Chpte Angul omentum Conceptul Polems Tue o lse: () two vectos e exctl opposte n decton, the vecto poduct must e zeo. () The mgntude o the vecto poduct o two vectos s t mnmum when the two vectos e pependcul. (c) Knowng the mgntude o the vecto poduct o two nonzeo vectos nd the vectos ndvdul mgntudes unquel detemnes the ngle etween them. Detemne the Concept The vecto poduct o A nd B s dened to e A B AB snφ nˆ whee nˆ s unt vecto noml to the plne dened A nd B nd φ s the ngle etween A nd B. () Tue. A nd B e n opposte decton, then snφ sn 8. () Flse. A nd B e pependcul, then snφ sn 9 nd the vecto poduct o A nd B s mxmum. (c) Flse. Knowng the mgntude o the vecto poduct nd the vectos ndvdul mgntudes onl gves the mgntude o the sne o the ngle etween the vectos. t does not detemne the ngle unquel, no does ths knowledge tell us the sne o the ngle s postve o negtve. Consde two nonzeo vectos A nd B. The vecto poduct hs the getest mgntude A nd B e () pllel, () pependcul, (c) ntpllel, (d) t n ngle o 45 to ech othe. Detemne the Concept The vecto poduct o the vectos A nd B s dened to e A B ABsnφ nˆ whee nˆ s unt vecto noml to the plne dened A nd B nd φ s the ngle etween A nd B. Hence, the vecto poduct o A nd B s mxmum when snφ. Ths condton s stsed povded A nd B e pependcul. () s coect. 969
97 Chpte 3 Wht s the ngle etween oce vecto F nd toque vecto τ poduced F? Detemne the Concept Becuse τ F F snφ nˆ, whee nˆ s unt vecto noml to the plne dened nd F, the ngle etween F nd τ s 9. 4 A pont ptcle o mss m s movng wth constnt speed v long stght lne tht psses though pont P. Wht cn ou s out the ngul momentum o the ptcle eltve to pont P? () ts mgntude s mv. () ts mgntude s zeo. (c) ts mgntude chnges sgn s the ptcle psses though pont P. (d) t ves n mgntude s the ptcle ppoches pont P. Detemne the Concept nd p e elted ccodng to p nd the mgntude o s psnφ whee φ s the ngle etween nd p. Becuse the moton s long lne tht psses though pont P, nd so s. () s coect. 5 [SS] A ptcle tvels n ccul pth nd pont P s t the cente o the ccle. () the ptcle s lne momentum p s douled wthout chngng the dus o the ccle, how s the mgntude o ts ngul momentum out P ected? () the dus o the ccle s douled ut the speed o the ptcle s unchnged, how s the mgntude o ts ngul momentum out P ected? Detemne the Concept nd p e elted ccodng to p nd the mgntude o s psnφ whee φ s the ngle etween nd p. () Becuse s dectl popotonl to p, s douled. () Becuse s dectl popotonl to, s douled. 6 A ptcle moves long stght lne t constnt speed. How does ts ngul momentum out n xed pont v wth tme? Detemne the Concept We cn detemne how the ngul momentum o the ptcle out n xed pont ves wth tme exmnng the devtve o the coss poduct o nd p. The ngul momentum o the ptcle s gven : p
Deentte wth espect to tme to otn: Becuse d v : dt dp p mv, Fnet, nd dt Becuse the ptcle moves long stght lne t constnt speed: Becuse v nd p( mv ) Angul omentum 97 d dp d + p dt dt dt d dt ( F ) + ( v p) net e pllel: v p Fnet F net () Susttute n equton () to otn: d dt constnt An ltente soluton The ollowng dgm shows ptcle whose mss s m movng long stght lne t constnt speed. The pont dented s P s n xed pont nd the vecto s the poston vecto, eltve to ths xed pont, o the ptcle movng wth constnt veloct. m φ v d Fom the denton o ngul momentum, the mgntude o the ngul momentum o the ptcle, eltve to the xed pont t P, s gven : Becuse P d snφ : mvd φ ( ) mv snφ mv snφ o, ecuse m, v nd d e constnts, constnt 7 Tue o lse: the net toque on ottng sstem s zeo, the ngul veloct o the sstem cnnot chnge. ou nswe s lse, gve n exmple o such stuton.
97 Chpte Flse. The net toque ctng on ottng sstem equls the chnge n the sstem s ngul momentum; tht s, τ net d dt whee. Hence, τ net s zeo, ll we cn s o sue s tht the ngul momentum (the poduct o nd ) s constnt. chnges, so must. An exmple s hgh dve gong om tucked to lout poston. 8 You e stndng on the edge o tuntle wth ctonless engs tht s ntll ottng when ou ctch ll tht s movng n the sme decton ut ste thn ou e movng nd on lne tngent to the edge o the tuntle. Assume ou do not move eltve to the tuntle. () Does the ngul speed o the tuntle ncese, decese, o emn the sme dung the ctch? () Does the mgntude o ou ngul momentum (out the otton xs o the tle) ncese, decese, o emn the sme te the ctch? (c) How does the ll s ngul momentum (out the otton xs o the tle) chnge te the ctch? (d) How does the totl ngul momentum o the sstem, ou-tle-ll (out the otton xs o the tle) chnge te the ctch? Detemne the Concept You cn ppl consevton o ngul momentum to the ou-tle-ll sstem to nswe ech o these questons. () Becuse the ll s movng n the sme decton tht ou e movng, ou ngul speed wll ncese when ou ctch t. You Newton s 3 d lw ntecton wth the ll cuses toque tht cts on the ou-tle-ll sstem to ncese. () The ll hs ngul momentum eltve to the otton xs o the tle eoe ou ctch t nd so ctchng t nceses ou ngul momentum eltve to the otton xs o the tle. (c) The ll wll slow down s esult o ou ctch nd so ts ngul momentum eltve to the cente o the tle wll decese. (d) Becuse thee s zeo net toque on the ou-tle-ll sstem, ts ngul momentum emns the sme. 9 the ngul momentum o sstem out xed pont P s constnt, whch one o the ollowng sttements must e tue? () () (c) (d) (e) No toque out P cts on n pt o the sstem. A constnt toque out P cts on ech pt o the sstem. Zeo net toque out P cts on ech pt o the sstem. A constnt extenl toque out P cts on the sstem. Zeo net extenl toque out P cts on the sstem.
Angul omentum 973 Detemne the Concept s constnt, we know tht the net toque ctng on the sstem s zeo. Thee m e multple constnt o tme-dependent toques ctng on the sstem s long s the net toque s zeo. (e) s coect. A lock sldng on ctonless tle s ttched to stng tht psses though now hole though the tletop. ntll, the lock s sldng wth speed v n ccle o dus. A student unde the tle pulls slowl on the stng. Wht hppens s the lock spls nwd? Gve suppotng guments o ou choce. (The tem ngul momentum ees to the ngul momentum out vetcl xs though the hole.) () ts eneg nd ngul momentum e conseved. () ts ngul momentum s conseved nd ts eneg nceses. (c) ts ngul momentum s conseved nd ts eneg deceses. (d) ts eneg s conseved nd ts ngul momentum nceses. (e) ts eneg s conseved nd ts ngul momentum deceses. Detemne the Concept The pull tht the student exets on the lock s t ght ngles to ts moton nd exets no toque (ecll tht τ F nd τ F snφ ). Theeoe, we cn conclude tht the ngul momentum o the lock s conseved. The student does, howeve, do wok n dsplcng the lock n the decton o the dl oce nd so the lock s eneg nceses. () s coect. [SS] One w to tell n egg s hdoled o uncooked wthout ekng the egg s to l the egg lt on hd suce nd t to spn t. A hdoled egg wll spn esl, whle n uncooked egg wll not. Howeve, once spnnng, the uncooked egg wll do somethng unusul; ou stop t wth ou nge, t m stt spnnng gn. xpln the deence n the ehvo o the two tpes o eggs. Detemne the Concept The hdoled egg s sold nsde, so evethng ottes wth unom ngul speed. B contst, when ou stt n uncooked egg spnnng, the olk wll not mmedtel spn wth the shell, nd when ou stop t om spnnng the olk wll contnue to spn o whle. xpln wh helcopte wth just one mn oto hs second smlle oto mounted on hozontl xs t the e s n Fgue -4. Desce the esultnt moton o the helcopte ths e oto ls dung lght. Detemne the Concept The pupose o the second smlle oto s to pevent the od o the helcopte om ottng. the e oto ls, the od o the helcopte wll tend to otte on the mn xs due to ngul momentum eng conseved.
974 Chpte 3 The spn ngul-momentum vecto o spnnng wheel s pllel wth ts xle nd s ponted est. To cuse ths vecto to otte towd the south, n whch decton must oce e exeted on the est end o the xle? () up, () down, (c) noth, (d) south, (e) est. Detemne the Concept The vecto Δ (nd the toque tht s esponsle o ths chnge n the decton o the ngul momentum vecto) ntll ponts to the south. One cn use ght-hnd ule to detemne the decton o ths toque, nd hence the oce exeted on the est end o the xle, equed to cuse the ngul-momentum vecto to otte towd the south. ettng the nges o ou ght hnd pont est, otte ou wst untl ou thum ponts south. Note tht ou culed nges, whch pont n the decton o the oce tht must e exeted on the est end o the xle, pont upwd. () s coect. 4 You e wlkng towd the noth nd n ou let hnd ou e cng sutcse tht contns mssve spnnng wheel mounted on n xle ttched to the ont nd ck o the cse. The ngul veloct o the goscope ponts noth. You now egn to tun to wlk towd the est. As esult, the ont end o the sutcse wll () esst ou ttempt to tun t nd wll t to mntn ts ognl oentton, () esst ou ttempt to tun nd wll pull to the west, (c) se upwd, (d) dp downwd, (e) show no eect whtsoeve. Detemne the Concept n tunng towd the est, ou edect the ngul momentum vecto om noth to est exetng toque on the spnnng wheel. The oce tht ou must exet to poduce ths toque (use ght-hnd ule wth ou thum pontng ethe est o noth nd note tht ou nges pont upwd) s upwd. Tht s, the oce ou exet on the ont end o the sutcse s upwd nd the oce the sutcse exets on ou s downwd. Consequentl, the ont end o the sutcse wll dp downwd. (d) s coect. 5 [SS] The ngul momentum o the popelle o smll sngleengne plne ponts owd. The popelle ottes clockwse vewed om ehnd. () Just te lto, s the nose o the plne tlts upwd, the plne tends to vee to one sde. To whch sde does t tend to vee nd wh? () the plne s lng hozontll nd suddenl tuns to the ght, does the nose o the plne tend to vee upwd o downwd? Wh? () The plne tends to vee to the ght. The chnge n ngul momentum o the popelle s upwd, so the net toque τ on the popelle s upwd s well. The popelle must exet n equl ut opposte toque on the plne. Ths downwd toque exeted on the plne the popelle tends to cuse downwd Δ pop
Angul omentum 975 chnge n the ngul momentum o the plne. Ths mens the plne tends to otte clockwse s vewed om ove. () The nose o the plne tends to vee downwd. The chnge n ngul momentum Δ pop o the popelle s to the ght, so the net toque τ on the popelle s towd the ght s well. The popelle must exet n equl ut opposte toque on the plne. Ths letwd dected toque exeted the popelle on the plne tends to cuse letwd-dected chnge n ngul momentum o the plne. Ths mens the plne tends to otte clockwse s vewed om the ght. 6 You hve desgned c tht s poweed the eneg stoed n sngle lwheel wth spn ngul momentum. n the monng, ou plug the c nto n electcl outlet nd moto spns the lwheel up to speed, ddng huge mount o ottonl knetc eneg to t eneg tht wll e chnged nto tnsltonl knetc eneg o the c dung the d. Hvng tken phscs couse nvolvng ngul momentum nd toques, ou elze tht polems would se dung vous mneuves o the c. Dscuss some o these polems. Fo exmple, suppose the lwheel s mounted so ponts vetcll upwd when the c s on hozontl od. Wht would hppen s the c tvels ove hlltop? Though vlle? Suppose the lwheel s mounted so ponts owd, o to one sde, when the c s on hozontl od. Then wht would hppen s the c ttempts to tun to the let o ght? n ech cse tht ou exmne, consde the decton o the toque exeted on the c the od. Detemne the Concept ponts upwd nd the c tvels ove hll o though vlle, the oce the od exets on the wheels on one sde (o the othe) wll ncese nd c wll tend to tp. ponts owd nd the c tuns let o ght, the ont (o e) o the c wll tend to lt. These polems cn e veted hvng two dentcl lwheels tht otte on the sme sht n opposte dectons. 7 [SS] You e sttng on spnnng pno stool wth ou ms olded. () When ou extend ou ms out to the sde, wht hppens to ou knetc eneg? Wht s the cuse o ths chnge? () xpln wht hppens to ou moment o net, ngul speed nd ngul momentum s ou extend ou ms. Detemne the Concept The ottonl knetc eneg o the ou-stool sstem s gven K ot. Becuse the net toque ctng on the ou-stool sstem s zeo, ts ngul momentum s conseved.
976 Chpte () You knetc eneg deceses. ncesng ou moment o net whle consevng ou ngul momentum deceses ou knetc eneg K. () xtendng ou ms out to the sde nceses ou moment o net, deceses ou ngul speed. The ngul momentum o the sstem s unchnged. 8 A unom od o mss nd length ests on hozontl ctonless tle. A lo o putt o mss m /4 moves long lne pependcul to the od, stkes the od ne ts end, nd stcks to the od. Desce qulttvel the susequent moton o the od nd putt. Detemne the Concept The cente o mss o the od-nd-putt sstem moves n stght lne, nd the sstem ottes out ts cente o mss. stmton nd Appoxmton 9 [SS] An ce-skte stts he pouette wth ms outstetched, ottng t.5 ev/s. stmte he ottonl speed (n evolutons pe second) when she ngs he ms lt gnst he od. Pctue the Polem Becuse we hve no nomton egdng the mss o the skte, we ll ssume tht he od mss (not ncludng he ms) s 5 kg nd tht ech m hs mss o 4. kg. et s lso ssume tht he ms e. m long nd tht he od s clndcl wth dus o cm. Becuse the net extenl toque ctng on he s zeo, he ngul momentum wll emn constnt dung he pouette. Becuse the net extenl toque ctng on he s zeo: xpess he totl moment o net wth he ms outstetched: Δ o () ms n msn msout msout + msout od ms Tetng he od s though t s clndcl, clculte the moment o net o he od, mnus he ms: od m. kg m ( 5kg)(.m)
odelng he ms s though the e ods, clculte the moment o net when the e outstetched: ms Angul omentum 977 [ ( 4kg)(.m) ] 3.67 kg m Susttute to detemne he totl moment o net wth he ms outstetched: xpess he totl moment o net wth he ms lt gnst he od: ms out ms n. kg m 3.67 kg m + od + ms. kg m.3 kg m +.67 kg m [( 4.kg)(.m) ] Solve equton () o ms n to otn: msout ms n msn msout Susttute numecl vlues nd evlute : ms n ms n 3.67 kg m.3 kg m 4ev/s (.5ev/s) stmte the to o ngul veloctes o the otton o dve etween the ull tuck poston nd the ull-lout poston. Pctue the Polem Becuse the net extenl toque ctng on the dve s zeo, the dve s ngul momentum wll emn constnt s she ottes om the ull tuck to the ull lout poston. Assume tht, n lout poston, the dve s thn od o length.5 m nd tht, n the ull tuck poston, the dve s sphee o dus.5 m. Becuse the net extenl toque ctng on the dve s zeo: Δ o lout tuck lout lout tucktuck Solvng o the to o the ngul veloctes gves: tuck lout lout tuck
978 Chpte Susttutng o the moment o net o thn od eltve to n xs though ts cente o mss nd the moment o net o sphee eltve to ts cente o mss nd smplng elds: tuck lout 5 m m l 5l 4 Susttute numecl vlues nd evlute : tuck lout tuck lout 5 4 (.5 m) (.5 m) 5 The ds on s nd th e o nel dentcl length. th s mss s 9.35 tmes s s mss, th s dus s.88 tmes s s dus, nd s s on vege.5 tmes the w om the Sun thn th s. The tn e s.88 tmes longe thn th s e. Assume tht the e oth unom sphees nd tht the ots out the Sun e ccles. stmte the to (th to s) o () the spn ngul moment, () the spn knetc eneges, (c) the otl ngul moment, nd (d) the otl knetc eneges. Pctue the Polem We cn use the dentons o spn ngul momentum, spn knetc eneg, otl ngul momentum, nd otl knetc eneg to evlute these tos. () The to o the spn ngul moment o th nd s s: spn Becuse s nd th hve nel dentcl lengths o ds, : spn Susttutng o the moments o net nd smplng elds: spn 5 5 R R R R Susttute numecl vlues o the tos nd evlute : spn spn 9.35.88 ( ) 33 () The to o the spn knetc eneges o th nd s s: K K spn
Angul omentum 979 Becuse spn : spn spn K K Fom () 33 spn. Hence: 33 spn K K (c) Tetng th nd s s pont ojects, the to o the otl ngul moment s: o, o, o Susttutng o the moments o net nd otl ngul speeds elds: o T T π π whee nd e the d o the ots o th nd s, espectvel. Smpl to otn: o T T Susttute numecl vlues o the thee tos nd evlute o : ( ) ( ) 8.88.5 9.35 o (d) The to o the otl knetc eneges o th nd s s: o, o, o K K Susttutng o the moments o net nd ngul speeds nd smplng gves: o T T T T K K π π Susttute numecl vlues o the tos nd evlute o K K : ( ) ( ) 4.88.5 9.35 o K K
98 Chpte The pol ce cps contn out.3 9 kg o ce. Ths mss contutes neglgl to the moment o net o th ecuse t s locted t the poles, close to the xs o otton. stmte the chnge n the length o the d tht would e expected the pol ce cps wee to melt nd the wte wee dstuted unoml ove the suce o th. Pctue the Polem The chnge n the length o the d s the deence etween ts length when the ce cps hve melted nd the wte hs een dstuted ove the suce o th nd the length o the d eoe the ce cps melt. Becuse the net toque ctng on th dung ths pocess s zeo, ngul momentum s conseved nd we cn elte the ngul speed (whch e elted to the length o the d) o th eoe nd te the ce cps melt to the moments o net o the th-plus-sphecl shell the ce cps melt. xpess the chnge n the length o d s: ΔT T T () te eoe Becuse the net toque ctng on th dung ths pocess s zeo, ngul momentum s conseved: Susttutng o te nd eoe elds: Δ te eoe ( + ) sphee shell te sphee eoe Becuse π T π shell Tte o, smplng, sphee + shell T T π T : ( + ) sphee sphee eoe te sphee eoe Solve ot te to otn: Susttutng ot te n equton () nd smplng elds: T + shell te T eoe sphee ΔT + shell T sphee shell sphee eoe T eoe T eoe
Angul omentum 98 Susttute o shell nd sphee nd smpl to otn: m 5m ΔT Teoe T R 3 3 5 eoe Susttute numecl vlues nd evlute ΔT: 9 (.3 kg) ( 5 T 3 5.98 Δ 4 d 4h 36s d h.55s 3 [SS] A.-g ptcle moves t constnt speed o 3. mm/s ound ccle o dus 4. mm. () Fnd the mgntude o the ngul momentum o the ptcle. () l( l + )h, whee l s n ntege, nd the ( ) nd the ppoxmte vlue o l. (c) B how much does l chnge vlue o ll+ the ptcle s speed nceses one-mllonth o pecent, nd nothng else chngng? Use ou esult to expln wh the quntzton o ngul momentum s not notced n mcoscopc phscs. Pctue the Polem We cn use mv to nd the ngul momentum o the ptcle. n () we cn solve l( l + )h o l( l +) nd the ppoxmte vlue o l. () Use the denton o ngul momentum to otn: mv.4 3 3 3 (. kg)( 3. m/s)( 4. m) 8 kg m /s.4 8 kg m /s () Solve the equton l( l + )h o l ( l +) : l + () h ( l ) Susttute numecl vlues nd l + evlute l ( ): l( l ).4 kg m /s.5 J s 8 + 34 5. 5 Becuse l >>, ppoxmte ts vlue wth the sque oot o l l + : ( ) l.3 6 (c) The chnge n l s: Δ l l l () new
98 Chpte the ptcle s speed nceses one-mllonth o pecent whle nothng else chnges: v v + 8 nd + 8 v ( + )v 8 8 ( + ) quton () ecomes: l new nd l new 8 ( ) [( + ) ] l + new 8 ( + ) 8 Susttutng n equton () elds: ( + ) Δl l h h h 8 new l h h Susttute numecl vlues nd evlute Δ l : Δl.4 kg m /s.5 J s 8 8 34.3 nd l.3 l.3 8 8 Δ 6 6 % The quntzton o ngul momentum s not notced n mcoscopc phscs ecuse no expement cn detect ctonl chnge n l o 6 %. 4 Astophscsts n the 96s ted to expln the exstence nd stuctue o pulss extemel egul stonomcl souces o do pulses whose peods nged om seconds to mllseconds. At one pont, these do souces wee gven the conm G (ttle Geen en), eeence to the de tht the mght e sgnls o extteestl cvlztons. The explnton gven tod s no less nteestng. Consde the ollowng. Ou Sun, whch s l tpcl st, hs mss o.99 3 kg nd dus o 6.96 8 m. Although t does not otte unoml, ecuse t s not sold od, ts vege te o otton s out ev/5 d. Sts lge thn the Sun cn end the le n spectcul explosons clled supenove, levng ehnd collpsed emnnt o the st clled neuton st. These neuton-sts hve msses comple to the ognl msses o the sts ut d o onl ew klometes! The hgh otton tes e due to the consevton o ngul momentum dung the collpses. These sts emt ems o do wves. Becuse o the pd ngul speed o the sts, the em sweeps pst th t egul, ve shot, ntevls. To poduce the oseved do-wve pulses, the st hs to otte t tes tht nge om out ev/s to ev/s. () Usng dt om the textook, estmte the otton te o the Sun t wee to collpse nto neuton st o dus km. The Sun s not unom sphee o
Angul omentum 983 gs, nd ts moment o net s gven.59r. Assume tht the neuton st s sphecl nd hs unom mss dstuton. () s the ottonl knetc eneg o the Sun gete o smlle te the collpse? B wht cto does t chnge, nd whee does the eneg go to o come om? Pctue the Polem We cn use consevton o ngul momentum n Pt () to elte the eoe-nd-te collpse otton tes o the Sun. n Pt (), we cn expess the ctonl chnge n the ottonl knetc eneg o the Sun s t collpses nto neuton st to decde whethe ts ottonl knetc eneg s gete ntll o te the collpse. () Use consevton o ngul momentum to elte the ngul moment o the Sun eoe nd te ts collpse: () Usng the gven omul, ppoxmte the moment o net o the Sun eoe collpse: sun 3 5 46 ( kg)( 6.96 km) 5.69 kg m.59r.59.99 Fnd the moment o net o the Sun when t hs collpsed nto sphecl neuton st o dus km nd unom mss dstuton: R 5 5 3 (.99 kg)( km) 7.96 37 kg m Susttute numecl vlues n equton () nd smpl to otn: Gven tht ev/5 d, evlute : 5.69 7.96 46 37 8 7.5.9 kg m kg m 8 ev 7.5.86 ev/d 5d 7 ev/d Note tht the ottonl peod deceses the sme cto o / nd ecomes: T π.86 π ev π d d h d ev 4h 36s 3 7 3. s
984 Chpte () xpess the ctonl chnge n the Sun s ottonl knetc eneg s consequence o ts collpse: ΔK K K K K K K Susttutng o the knetc eneges ΔK nd smplng elds: K Susttute numecl vlues nd evlute ΔK/K : ΔK K 7.5 8 7.86 ev/d ev/5d 7. 8 Tht s, the ottonl knetc eneg nceses cto o ppoxmtel 7 8. The ddtonl ottonl knetc eneg comes t the expense o gvttonl potentl eneg, whch deceses s the Sun gets smlle. 5 The moment o net o th out ts spn xs s ppoxmtel 8.3 37 kg m. () Becuse th s nel sphecl, ssume tht the moment o net cn e wtten s CR, whee C s dmensonless constnt, 5.98 4 kg s the mss o th, nd R 637 km s ts dus. Detemne C. () th s mss wee dstuted unoml, C would equl /5. Fom the vlue o C clculted n Pt (), s th s denst gete ne ts cente o ne ts suce? xpln ou esonng. Pctue the Polem We cn solve CR o C nd susttute numecl vlues n ode to detemne n expementl vlue o C o th. We cn then compe ths vlue to those o sphecl shell nd sphee n whch the mss s unoml dstuted to decde whethe th s mss denst s getest ne ts coe o ne ts cust. () xpess the moment o net o th n tems o the constnt C: CR C R Susttute numecl vlues nd 8.3 evlute C: ( 4 5.98 kg )( 637 km ).33 37 kg m () Becuse expementll C <.4, the mss denst must e gete ne the cente o th. 6 stmte Tmoth Goeel s ntl tkeo speed, ottonl veloct, nd ngul momentum when he peoms quduple utz (Fgue -4). ke n ssumptons ou thnk esonle, ut just them. Goeel s mss s out 6 kg nd the heght o the jump s out.6 m. Note tht hs ngul speed wll C
Angul omentum 985 chnge qute t dung the jump, s he egns wth ms outstetched nd pulls them n. You nswe should e ccute to wthn cto o, ou e ceul. Pctue the Polem We ll ssume tht he lunches hmsel t n ngle o 45 wth the hozontl wth hs ms sped wde, nd then pulls them n to ncese hs ottonl speed dung the jump. We ll lso ssume tht we cn model hm s.-m long clnde wth n vege dus o.5 m nd mss o 6 kg. We cn then nd hs tke-o speed nd tme usng constnt-cceleton equtons, nd use the ltte, togethe wth the denton o ottonl veloct, to nd hs ntl ottonl veloct. Fnll, we cn ppl consevton o ngul momentum to nd hs ntl ngul momentum. Usng constnt-cceleton equton, elte hs tkeo speed v to hs mxmum elevton Δ: v v + Δ o, ecuse v v sn(45 ), v, nd g, v sn 45 gδ Solvng o v nd smplng elds: v gδ sn 45 gδ sn 45 Susttute numecl vlues nd evlute v : v ( )(.6m) 9.8m/s sn45 4.9 m/s Use ts denton to expess Goeel s ngul veloct: Δ θ Δt Use constnt-cceleton equton to expess Goeel s tme Δt: Δt Δtse.6 m Δ g Susttute numecl vlues nd evlute Δt: (.6m) Δ t.699s 9.8m/s Susttute numecl vlues nd evlute : Use consevton o ngul momentum to elte hs tke-o ngul veloct to hs vege ngul veloct s he peoms quduple utz: 4ev.699s π d ev 36 d/s
986 Chpte Assumng tht he cn chnge hs moment o net cto o pullng hs ms n, solve o nd evlute : xpess hs tke-o ngul momentum: Assumng tht we cn model hm s sold clnde o length l wth n vege dus nd mss m, expess hs moment o net wth ms outstetched (hs tke-o conguton): Susttutng o gves: Susttute numecl vlues nd evlute : ( 36 d/s) 8d/s ( m ) m whee the cto o epesents ou ssumpton tht he cn doule hs moment o net extendng hs ms. m ( 6kg)(.5m) ( 8d/s) 4kg m /s The Vecto Poduct nd the Vecto Ntue o Toque nd Rotton 7 [SS] A oce o mgntude F s ppled hozontll n the negtve x decton to the m o dsk o dus R s shown n Fgue -4. Wte F nd n tems o the unt vectos ˆ, j ˆ, nd k ˆ, nd compute the toque poduced ths oce out the ogn t the cente o the dsk. Pctue the Polem We cn expess F nd n tems o the unt vectos î nd ĵ nd then use the denton o the vecto poduct to nd τ. xpess F n tems o F nd the unt vecto î : F Fˆ xpess n tems o R nd the unt vecto ĵ : Rˆj ( ) ( ) F τ F FR ˆj ˆ FR ˆ ˆj Clculte the vecto poduct o nd : FR kˆ
Angul omentum 987 8 Compute the toque out the ogn o the gvttonl oce F mgˆ j ctng on ptcle o mss m locted t x ˆ + j ˆ nd show tht ths toque s ndependent o the coodnte. Pctue the Polem We cn nd the toque om the vecto poduct o nd F. ( )( ) F τ F xˆ + ˆj mgˆj mgx( ˆ ˆj ) mg( ˆj ˆj ) Compute the vecto poduct o nd : mgxkˆ 9 Fnd A B o the ollowng choces: () A 4ˆ nd B 6ˆ + 6j ˆ, () A 4ˆ nd B 6ˆ + 6k ˆ, nd (c) A ˆ + 3ˆ j nd B 3ˆ + ˆ j. Pctue the Polem We cn use the dentons o the vecto poducts o the unt vectos î, ĵ, nd kˆ to evlute A B n ech cse. () vlute A B o A 4 î nd B 6 î + 6 ĵ : A B 4ˆ 4 4 ( 6ˆ + 6 ˆj ) ( ˆ ˆ ) + 4( ˆ ˆj ) ( ) 4kˆ + 4kˆ () vlute A B o A 4 î nd B 6 î + 6 kˆ : (c) vlute A B o A î + 3 ĵ nd B 3 î + ĵ : A B 4ˆ A B 4 4 ( 6ˆ + 6kˆ ) ( ˆ ˆ ) + 4( ˆ kˆ ) ( ) + 4( ˆj ) 4 ˆj ( ˆ + 3 ˆj ) ( 3ˆ + ˆj ) 6( ˆ ˆ ) + 4( ˆ ˆj ) + 9( ˆj ˆ ) + 6( ˆj ˆj ) 6() + 4( kˆ ) + 9( kˆ ) + 6() 5kˆ 3 Fo ech cse n Polem 3, compute A B. Compe t to A B to estmte whch o the ps o vectos e closest to eng pependcul. Ve ou nswes clcultng the ngle usng the dot poduct.
988 Chpte Pctue the Polem Becuse A B A B snφ, vectos A nd B e A B pependcul, then A B A B o. The scl poduct o vectos A B A nd B s A B A B cosφ. We cn ve ou estmtons usng ths denton to clculte φ o ech p o vectos. () Fo A 4 î nd B 6 î + 6 ĵ : A B 4ˆ ( 6ˆ + 6 ˆj ) 4kˆ A B ( 4)( 6 ) 4.77 nd the vectos A nd B e not pependcul. The ngle etween A nd B s: 4ˆ ( 6ˆ 6 ˆ B + j) A φ cos cos A B 4 4 cos 45, 4 esult conmng tht otned ove. () Fo A 4 î nd B 6 î + 6 kˆ : A B 4ˆ ( 6ˆ + 6kˆ ) A B ( 4)( 6 ) 4 ˆj 4.77 nd the vectos A nd B e not pependcul. The ngle etween A nd B s: 4ˆ ( 6ˆ 6 ˆ B + k) A φ cos cos A B 4 4 cos 45, 4 esult conmng tht otned ove.
(c) Fo A î + B 3 î + ĵ : 3 ĵ nd The ngle etween A nd B s: Angul omentum 989 ( ˆ + 3 ˆj ) ( 3ˆ + ˆj ) A B 5kˆ A B 3 3 3 5 3.385 nd the vectos A nd B e not pependcul. cos A B φ A B ( ˆ 3 ˆ) ( 3ˆ ˆ ) cos + j + j 3 3 cos 3, 3 esult conmng tht otned ove. Whle none o these sets o vectos e pependcul, those n () nd () e the closest, wth φ 45, to eng pependcul. 3 A ptcle moves n ccle tht s centeed t the ogn. The ptcle hs poston nd ngul veloct. () Show tht ts veloct s gven v. v () Show tht ts centpetl cceleton s gven ( ) Pctue the Polem et e n the x plne nd pont n the +x decton. Then ponts n the +z decton. We cn estlsh the esults clled o n ths polem omng the ppopte vecto poducts nd deenttng v. c. () xpess usng unt vecto notton: kˆ xpess usng unt vecto notton: ˆ Fom the vecto poduct o nd : kˆ ˆ ( kˆ ˆ ) ˆj vˆj nd v
Chpte 99 () Deentte v wth espect to t to expess : ( ) ( ) c t t v v + + + + dt d dt d dt d dt d dt d whee ( ) c nd c t nd e the tngentl nd centpetl cceletons, espectvel. 3 You e gven thee vectos nd the components n the om: A x ˆ + ˆ j + z ˆ k, B x ˆ + ˆ j + z ˆ k, nd C c x ˆ + c ˆ j + c z ˆ k. Show tht the ollowng equltes hold: ( ) ( ) ( ) A C B B A C C B A Pctue the Polem We cn estlsh these equltes cng out the detls o the vecto- nd scl-poducts nd compng the esults o these opetons. vlute the vecto poduct o B nd C to otn: ( ) ( ) ( )k j C B ˆ ˆ ˆ x x z x x z z z c c c c c c + + Fom the scl poduct o A wth B C to otn: ( ) x z x z z x x z z x z x c c c c c c + + C B A () vlute the vecto poduct o A nd B to otn: ( ) ( ) ( )k j B A ˆ ˆ ˆ x x z x x z z z + + Fom the scl poduct o C wth A B to otn: ( ) x z x z z x x z z x z x c c c c c c + + B A C () vlute the vecto poduct o C nd A to otn: ( ) ( ) ( )k j A C ˆ ˆ ˆ x x z x x z z z c c c + + Fom the scl poduct o B wth C A to otn: ( ) x z x z z x x z z x z x c c c c c c + + A C B (3)
The eqult o equtons (), (), nd (3) estlshes the equltes. 33 A 3ˆ j, A B 9ˆ, nd A B, nd B. Pctue the Polem We cn wte B n the om Angul omentum 99 B B ˆ + B ˆj + B kˆ nd use the scl poduct o A nd B to nd B nd the vecto poduct to nd B x nd B z. xpess B n tems o ts components: B B ˆ + B ˆj + B kˆ vlute A B : A B 3 B B 4 vlute A : B A B 3 ˆj ( B ˆ + 4 ˆj + B kˆ ) x 3B x x x z kˆ + 3B ˆ Becuse A B 9 î : B x nd B z 3. Susttute o B nd B z n equton () to otn: 34 A 4ˆ, B z, B 5 B 4 ˆj + 3kˆ, nd A B ˆ k, detemne B. Pctue the Polem Becuse B z, we cn expess B s B B ˆ + om ts vecto poduct wth A to detemne B x nd B. xpess B n tems o ts components: B B ˆ + x B ˆj z z z () x B xpess A B : A B ( B B j) B k k Solvng o B elds: B 3 ˆj () nd 4 ˆ ˆ + ˆ 4 ˆ ˆ x Relte B to B x nd B : B B x + B Solve o nd evlute B x : B B B 5 3 4 x Susttute o B x nd B n equton () to otn: B 4 ˆ + 3 ˆj
99 Chpte 35 Gven thee noncopln vectos A, B, nd C, show tht A s the volume o the pllelepped omed the thee vectos. B C ( ) Pctue the Polem et, wthout loss o genelt, the vecto C le long the x xs nd the vecto B le n the x plne s shown elow to the let. The dgm to the ght shows the pllelepped spnned the thee vectos. We cn ppl the dentons o the vecto- nd scl-poducts to show tht A ( B C ) s the volume o the pllelepped. B C Acosθ θ A C Bsnφ φ B The mgntude o the vecto poduct o B nd C s: Fomng the scl-poduct o A wth the vecto-poduct o B nd C gves: B C BC snφ nd B C Bsnφ C A ( ) e o the se pllelogm ( B C ) A( Bsnφ) ( BC snφ)( Acosθ ) ( e o se)( heght) V pllelepped C cosθ 36 Usng the coss poduct, pove the lw o snes o the tngle shown n Fgue -43. Tht s, A, B, nd C e the lengths o ech sde o the tngle, show tht A/sn B/sn C/sn c. Pctue the Polem Dw the tngle usng the thee vectos s shown elow. Note tht A + B C. We cn nd the mgntude o the vecto poduct o A nd B nd o A nd C nd then use the vecto poduct o A nd C, usng
Angul omentum 993 B C A + B C, to show tht AC sn ABsn c o. Poceedng smll, sn sn c we cn extend the lw o snes to the thd sde o the tngle nd the ngle opposte t. A c B xpess the mgntude o the vecto poduct o A nd : xpess the mgntude o the vecto poduct o A nd C : Fom the vecto poduct o A wth C to otn: C A B B AB ( c) AB c A C AC sn sn 8 ( A + B) A C A A A + A B A B ecuse A A. Becuse A C A B : A C A B nd AC sn ABsn c sn Smpl nd ewte ths expesson to otn: B C sn sn c Poceed smll to extend ths esult to the lw o snes: A sn B sn C sn c Toque nd Angul omentum 37 [SS] A.-kg ptcle moves dectl estwd t constnt speed o 4.5 m/s long n est-west lne. () Wht s ts ngul momentum (ncludng decton) out pont tht les 6. m noth o the lne? () Wht s ts ngul momentum (ncludng decton) out pont tht les 6. m south o the lne?
994 Chpte (c) Wht s ts ngul momentum (ncludng decton) out pont tht les 6. m dectl est o the ptcle? Pctue the Polem The ngul momentum o the ptcle s p whee s the vecto loctng the ptcle eltve to the eeence pont nd p s the ptcle s lne momentum. () The mgntude o the ptcle s ngul momentum s gven : Susttute numecl vlues nd evlute : psnφ mvsnφ mv (.kg)( 4.5m/s)( 6.m) 54kg m /s ( snφ) Use ght-hnd ule to estlsh the decton o : 54kg m /s, upwd () Becuse the dstnce to the lne long whch the ptcle s movng s the sme, onl the decton o des: (c) Becuse p o pont on the lne long whch the ptcle s movng: 54kg m /s, downwd 38 You oseve.-kg ptcle movng t constnt speed o 3.5 m/s n clockwse decton ound ccle o dus 4. m. () Wht s ts ngul momentum (ncludng decton) out the cente o the ccle? () Wht s ts moment o net out n xs though the cente o the ccle nd pependcul to the plne o the moton? (c) Wht s the ngul veloct o the ptcle? Pctue the Polem The ngul momentum o the ptcle s p whee s the vecto loctng the ptcle eltve to the eeence pont nd p s the ptcle s lne momentum. () The mgntude o the ptcle s ngul momentum s gven : Susttute numecl vlues nd evlute the mgntude o : psnφ mvsnφ mv (.kg)( 3.5m/s)( 4.m) 8kg m /s ( snφ)
Use ght-hnd ule to estlsh the decton o : Angul omentum 995 8kg m /s, w om ou () Tet the.-kg ptcle s pont ptcle to otn: m Susttute numecl vlues nd evlute : (.kg)( 4.m) 3kg m (c) Becuse, the ngul speed o the ptcle s the to o ts ngul momentum nd ts moment o net: Susttute numecl vlues nd evlute : 8kg m /s 3kg m.88d/s 39 () A ptcle movng t constnt veloct hs zeo ngul momentum out ptcul pont. Use the denton o ngul momentum to show tht unde ths condton the ptcle s movng ethe dectl towd o dectl w om the pont. () You e ght-hnded tte nd let wst-hgh stll go pst ou wthout swngng. Wht s the decton o ts ngul momentum eltve to ou nvel? (Assume the ll tvels n stght hozontl lne s t psses ou.) Pctue the Polem nd p e elted ccodng to p., then exmnton o the mgntude o p wll llow us to conclude tht sn φ nd tht the ptcle s movng ethe dectl towd the pont, dectl w om the pont, o though the pont. () Becuse : p m v m v o v xpess the mgntude o v : v vsn φ Becuse nethe no v s zeo: sn φ whee φ s the ngle etween nd v. Solvng o φ elds: φ ( ) sn o 8
996 Chpte () Use the ght-hnd ule to estlsh tht the ll s ngul momentum s downwd. 4 A ptcle tht hs mss m s tvelng wth constnt veloct v long stght lne tht s dstnce om the ogn O (Fgue -44). et da e the e swept out the poston vecto om O to the ptcle dung tme ntevl dt. Show tht da/dt s constnt nd s equl to m, whee s the mgntude o the ngul momentum o the ptcle out the ogn. Pctue the Polem The e swept out the poston vecto (the shded e n Fgue -44) s the deence etween the e o tpezod nd the e o tngle. et x e the x component o nd x + Δx e the x component o. Use the omuls o the es o tpezod nd tngle to expess ΔA nd then tke the lmt s Δt to expess da/dt. ettng θ e the ngle etween nd the hozontl xs, we cn expess s uncton o nd θ. The e swept out the poston vecto (the shded e n Fgue - 44) s gven : n the lmt s Δt : Becuse snθ : ΔA Atp A Δx da dt da dt tngle [ Δx + ( x + Δx) ] ( x + Δx) dx v dt v m ( snθ ) ( psnθ ) constnt v m ( snθ ) m mv 4 A 5-g con tht hs dmete o.5 cm s spnnng t ev/s out xed vetcl xs. The con s spnnng on edge wth ts cente dectl ove the pont o contct wth the tletop. As ou look down on the tletop, the con spns clockwse. () Wht s the ngul momentum (ncludng decton) o the con out ts cente o mss? (To nd the moment o net out the xs, see Tle 9-.) odel the con s clnde o length nd tke the lmt s ppoches zeo. () Wht s the con s ngul momentum (ncludng decton) out pont on the tletop cm om the xs? (c) Now the con s cente o mss tvels t 5. cm/s n stght lne est coss the tletop, whle spnnng the sme w s n Pt (). Wht s the ngul momentum (ncludng decton) o the con out pont on the lne o moton o the cente o mss? (d) When t s oth spnnng nd sldng, wht s the ngul momentum o the con (ncludng decton) out pont cm noth o the lne o moton o the cente o mss?
Angul omentum 997 Pctue the Polem We cn nd the totl ngul momentum o the con om the sum o ts spn nd otl ngul moment. () The spn ngul momentum o the con s: Fom Tle 9-, o neglgle comped to R: Susttute o to otn: spn spn R spn 4 4 R spn Susttute numecl vlues nd evlute spn : spn 4 ev s π d ev 5 (.5kg)(.75m).33 kg m /s Use ght-hnd ule to estlsh the decton o spn : spn.3 5 kg m /s, w om ou. ()The totl ngul momentum o the con s the sum o ts otl nd spn ngul moment: + totl otl spn Susttute numecl vlues nd evlute totl : totl + spn.3 5 kg m /s Use ght-hnd ule to estlsh the decton o totl : totl.3 5 kg m /s, w om ou (c) Becuse otl : totl.3 5 kg m /s, w om ou (d) When t s oth spnnng nd sldng, the totl ngul momentum o the con s: + totl otl spn
998 Chpte The otl ngul momentum o the con s: The spn ngul momentum o the con s: vr otl spn spn spn 4 R spn Susttutng o otl nd spn elds: totl vr + R spn Susttute numecl vlues nd evlute totl : 4 totl (.5kg)(.5 m/s)(.m) 8.8 5 ev π d + 4 (.5 kg)(.75 m) s ev kg m /s, pontng towd ou 4 () Two sts o msses m nd m e locted t nd eltve to some ogn O, s shown n Fgue -45. The exet equl nd opposte ttctve gvttonl oces on ech othe. Fo ths two-st sstem, clculte the net toque exeted these ntenl oces out the ogn O nd show tht t s zeo onl oth oces le long the lne jonng the ptcles. ()The ct tht the Newton s thd-lw p o oces e not onl equl nd oppostel dected ut lso le long the lne connectng the two ojects s sometmes clled the stong om o Newton s thd lw. Wh s t mpotnt to dd tht lst phse? Hnt: Consde wht would hppen to these two ojects the oces wee oset om ech othe. Pctue the Polem Both the oces ctng on the ptcles exet toques wth espect to n xs pependcul to the pge nd though pont O nd the net toque out ths xs s the vecto sum. () The net toque out n xs pependcul to the pge nd though pont O s gven : τ net τ F + F o, ecuse F F, τ F net ( ) Becuse ponts long F : τ ( ) F net
Angul omentum 999 () the oces e not long the sme lne, thee wll e net toque (ut stll no net oce) ctng on the sstem. Ths net toque would cuse the sstem to ccelete ngull, cont to osevton, nd hence mkes no sense phscll. 43 A.8-kg ptcle moves n ccle o dus 3.4 m. As ou look down on the plne o ts ot, t s ntll movng clockwse. we cll the clockwse decton postve, ts ngul momentum eltve to the cente o the ccle ves wth tme ccodng to () t N m s ( 4. N m)t. () Fnd the mgntude nd decton o the toque ctng on the ptcle. () Fnd the ngul veloct o the ptcle s uncton o tme. Pctue the Polem The ngul momentum o the ptcle chnges ecuse net toque cts on t. Becuse we know how the ngul momentum depends on tme, we cn nd the net toque ctng on the ptcle deenttng ts ngul momentum. We cn use constnt-cceleton equton nd Newton s second lw to elte the ngul speed o the ptcle to ts ngul cceleton. () The mgntude o the toque ctng on the ptcle s the te t whch ts ngul momentum chnges: d τ net dt vlute d/dt to otn: τ net d t dt [ N m s ( 4. N m) ] 4. N m Note tht, ecuse deceses s the ptcle ottes clockwse, the ngul cceleton nd the net toque e oth upwd. () The ngul speed o the ptcle s gven : otl otl otl Tetng the.8-kg ptcle s pont ptcle, expess ts moment o net eltve to n xs though the cente o the ccle nd noml to t: R otl Susttute o otl nd otl to otn: otl N m s R ( 4. N m) t
Chpte Susttute numecl vlues nd evlute otl : otl m s ( 4. N m) (.8kg)( 3.4m) (.9 d/s ) N t.48 d/s t Note tht the decton o the ngul veloct s downwd. 44 You e desgnng lthe moto nd pt o t conssts o unom clnde whose mss s 9 kg nd dus s.4 m tht s mounted so tht t tuns wthout cton on ts xs, whch s xed. The clnde s dven elt tht wps ound ts pemete nd exets constnt toque. At t, the clnde s ngul veloct s zeo. At t 5 s, ts ngul speed s 5 ev/mn. () Wht s the mgntude o ts ngul momentum t t 5 s? () At wht te s the ngul momentum ncesng? (c) Wht s the mgntude o the toque ctng on the clnde? (d) Wht s the mgntude o the ctonl oce ctng on the m o the clnde? Pctue the Polem The ngul momentum o the clnde chnges ecuse net toque cts on t. We cn nd the ngul momentum t t 5 s om ts denton nd the mgntude o the net toque ctng on the clnde om the te t whch the ngul momentum s chngng. The mgntude o the ctonl oce ctng on the m cn e ound usng the denton o toque. () The ngul momentum o the clnde s gven : m Susttute numecl vlues nd evlute : ( 9kg)(.4m) 377 kg m /s 5 3.8 π d mn ev 6s kg m /s ev mn () The te t whch the ngul momentum o the clnde s ncesng s gven : d dt ( 377kg m /s) 5s 5kg m /s 5kg m /s
Angul omentum (c) Becuse the toque ctng on the unom clnde s constnt, the te o chnge o the ngul momentum s constnt nd hence the nstntneous te o chnge o the ngul momentum t n nstnt s equl to the vege te o chnge ove the tme dung whch the toque cts: d τ dt 5kg m /s (d) The mgntude o the ctonl oce ctng on the m s: τ l 5.kg m /s.4m 38N 45 [SS] n Fgue -46, the nclne s ctonless nd the stng psses though the cente o mss o ech lock. The pulle hs moment o net nd dus R. () Fnd the net toque ctng on the sstem (the two msses, stng, nd pulle) out the cente o the pulle. ()Wte n expesson o the totl ngul momentum o the sstem out the cente o the pulle. Assume the msses e movng wth speed v. (c) Fnd the cceleton o the msses usng ou esults o Pts () nd () nd settng the net toque equl to the te o chnge o the sstem s ngul momentum. Pctue the Polem et the sstem nclude the pulle, stng, nd the locks nd ssume tht the mss o the stng s neglgle. The ngul momentum o ths sstem chnges ecuse net toque cts on t. We ll tke clockwse to e postve to e consstent wth postve upwd veloct o the lock whose mss s m s ndcted n the gue. () xpess the net toque out the cente o mss o the pulle: τ net ( m g snθ ) R Rm g T R Rg( m sn m ) m gr + T R T R + T R + θ () xpess the totl ngul momentum o the sstem out n xs though the cente o the pulle: + mvr + mvr vr + m + m R
Chpte (c) xpess τ s the tme devtve o the ngul momentum: d τ dt R R d dt vr R + m + m + m + m qute ths esult to tht o Pt () nd solve o to otn: ( m θ m ) g R sn + m + m 46 Fgue -47 shows the e vew o spce cpsule tht ws let ottng pdl out ts longtudnl xs t 3 ev/mn te collson wth nothe cpsule. You e the lght contolle nd hve just moments to tell the cew how to stop ths otton eoe the ecome ll om the otton nd the stuton ecomes dngeous. You know tht the hve ccess to two smll jets mounted tngentll t dstnce o 3. m om the xs, s ndcted n the gue. These jets cn ech eject g/s o gs wth nozzle speed o 8 m/s. Detemne the length o tme these jets must un to stop the otton. n lght, the moment o net o the shp out ts xs (ssumed constnt) s known to e 4 kg m. Pctue the Polem The oces esultng om the elese o gs om the jets wll exet toque on the spceshp tht wll slow nd eventull stop ts otton. We cn elte ths net toque to the ngul momentum o the spceshp nd to the tme the jets must e. Relte the ng tme o the jets to the desed chnge n ngul momentum: xpess the mgntude o the net toque exeted the jets: ettng Δm/Δt epesent the mss o gs pe unt tme exhusted om the jets, elte the oce exeted the gs on the spceshp to the te t whch the gs escpes: Δ Δ Δt () τ τ net FR Δm F v Δt' net τ net Susttutng o F elds: τ net Δm vr Δ t'
Angul omentum 3 Susttute o τ net n equton () to otn: Δ Δt Δm vr Δt' Susttute numecl vlues nd evlute Δt: Δt ( 4 kg m ) ( kg/s)( 8 m/s)( 3.m) 3 ev mn π d mn ev 6s.6 s 47 A pojectle (mss ) s lunched t n ngle θ wth n ntl speed v. Consdeng the toque nd ngul momentum out the lunch pont, explctl show tht d/dt τ. gnoe the eects o esstnce. (The equtons o pojectle moton e ound n Chpte 3.) Pctue the Polem We cn use constnt-cceleton equtons to expess the pojectle s poston nd veloct coodntes s unctons o tme. We cn use these coodntes to expess the ptcle s poston nd veloct vectos nd v. Usng ts denton, we cn expess the pojectle s ngul momentum s uncton o tme nd then deentte ths expesson to otn d dt. Fnll, we cn use the denton o the toque, eltve to n ogn locted t the lunch poston, the gvttonl oce exets on the pojectle to expess τ nd complete the demonstton tht d dt τ. Usng ts denton, expess the ngul momentum vecto o the pojectle: mv () Usng constnt-cceleton equtons, expess the poston coodntes o the pojectle s uncton o tme: x v xt ( v cosθ )t nd + v t + t ( v snθ ) t gt [( v cosθ ) t] ˆ + ( v snθ ) xpess the pojectle s poston vecto : [ ]j ˆ t gt Usng constnt-cceleton equtons, expess the veloct o the pojectle s uncton o tme: vx v x v cosθ nd v v + t v snθ gt xpess the pojectle s veloct vecto : v cos ˆ θ + v snθ gt v v [ ] [ ] j ˆ
4 Chpte Susttutng n equton () nd smplng elds: {[( ) ] ˆ V cosθ t + [( V snθ ) t gt ] ˆj } m [ V cosθ ] ˆ + [ V snθ gt] ( mgt V cosθ )kˆ { ˆj } Deentte wth espect to t to d d ( cos ) otn: mgt V θ dt dt ( mgtv cosθ )kˆ Usng ts denton, expess the toque ctng on the pojectle: kˆ () τ ( mgtv cosθ )kˆ ( mg) j [( v cosθ ) t] ˆ + ( v snθ ) [ t gt ] ˆj ( mg) ˆ ˆj (3) Compng equtons () nd (3) we see tht: d τ dt Consevton o Angul omentum 48 A plnet moves n n ellptcl ot out the Sun, wth the Sun t one ocus o the ellpse, s n Fgue -48. () Wht s the toque out the cente o the Sun due to the gvttonl oce o ttcton o the Sun on the plnet? () At poston A, the plnet hs n otl dus nd s movng wth speed v pependcul to the lne om the sun to the plnet. At poston B, the plnet hs n otl dus nd s movng wth speed v, gn pependcul to the lne om the sun to the plnet. Wht s the to o v to v n tems o nd? Pctue the Polem et m epesent the mss o the plnet nd ppl the denton o toque to nd the toque poduced the gvttonl oce o ttcton. We cn use Newton s second lw o moton n the om τ d dt to show tht s constnt nd ppl consevton o ngul momentum to the moton o the plnet t ponts A nd B. () xpess the toque poduced the gvttonl oce o ttcton o the sun o the plnet: () Becuse τ : τ F ecuse F cts long the decton o. d mv constnt dt
Angul omentum 5 Notng tht t ponts A nd B v v, expess the eltonshp etween the dstnces om the sun nd the speeds o the plnets: v v v v 49 [SS] You stnd on ctonless pltom tht s ottng t n ngul speed o.5 ev/s. You ms e outstetched, nd ou hold hev weght n ech hnd. The moment o net o ou, the extended weghts, nd the pltom s 6. kg m. When ou pull the weghts n towd ou od, the moment o net deceses to.8 kg m. () Wht s the esultng ngul speed o the pltom? () Wht s the chnge n knetc eneg o the sstem? (c) Whee dd ths ncese n eneg come om? Pctue the Polem et the sstem consst o ou, the extended weghts, nd the pltom. Becuse the net extenl toque ctng on ths sstem s zeo, ts ngul momentum emns constnt dung the pullng n o the weghts. () Usng consevton o ngul momentum, elte the ntl nd nl ngul speeds o the sstem to ts ntl nd nl moments o net: Susttute numecl vlues nd evlute : 6.kg m.8kg m (.5ev/s) 5.ev/s () xpess the chnge n the knetc eneg o the sstem: Δ K K K Susttute numecl vlues nd evlute ΔK: ΔK ev π d (.8kg m ) 5. ( 6.kg m ).6 kj s ev ev π d.5 s ev (c) Becuse no extenl gent does wok on the sstem, the eneg comes om ou ntenl eneg. 5 A smll lo o putt o mss m lls om the celng nd lnds on the oute m o tuntle o dus R nd moment o net tht s ottng eel wth ngul speed out ts vetcl xed-smmet xs. () Wht s the postcollson ngul speed o the tuntle-putt sstem? () Ate sevel tuns, the
6 Chpte lo les o the edge o the tuntle. Wht s the ngul speed o the tuntle te the lo s deptue? Pctue the Polem et the sstem consst o the lo o putt nd the tuntle. Becuse the net extenl toque ctng on ths sstem s zeo, ts ngul momentum emns constnt when the lo o putt lls onto the tuntle. () Usng consevton o ngul momentum, elte the ntl nd nl ngul speeds o the tuntle to ts ntl nd nl moments o net nd solve o : xpess the nl ottonl net o the tuntle-plus-lo: () + + mr lo Susttute o n equton () nd smpl to otn: + mr mr + () the lo les o tngentll to the tuntle, ts ngul momentum doesn t chnge (wth espect to n xs though the cente o tuntle). Becuse thee s no extenl toque ctng on the lo-tuntle sstem, the totl ngul momentum o the sstem wll emn constnt nd the ngul momentum o the tuntle wll not chnge. The tuntle wll contnue to spn t '. 5 [SS] A lz Susn conssts o hev plstc dsk mounted on ctonless eng estng on vetcl sht though ts cente. The clnde hs dus R 5 cm nd mss.5 kg. A cockoch (mss m.5 kg) s on the lz Susn, t dstnce o 8. cm om the cente. Both the cockoch nd the lz Susn e ntll t est. The cockoch then wlks long ccul pth concentc wth the cente o the z Susn t constnt dstnce o 8. cm om the xs o the sht. the speed o the cockoch wth espect to the lz Susn s. m/s, wht s the speed o the cockoch wth espect to the oom? Pctue the Polem Becuse the net extenl toque ctng on the lz Susncockoch sstem s zeo, the net ngul momentum o the sstem s constnt (equl to zeo ecuse the lz Susn s ntll t est) nd we cn use consevton o ngul momentum to nd the ngul veloct o the lz Susn. The speed o the cockoch eltve to the loo v s the deence etween ts speed wth espect to the lz Susn nd the speed o the lz Susn t the locton o the cockoch wth espect to the loo.
Angul omentum 7 Relte the speed o the cockoch wth espect to the loo v to the speed o the lz Susn t the locton o the cockoch: Use consevton o ngul momentum to otn: xpess the ngul momentum o the lz Susn: v v () () S C S S R xpess the ngul momentum o the cockoch: C v CC m Susttute o S nd C n equton v () to otn: R m Solvng o elds: mv R + m Susttute o n equton () to m v otn: v v R + m Susttute numecl vlues nd evlute v : v (.5kg)(.8 m) (. m/s) (.5m)(.5m) + (.5kg)(.8 m). m/s mm/s Remks: Becuse the moment o net o the lz Susn s so much lge thn the moment o net o the cockoch, te the cockoch egns movng, the ngul speed o the lz Susn s ve smll. Theeoe, the speed o the cockoch eltve to the loo s lmost the sme s the speed eltve to the lz Susn. 5 Two dsks o dentcl mss ut deent d ( nd ) e spnnng on ctonless engs t the sme ngul speed ut n opposte dectons (Fgue -49). The two dsks e ought slowl togethe. The esultng ctonl oce etween the suces eventull ngs them to common ngul veloct. () Wht s the mgntude o tht nl ngul veloct n tems o? () Wht s the chnge n ottonl knetc eneg o the sstem? xpln.
8 Chpte Pctue the Polem The net extenl toque ctng on ths sstem s zeo nd so we know tht ngul momentum s conseved 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 consevton o ngul momentum, elte the ntl ngul speeds o the dsks to the common nl speed nd to the moments o net: Solvng o elds: o + ( ) () + xpess nd : ( ) m m nd m Susttute o nd n equton () m m 3 5 nd smpl to otn: m + m () The chnge n knetc eneg o the sstem s gven : ΔK K K () The ntl knetc eneg o the sstem s the sum o the knetc eneges o the two dsks: K K + K ( + ) + Susttutng o K nd K n equton () elds: Susttute o om pt () nd smpl to otn: ( + ) ( ) ΔK + ΔK 3 ( + )( 5 ) ( + ) 6 ( + ) 5 [ ] Notng tht the quntt n ckets s K, susttute to otn: ΔK 6 5 K The ctonl oce etween the suces s esponsle o some o the ntl knetc eneg eng conveted to theml eneg s the two dsks come togethe.
Angul omentum 9 53 A lock o mss m sldng on ctonless tle s ttched to stng tht psses though now hole though the cente o the tle. The lock s sldng wth speed v n ccle o dus. Fnd () the ngul momentum o the lock, () the knetc eneg o the lock, nd (c) the tenson n the stng. (d) A student unde the tle now slowl pulls the stng downwd. How much wok s equed to educe the dus o the ccle om to /? Pctue the Polem () nd () We cn expess the ngul momentum nd knetc eneg o the lock dectl om the dentons. (c) The tenson n the stng povdes the centpetl oce equed o the unom ccul moton nd cn e expessed usng Newton s second lw. (d) Fnll, we cn use the wokknetc eneg theoem to expess the wok equed to educe the dus o the ccle cto o two. () xpess the ntl ngul momentum o the lock: () xpess the ntl knetc eneg o the lock: mv K mv (c) Usng Newton s second lw, elte the tenson n the stng to the centpetl oce equed o the ccul moton: T F c v m (d) Use the wok-knetc eneg theoem to elte the equed wok to the chnge n the knetc eneg o the lock: W ΔK K K ( ) m m 4 3 m m m Susttutng o om Pt () nd smplng gves: ( mv ) W 3 m 3 mv 54 A.-kg pont mss movng on ctonless hozontl suce s ttched to ue nd whose othe end s xed t pont P. The ue nd exets oce whose mgntude s F x, whee x s the length o the ue nd