 9.8m/s. m

46 78 Chpte Susttute n equton ( to epess the eneg dsspted n the ollson: ( M Mg E + mg Susttute numel vlues nd evlute E: E 6 [ ] (.75 kg(.m ( d/s ( 9.8m/s (.75 kg(.m + (.4kg(.8m.8J 76 Ptue the Polem et v e the speed o the ptle mmedtel te the ollson nd nd e the ngul velotes o the od mmedtel eoe nd mmedtel te the elst ollson wth the mss m. Choose the eo o gvttonl potentl eneg e t dstne elow the pvot. Beuse the net etenl toque tng on the sstem s eo, ngul momentum s onseved n ths elst ollson. The od, on ts downwd swng, ques ottonl knet eneg. Angul momentum s onseved n the elst ollson wth the ptle nd the knet eneg o the te-ollson sstem s then tnsomed nto gvttonl potentl eneg s the od-plus-ptle swng upwd. et the eo o gvttonl potentl eneg e t dstne elow the pvot nd use oth ngul momentum nd mehnl eneg onsevton to elte the dstnes nd nd the mss M to m. Use eneg onsevton to elte the eneges o the sstem mmedtel eoe nd te the elst ollson: Susttute o K, U, nd U to otn: Solve o mv : Appl onsevton o eneg to epess the ngul speed o the od just eoe the ollson: K K + U U o, euse K, K + U U mv + Mg θ ( os Mg m mv Mg osθm ( K K + U U o, euse K, K + U U Susttute o K, U, nd U to otn: ( + Mg Mg M Solve o : g

47 Consevton o Angul Momentum 78 Appl onsevton o eneg to the od te the ollson: ( M Mg ( osθ m Solve o :.6g Appl onsevton o ngul momentum to the ollson: o M M + mv ( ( Solve o mv: M ( mv Susttute o nd to otn: g.6g M mv ( Dvde equton ( equton ( to elmnte m nd solve o v: v Mg osθm g.6g M g g osθ m.6g Susttute numel vlues nd evlute v: v ( (.8m ( (.m.6( 9.8m/s (.m 9.8m/s 9.8m/s os7 5.7 m/s Solve equton ( o m: m Mg osθ v m Susttute o v n the epesson o mv nd solve o m: m ( kg( 9.8m/s (.m ( 5.7 m/s.575kg os7 Beuse the ollson ws elst: E

48 78 Chpte 77 Ptue the Polem We n detemne the ngul momentum o the wheel nd the ngul velot o ts peesson om the dentons. The peod o the peessonl moton n e ound om ts ngul velot nd the ngul momentum ssoted wth the moton o the ente o mss om ts denton. ( Usng the denton o ngul momentum, epess the ngul momentum o the spnnng wheel: Susttute numel vlues nd evlute : MR 8. J s w g R N (.8m 9.8m/s ev π d s ev ( Usng ts denton, epess the ngul velot o peesson: d φ p dt MgD Susttute numel vlues nd evlute p : ( N(.5m p 8.J s.44 d/s ( Epess the peod o the peessonl moton s unton o the ngul velot o peesson: T π π.44 d/s p 5.s (d Epess the ngul momentum o the ente o mss due to the peesson: MD p m p p Susttute numel vlues nd evlute p : p N 9.8m/s.79J s (.5m (.44 d/s The deton o p s ethe up o down, dependng on the deton o. *78 Ptue the Polem The ngul velot o peesson n e ound om ts denton. Both the speed nd eleton o the ente o mss dung peesson e elted to the ngul velot o peesson. We n use Newton s nd lw to nd the vetl nd

49 hoontl omponents o the oe eeted the pvot. Consevton o Angul Momentum 78 ( Usng ts denton, epess the ngul velot o peesson: p dφ dt MgD s s MgD MR s gd R s Susttute numel vlues nd evlute p : (.64 m ( (.5m 9.8m/s ev π d mn 7 mn ev 6s p.7 d/s ( Epess the speed o the ente o mss n tems o ts ngul velot o peesson: v m D p.64 m/s (.5m(.7 d/s ( Relte the eleton o the ente o mss to ts ngul velot o peesson: (d Use Newton s nd lw to elte the vetl omponent o the oe eeted the pvot to the weght o the dsk: Relte the hoontl omponent o the oe eeted the pvot to the eleton o the ente o mss: F F m v v D p Mg.55m/s 4.5 N M m.4 N (.5m(.7 d/s (.5kg( 9.8m/s (.5kg(.55m/s Genel Polems 79 Ptue the Polem Whle the -kg ptle s movng n stght lne, t hs ngul momentum gven p whee s ts poston veto nd p s ts lne momentum. The toque due to the ppled oe s gven τ F. ( Epess the ngul momentum o the ptle: p m ˆ + 5.m Epess the vetos nd p : ( ( j ˆ

50 784 Chpte nd p mvˆ kg ( 9kg m/sˆ ( ( m/s Susttute nd smpl to nd : ( m ˆ + ( 5.m ˆ [ ˆj ] ( 9kg m/s ( 47.7 kg m /s( ˆj ˆ ( 47.7 kg m /skˆ ˆ ( Usng ts denton, epess the toque due to the oe: τ F [ ˆj ] ( N ( 5.9 N m( ˆj ˆ Susttute nd smpl to nd τ : τ ( m ˆ + ( 5.m ( 5.9 N mkˆ ˆ 8 Ptue the Polem The ngul momentum o the ptle s gven p whee s ts poston veto nd p s ts lne momentum. The toque tng on the ptle s gven τ d dt. Epess the ngul momentum o the ptle: d Evlute : dt p mv m v d m dt d 6tˆj dt [ { ˆj }] Susttute nd smpl to nd : ( ( ˆ kg 4m + ( t m/s ( 6t m/s ˆj ( 7.t J skˆ Fnd the toque due to the oe: [( 7. J s kˆ ] d d τ t dt dt ( 7. N mkˆ

51 Consevton o Angul Momentum Ptue the Polem The e sktes otte out the ente o mss; pont we n lote usng ts denton. Knowng the loton o the ente o mss we n detemne the moment o net wth espet to n s though ths pont. The ngul momentum o the sstem s then gven m nd ts knet eneg n e ound om K. m ( Epess the ngul momentum o the sstem out the ente o mss o the sktes: m Usng ts denton, lote the ente o mss, eltve to the 85-kg skte, o the sstem: m ( 55kg(.7 m + ( 85kg(.668m 55kg + 85kg Clulte m : m ( 55kg(.7 m.668m + ( 85kg(.668m 96.5kg m Susttute to detemne : ( 96.5kg m 4J s ev π d.5s ev ( Relte the totl knet eneg o the sstem to ts ngul momentum nd evlute K: K m Susttute numel vlues nd evlute K: K ( 4J s ( m 96.5kg 6J

52 786 Chpte *8 Ptue the Polem et the ogn o the oodnte sstem e t the pvot (pont P. The dgm shows the oes tng on the ll. We ll ppl Newton s nd lw to the ll to detemne ts speed. We ll then use the devtve o ts poston veto to epess ts velot nd the denton o ngul momentum to show tht hs oth hoontl nd vetl omponents. We n use the devtve o wth espet to tme to show tht the te t whh the ngul momentum o the ll hnges s equl to the toque, eltve to the pvot pont, tng on t. ( Epess the ngul momentum o the ll out the pont o suppot: Appl Newton s nd lw to the ll: p m v ( v F T snθ m snθ nd F T osθ mg Elmnte T etween these equtons nd solve o v: v g snθ tnθ Susttute numel vlues nd evlute v: Epess the poston veto o the ll: Fnd the velot o the ll: Evlute : v (.5m( 9.8m/s.6 m/s sn tn t (.5m sn ( ost ˆ + sn ˆj (.5m os kˆ whee ˆk. d v dt (.75 m/s( snt ˆ + ost ˆj.6 m/s.75d/s sn (.5m

53 Consevton o Angul Momentum 787 Susttute o to otn: v (.6 m/s( sn t ˆ + ostˆj Susttute n equton ( nd evlute : ( kg( [.5m sn ( ost ˆ + snt ˆj (.5m os kˆ ] [(.6 m/s ( snt ˆ + ost ˆj ] [ 5.6( ost ˆ + snt ˆj +.9kˆ ] J s The hoontl omponent o s: 5.6( os t + snt j J s The vetl omponent o s:.9k ˆ J s ( Evlute dt d : d [ 5.6 ( snt ˆ + ostˆj ] J Evlute the mgntude o dt dt d : ( 5.6 N m s (.75d/s d dt ˆ 4.7 N m ˆ Epess the mgntude o the toque eeted gvt out the pont o suppot: Susttute numel vlues nd evlute τ : τ mg snθ τ ( kg( 9.8m/s ( 4.7 N m.5m sn 8 Ptue the Polem n pt ( we need to dede whethe net toque ts on the ojet. n pt ( the ssue s whethe n etenl oes t on the ojet. n pt ( we n ppl the denton o knet eneg to nd the speed o the ojet when the unwpped length hs shotened to /. ( Consde the ovehed vew o the lndl post nd the ojet shown n the djonng gue. The ojet ottes ountelokwse. The toque out the ente o the lnde s lokwse nd o mgntude RT, whee R s the dus o the lnde nd T s the tenson. So

54 788 Chpte must deese. ( Beuse, n ths tonless envonment, no net etenl oes t on the ojet: No, deeses. ts knet eneg s onstnt. ( Epess the knet eneg o the v K m mv ( ojet s t spls nwd: v.(the knet eneg emns onstnt. 84 Ptue the Polem Beuse the net toque tng on the sstem s eo; we n use onsevton o ngul momentum to elte the ntl nd nl ngul velotes o the sstem. Usng onsevton o ngul momentum, elte the ntl nd nl ngul velotes to the ntl nd nl moments o net: o Solve o : Epess : M + ( m l Epess : M + ( Susttute to epess m n tems o : M + ( 4 ml M + ( 4 m 4 4 l M + 5m M + 5m Epess the ntl knet eneg o the sstem: K [ ( 4 ] M + ml ( M + 5ml

55 Consevton o Angul Momentum 789 Epess the nl knet eneg o the sstem nd smpl to otn: K ( M + 5m [ M + ( m ] ( M + 5m ( M + 5ml M + 5m 4 l M + 5m M + 5m l M + 5m M + 5m 85 Detemne the Conept Yes. The net etenl toque s eo nd ngul momentum s onseved s the sstem evolves om ts ntl to ts nl stte. Beuse the dsks ome to the sme nl poston, the ntl nd nl ongutons e the sme s n Polem 84. Theeoe, the nswes e the sme s o Polem Ptue the Polem Beuse the net toque tng on the sstem s eo; we n use onsevton o ngul momentum to elte the ntl nd nl ngul velotes o the sstem. Usng onsevton o ngul momentum, elte the ntl nd nl ngul velotes to the ntl nd nl moments o net: Solve o : Relte the tenson n the stng to the ngul speed o the sstem nd solve o nd evlute : o ( T m nd T ml. d/s l m ( 8 N (.4kg(.6m

56 79 Chpte Epess nd evlute : M + ( ml 4 (.8kg( m + (.4kg(.6m.9 kg m Epess nd evlute : M + ( m (.8kg( m + (.4kg( m. kg m 4 Susttute n equton ( nd solve o :.9 kg m. kg m.5d/s (. d/s Epess nd evlute the ntl knet eneg o the sstem: (.9 kg m (. d/s K 76J Epess nd evlute the nl knet eneg o the sstem: (. kg m (.5d/s K 6.7 J 87 Ptue the Polem Untl the nelst ollson o the lndl ojets t the ends o the lnde, oth ngul momentum nd eneg e onseved. et K epesent the knet eneg o the sstem just eoe the dsks eh the end o the lnde nd use onsevton o eneg to elte the ntl nd nl knet eneges to the nl dl velot. Usng onsevton o mehnl eneg, elte the ntl nd nl knet eneges o the dsks: K K' o + mv ( Solve o v : v ( m Usng onsevton o ngul momentum, elte the ntl nd nl ngul velotes to the ntl o