. The torque at the rear sprocket is what actually accelerates the bicycle, and so R R

Transcription

1 CHAPTE 8: anal Mn Answes Quesns. The dmee desgned 7-nch wheels nceases s eadng by he ccumeence a 7-nch wheel 7 " eey elun he wheel. a 4-nch wheel s used, he dmee wll sll egse 7 " eey elun, bu nly 4 " lnea dsance wll hae been aeled. Thus he dmee wll ead a dsance ha s uhe han yu acually aeled, by a ac The dmee wll ead.5% hgh.. a dsk aes a cnsan angula elcy, a pn n he m has adal accelean nly n angenal accelean. he dsk s angula elcy nceases unmly, he pn wll hae bh adal and angenal accelean. he dsk aes a cnsan angula elcy, nehe cmpnen lnea accelean s changng bh adal and angenal accelean ae cnsan. he dsk aes wh a unmly nceasng angula elcy, hen he adal accelean s changng, bu he angenal accelean s a cnsan nn-ze alue.. A nn-gd bdy cann be descbed by a sngle alue angula elcy. Snce he bdy s nngd, he angula psn ne pa he bdy changes wh espec he pas he bdy. Cnsde he sla sysem as an example a nn-gd bdy sysem. Each plane bs n bascally he same decn aund he Sun, bu each plane has s wn angula elcy whch s deen han ha he he planes. 4. Snce he que nles he pduc ce mes lee am, a small ce can exe a geae que han a lage ce he small ce has a lage enugh lee am. 5. he lee am s ze, hen he ce des n exe any que and s cann pduce an angula accelean. Thee wll be n change n he angula sae mn. Hwee, he ce wll add he ne ce n he bdy and s wll change he lnea accelean he bdy. The bdy s lnea sae mn wll change. 6. When yu d a s-up, que m yu abdmen muscles mus ae he uppe hal he bdy m a layng-dwn psn a sng-up psn. The lage he mmen nea he uppe hal he bdy, he me que s needed, and hus he hade he s-up s d. Wh he hands behnd he head, he mmen nea he uppe hal he bdy s lage han wh he hands useched n n. 7. The ensn ce n he bcycle chan can be assumed be he same a bh he n and ea spckes. The ce s elaed he que a each spcke by F, and s F F. The que a he ea spcke s wha acually acceleaes he bcycle, and s F F. We see ha, achee a gen que a he back spcke, a lage n que (due pedalng) mus be pesen when he ea spcke s small. Thus s hade pedal wh a small ea spcke. Lkewse, achee a gen que a he back spcke, a lage n que (due pedalng) mus be pesen when he n spcke s lage. Thus s hade pedal wh a lage n spcke. 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 9

2 Gancl Physcs: Pncples wh Applcans, 6 h Edn 8. The legs hae a lwe mmen nea when he leg mass s cncenaed nex he bdy. Tha means he legs wll eque less que hae a gen angula accelean,, alenaely, a hghe angula accelean can be deelped. Thus he anmal can un as. 9. The lng beam nceases he anal nea he walke. he walke ges -cene m he ghpe, gay wll exe a que n he walke causng he walke ae wh he ee as a p pn. Wh a lage anal nea, he angula accelean caused by ha gaanal que wll be smalle, and he walke wll heee hae me me cmpensae. The lng sze he beam allws he walke make elaely small shs n he cene mass bng hem back beng ceneed n he ghpe. s much ease he walke me a lng, naw bjec wh he pecsn needed small adjusmens han a sh, heay bjec lke a babell.. Jus because he ne ce n a sysem s ze, he ne que need n be ze. Cnsde a unm bjec wh w equal ces n, and shwn n he s dagam. The ne ce n he bjec s ze ( wuld n sa anslae unde he acn hese ces), bu hee s a ne cuneclckwse que abu he cene he d ( wuld sa ae unde he acn hese ces). Jus because he ne que n a sysem s ze, he ne ce need n be ze. Cnsde an bjec wh w equal ces n, as shwn n he secnd dagam. The ne que n he bjec s ze ( wuld n sa ae unde he acn hese ces), bu hee s a ne dwnwad ce n he d ( wuld sa anslae unde he acn hese ces).. Applyng cnsean enegy a he p and bm he nclne, assumng ha hee s n wk dne by cn, ges E E Mgh M. F a sld ball, M. he p bm ball lls whu slppng (n wk dne by cn) hen Mgh M M gh., and s 7 5 Ths speed s ndependen he angle he nclne, and s bh balls wll hae he same speed a he bm. The ball n he nclne wh he smalle angle wll ake me me each he bm han he ball n he nclne wh he lage angle.. Applyng cnsean enegy a he p and bm he nclne, and assumng ha hee s n wk dne by cn, ges E E Mgh M. F a sld ball, p bm M. he ball lls whu slppng (n wk dne by cn) hen, and s 5 Mgh M M gh 7 5 Ths speed s ndependen he mass and adus he ball, and s bh balls wll hae he same speed a he bm. n ac, hs s ue ANY hegh all, s he w balls wll hae dencal nsananeus speeds all alng he descen, and s bh balls wll ake he same me each he 7 bm. The al knec enegy s KE KE KE M M M, and s ans 5 he ball wh he lage mass has he geae al knec enegy. Anhe way cnsde hs s ha he nal penal enegy Mgh s all cneed knec enegy. The lage mass has me penal enegy begn wh (due he lage mass), and s has me knec enegy a he bm. 5 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 9

3 Chape 8 anal Mn. Applyng cnsean enegy a he p and bm he nclne, assumng ha hee s n wk dne by cn, ges E E Mgh M. he bjecs ll whu slppng, hen p bm, and s M, and a cylnde, 5 Mgh Mgh M M. Thus 7 sphee M. F a sld ball, gh and 4gh. Snce cyl, he sphee has he geae speed a he bm. Tha s ue any amun hegh sphee cyl change, and s he sphee s always mng ase han he cylnde ae hey sa me. Thus he sphee wll each he bm s. Snce bh bjecs saed wh he same penal enegy, bh hae he same al knec enegy a he bm. Bu snce bh bjecs hae he same mass and he cylnde s mng slwe, he cylnde has he smalle anslanal KE and hus he geae anal KE. 4. Mmenum and angula mmenum ae cnseed clsed sysems sysems n whch hee ae n exenal ces ques appled he sysem. Pbably n macscpc sysems n Eah ae uly clsed, and s exenal ces and ques (lke hse appled by a cn, example) aec he sysems e me. 5. a lage numbe peple wen he equa, he anal nea he Eah wuld ncease, snce he peple wuld be uhe m he axs an. Angula mmenum wuld be cnseed n such an neacn, and s snce he anal nea nceased, he angula elcy wuld decease he Eah wuld slw dwn a small amun. The lengh a day wuld heee ncease. 6. n de d a smesaul, he de needs sme nal angula mmenum when she leaes he dng bad, because angula mmenum wll be cnseed dung he ee-all mn he de. She cann exe a que n hesel n slan, and s hee s n angula mmenum nally, hee wll be n an dung he es he de. 7. The mmen nea wll ncease, because ms he mass he dsk wll be uhe m he axs an han was wh he gnal axs psn. 8. Yu angula elcy wll n change. Bee yu le g he masses, yu bdy has a cean angula mmenum, whch s he pduc yu mmen nea and yu angula elcy. N ques ae pu upn yu by he ac dppng he masses, and s yu angula mmenum des n change. yu dn change yu mmen nea by changng he psn yu bdy, hen yu angula elcy wll n change. The masses, when dpped, wll hae a hznal mn ha s angenal he ccle n whch hey wee mng bee hey wee dpped. An bjec aelng hznally a sme dsance m a ecal lne (lke yu axs an) has angula mmenum elae ha ecal lne. The masses keep he angula mmenum ha hey had bee beng dpped. 9. The w sphees wuld hae deen anal neas. The sphee ha s hllw wll hae a lage anal nea han he sld sphee. he w sphees ae allwed ll dwn an nclne whu slppng, he sphee wh he smalle mmen nea (he sld ne) wll each he bm he amp s. See quesn numbe an explanan why hs happens. 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 94

4 Gancl Physcs: Pncples wh Applcans, 6 h Edn. Usng he gh hand ule, pn he nges n he decn he Eah s an, m wes eas. Then he humb pns nh. Thus he Eah s angula elcy pns alng s axs an, wads he Nh Sa.. See he dagam. T he le s wes, he decn he angula elcy. The decn he lnea elcy a pn n he p he wheel wuld be n he pape, whch s nh. he angula accelean s eas, whch s ppse he angula elcy, he wheel s slwng dwn s angula speed s deceasng. The angenal lnea accelean he pn n p wll be n he ppse decn s lnea elcy wll pn suh. ù. The angula mmenum he unable pesn sysem wll be cnseed, snce n exenal ques ae beng appled as he pesn walks he cene. As he pesn walks he cene, he eall mmen nea he sysem ges smalle, snce he pesn s clse he axs an. Snce he angula mmenum s cnsan, he angula elcy mus ncease. S he unable wll begn ae ase as yu walk he cene. Ths s smla he spnnng ce skae wh pulls he ams n ncease he angula speed.. The shsp, whle n md-a, cann exe a que n hmsel, and s hs angula mmenum wll be cnseed whle n he a. he uppe hal hs bdy aes n a cean decn dung he hwng mn, hen cnsee angula mmenum, he lwe hal hs bdy wll ae n he ppse decn. 4. Cnsde a helcpe n he a wh he spnnng. T change he s angula speed, a que mus be appled he. Tha que has cme m he helcpe, and s by Newn s d law, and equal and ppse que wll be appled by he he helcpe. Any change n speed wuld heee cause he bdy he helcpe spn n a decn ppse he change n he s angula elcy. Sme lage helcpes hae w sysems, spnnng n ppse decns. Tha makes any change n he speed he pa eque a ne que, and s he helcpe bdy wuld n end spn. Smalle helcpes hae a al whch aes n a ecal plane, causng a ce n he al he helcpe n he ppse decn he endency he al spn. Sluns Pblems. (a) (b) (c) (d) (e) ad 6 6 ad.5 ad 57 ad ad.99 ad 9 ad 6 ad.57 ad 6 ad 6 ad 6.8 ad 4 ad 6 7 ad 7. ad. The angle n adans s he damee he bjec dded by he dsance he bjec km Sun Sun 6 Eah Sun 49.6 km 9. ad 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 95

5 Chape 8 anal Mn.74 km Mn Mn 9.6 ad 84 km Eah Mn Snce hese angles ae paccally he same, sla eclpses ccu.. We nd he damee he sp m damee 5 8 damee.4 ad.8 m 5. m Eah Mn Eah Mn e ad mn 4. The nal angula elcy s 65 68ad s. Use he mn e 6 sec denn angula accelean. 68ad s 7 ad s. ad s. s 5. The ball lls d lnea dsance wh each elun. d m.5 m 5. e.5 m d 7.4 m e n each elun, he wheel mes wad a dsance equal s ccumeence, d. x 8 m x N d N.7 e e d.68 m 7. (a) 5 e ad mn mn e 6 s (b) 6.8 ad sec.75 m 46 m s 6.8 ad sec.6 ad sec 4 a 6.8 ad sec.75 m. m s 8. The angula speed he mey-g-und s ad 4. s.57 ad s (a).57 ad sec. m.9 m s (b) The accelean s adal. Thee s n angenal accelean. a.57 ad sec. m. m s wads he cene 9. (a) The Eah makes ne b aund he Sun n ne yea. ad y b ad s y.6 s (b) The Eah makes ne elun abu s axs n ne day. ad d ad s an d 86,4 s 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 96

6 Gancl Physcs: Pncples wh Applcans, 6 h Edn. Each lcan wll hae he same angula elcy ( elun pe day), bu he adus he ccula pah aes wh he lcan. Fm he dagam, we see cs, whee s he adus he Eah, and s he adus a laude. ad d 6 (a) 6.8 m 4.64 m s T d 864 s (b) (c) ad d T d 864 s ad d T d 864 s m cs m s 6 O 6.8 m cs m s. The cenpeal accelean s gen by a a. Sle he angula elcy., 9.8 m s ad e 6 s pm.7 m s ad mn. Cne he pm alues angula elces. e ad mn.6 ad s mn e 6 sec e ad mn 8 9. ad s mn e 6 sec (a) The angula accelean s und m Eq. 8-9a. 9.ad s.6 ad s.9 ad s.9 ad s 4. s (b) T nd he cmpnens he accelean, he nsananeus angula elcy s needed..6 ad s.9ad s. s.5 ad s The nsananeus adal accelean s gen by a. a.5 ad s.5 m.6 m s The angenal accelean s gen by a. an a.9ad s.5 m.4 m s an. The angenal speed he unable mus be equal he angenal speed he lle, hee s n slppage. 4. (a) The angula an can be und m Eq. 8-a. The nal angula equency s and he nal angula equency s pm. e ad. mn mn e 6 s ad s.5 ad s 7 s (b) Ae 5. mn ( s), he angula speed s as llws ad s s 4.6 ad s 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 97

7 Chape 8 anal Mn Fnd he cmpnens he accelean a pn n he ue skn m he angula speed and he adus. a an.454 ad s 4.5 m 6. m s 4 4 a ad 4.6 ad s 4.5 m 8. m s 5. The angula dsplacemen can be und m he llwng unm angula accelean elanshp. 4 5 e mn s mn 6 s.8 e 6. (a) F cnsan angula accelean: e mn 45 e mn e mn ad mn.5 s.5s e 6 s.4 ad s (b) F he angula dsplacemen, gen cnsan angula accelean: mn 45 e mn e mn.5 s. e 6 s 7. (a) The angula accelean can be und m e. mn 4. e mn wh. (b) The nal angula speed can be und m, wh. e. mn 8. Use Eq. 8-9d cmbned wh Eq. 8-a. 4. pm 4 pm 6 pm pm e mn 6.5 s.5 e mn 6 sec Each elun cespnds a ccumeence ael dsance.. m.5 e 4 m e 9. (a) The angula accelean can be und m. 85 e mn e ad mn ad e mn e 6 s s (b) The me cme a sp can be und m. 5 e 6 s 85 e mn mn s 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 98

8 Gancl Physcs: Pncples wh Applcans, 6 h Edn. Snce hee s n slppng beween he wheels, he angenal cmpnen he lnea accelean each wheel mus be he same. a a (a) an an small small lage lage small agel lage small small lage. cm 7. ad s.576 ad s.58 ad s 5. cm (b) Assume he pey wheel sas m es. Cne he speed an angula speed, and hen use Eq. 8-9a. e ad mn ad s mn e 6 s 6.8ad s.576 ad s. (a) The angula accelean can be und m und m. 4. ad s 4.ad s s 45 km h 95 km h.4 m 65 e, wh he angula elces beng m s.6 km h ad (b) The me sp can be und m, wh a nal angula elcy. 45 km h m s.6 km h.4 m 4.ad s. (a) The maxmum que wll be exeed by he ce he wegh, pushng angenal he ccle n whch he pedal mes. F mg 7.6 s.7 m 55 kg 9.8 m s 9 m N (b) She culd exe me que by pushng dwn hade wh he legs, asng he cene mass. She culd als pull upwads n he handle bas as she pedals, whch wll ncease he dwnwad ce he legs. e. The que s calculaed by F sn (a) F he s case, 9. (b) F he secnd case,. See he dagam, m he p ew. F sn.74 m 55 N sn 9 4 m N 45. F sn.74 m 55 N sn 45 9 m N F 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 99

9 Chape 8 anal Mn 4. Each ce s ened s ha s pependcula s lee am. Call cuneclckwse ques pse. The que due he hee appled ces s gen by 8 N.4 m 8 N.4 m 5 N. m.8 m N. appled ces Snce hs que s clckwse, we assume he wheel s ang clckwse, and s he cnal que s cuneclckwse. Thus he ne que s ne 8 N.4 m 8 N.4 m 5 N. m.4 m N.4 m N.4 m N, clckwse 5. Thee s a cuneclckwse que due he ce gay n he le blck, and a clckwse que due he ce gay n he gh blck. Call clckwse he pse decn. mgl mgl mg L L, clckwse 6. (a) The ce equed pduce he que can be und m F sn pependculaly he wench, s 9. Thus. The ce s appled 88 m N. N F.8 m (b) The ne que sll mus be 88 m N. Ths s pduced by 6 ces, ne a each he 6 pns. Thse ces ae als pependcula he lee am, and s 88 m N 6 F F. N ne pn pn pn m 7. F a sphee ang abu an axs hugh s cene, he mmen nea s gen by M.8 kg.648 m.8 kg m Snce all he sgncan mass s lcaed a he same dsance m he axs an, he mmen nea s gen by.667 M.5 kg m.9 kg m. The hub mass can be gned because s dsance m he axs an s ey small, and s has a ey small anal nea. 9. (a) The small ball can be eaed as a pacle calculang s mmen nea. M.65 kg. m.94 kg m (b) T keep a cnsan angula elcy, he ne que mus be ze, and s he que needed s he same magnude as he que caused by cn. F. N. m.4 m N appled appled. (a) The que exeed by he cnal ce s F sn. The cnal ce s assumed be angenal he clay, and s he angle s 9. F sn 6. m.5 N sn 9 9. m N al decn an F 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he

10 Gancl Physcs: Pncples wh Applcans, 6 h Edn (b) The me sp s und m, wh a nal angula elcy. The angula accelean can be und m al. The ne que (and angula accelean) s negae snce he bjec s slwng..6 e s ad e 9. m N. kg m. (a) T calculae he mmen nea abu he y-axs (ecal), use x s M m.5 m M.5 m m. m M. m m M.5 m. m 4.9 kg.5 m. m 6. kg m (b) T calculae he mmen nea abu he x-axs (hznal), use.5 m.6 kg m y M m M. (c) Because he lage alue, s hade acceleae he aay abu he ecal axs. The xygen mlecule has a dumbbell gemey, ang abu he dashed lne, as shwn n he dagam. he al mass s M, hen each am has a mass M/. he dsance beween hem s d, hen he dsance m he axs an each am s d/. Tea each am as a pacle calculang he mmen nea. 4 M d M d M d Md d M 4.9 kg m 5. kg. m. The ng ce he ckes wll ceae a ne que, bu n ne ce. Snce each cke es angenally, each ce has a lee am equal he adus he saelle, and each ce s pependcula he lee am. Thus 4F ne. Ths que wll cause an angula accelean accdng, whee M a cylnde. The angula accelean can be und m he knemacs by sle he ce. 4F M. Equang he w expessns he que and subsung enables us F M 6 kg 4. m e mn ad e mn 6 s mn 6 s mn. N. N 4. (a) The mmen nea a cylnde s und n Fgue 8-. M.58 kg 8.5 m.95 kg m. kg m (b) The wheel slws dwn n s wn m 5 pm es n 55.s. Ths s used calculae he cnal que. 5 e mn ad e mn 6 s.95 kg m 55. s m N 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he

11 Chape 8 anal Mn The ne que causng he angula accelean s he appled que plus he (negae) cnal que. appled appled.95 kg m m N 7. m N 5 e mn ad e mn 6 s 5. s 5. The que can be calculaed m. The anal nea a d abu s end s gen by ML. ML. e s ad e. kg.95 m 6 m N. s 6. The que needed s he mmen nea he sysem (mey-g-und and chlden) mes he angula accelean he sysem. Le he subscp mg epesen he mey-g-und. M m mg chlden mg chld 5 e mn ad e mn 6 s 76 kg 5 kg.5 m. s 4.5 m N 4. m N The ce needed s calculaed m he que and he adus. Assume ha he ce s all deced pependculaly he adus. F sn F 4.5 m N.5 m.7 N 7. The que n he wll cause an angula accelean gen by. The que and angula accelean wll hae he ppse sgn he nal angula elcy because he s beng bugh es. The anal nea s ha a sld cylnde. Subsue he expessns angula accelean and anal nea n he equan, and sle he angula dsplacemen. M 4 e ad mn 4.8 kg.7 m, mn e 6 s e 5865 ad 4. N m ad 99 e The me can be und m. M 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he

12 Gancl Physcs: Pncples wh Applcans, 6 h Edn 99 e 6 s, e mn mn.9 s 8. (a) The que ges angula accelean he ball nly, snce he am s cnsdeed massless. The angula accelean he ball s und m he gen angenal accelean. aan M M Ma.6 kg. m 7. m s an 7.8 m N 7.8 m N (b) The ceps muscle mus pduce he que equed, bu wh a lee am nly.5 cm, pependcula he ceps muscle ce. F F 7.8 m N.5 m. N 9. (a) The angula accelean can be und m. m s.m 9.7 ad s 9 ad s.5 s (b) The ce equed can be und m he que, snce F sn. n hs suan he ce s pependcula he lee am, and s 9. The que s als gen by, whee s he mmen nea he am-ball cmbnan. Equae he w expessns he que, and sle he ce. F sn F m d m L ball ball am am sn sn 9. kg. m.7 kg. m 9.7 ad s 7.9 N.5 m 4. (a) The mmen nea a hn d, ang abu s end, s gen n Fgue 8-(g). Thee ae hee blades add. al ML ML 6 kg.75 m 5 kg m. kg m (b) The que equed s he anal nea mes he angula accelean, assumed cnsan. 5. e/sec ad e 5 kg m 8.8 m N al al 8. s 4. We assume ha m m, and s m wll acceleae dwn, m wll acceleae up, and he pulley wll acceleae clckwse. Call he decn accelean he pse decn each bjec. The masses wll hae he same accelean snce hey ae cnneced by a cd. The m he pulley wll hae ha same accelean snce he cd s makng ae, and s a. Fm he ee-bdy pulley dagams each bjec, we hae he llwng. F F m g m a F m g m a y T T F m g F m a F m g m a y T T + + y F T F T m m mg mg F T F T + y 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he

13 Chape 8 anal Mn a F F T T Subsue he expessns he ensns n he que equan, and sle he accelean. a F F m g m a m g m a a g T T m m he mmen nea s gned, hen m he que equan we see ha FT accelean wll be a m m m m ncluded wll be smalle han he mmen nea s gned. a m T m F, and he g. We see ha he accelean wh he mmen nea 4. A p ew dagam he hamme s shwn, jus a he nsan elease, alng wh he accelean ecs. (a) The angula accelean s und m Eq. 8-9c. 8. m s. m.8 ad s 8 ad (b) The angenal accelean s und m he angula accelean and he adus. a.8 ad s. m. m s an (c) The cenpeal accelean s und m he speed and he adus. a 8. m s. m 65m s ad (d) The ne ce s he mass mes he ne accelean. s n he same decn as he ne accelean, als. F ma m a a 7. kg. m s 65 m s 4.77 N ne ne an ad (e) Fnd he angle m he w accelean ecs. aan. m s an an.4 a 65m s ad a ne a ad a an 4. The enegy equed bng he up speed m es s equal he nal anal KE he. KE e ad mn.75 kg m 85.4 J mn e 6 s Wk can be expessed n anal quanes as W, and s pwe can be expessed n W anal quanes as P. e ad mn 5 5 P 8 m N 8.4 W. W mn e 6 s hp 5.4 W.5 hp 746 W 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 4

14 Gancl Physcs: Pncples wh Applcans, 6 h Edn 45. The al knec enegy s he sum he anslanal and anal knec eneges. Snce he ball s llng whu slppng, he angula elcy s gen by. The anal nea a sphee abu an axs hugh s cene s m. 5 KE KE KE m m m m 7 al ans kg.m s 56 J 46. (a) F he daly an abu s axs, ea he Eah as a unm sphee, wh an angula equency ne elun pe day. KE M daly daly 5 Eah daly ad day 9 6. kg 6.4 m.6 J day 86,4 s (b) F he yealy elun abu he Sun, ea he Eah as a pacle, wh an angula equency ne elun pe yea. KE M yealy yealy Sun- yealy Eah 4 ad day 6. kg.5 m.7 J 65 day 86,4 s Thus he al KE s KE KE. The KE due he 9.6 J.6 J.6 J daly yealy daly mn s abu, smalle han ha due he yealy mn. 47. The wk equed s he change n anal knec enegy. The nal angula elcy s. ad 4 4 W KE M 64 kg 7.5 m.4 J 8. s 48. Apply cnsean enegy he sphee, as dne n Example 8-. (a) The wk Example 8- s exacly applcable hee. The symbl d s epesen he dsance he sphee lls alng he plane. The sphee s llng whu slppng, s. CM (b) KE KE gh gd sn 9.8 m s. m sn CM m s CM M 8.67 m s. m 4.8 ad s M ans CM CM CM CM M 5.5 (c) Only he angula speed depends n he adus. Nne he esuls depend n he mass. 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 5

15 Chape 8 anal Mn 49. The nly ce dng wk n hs sysem s gay, s mechancal enegy wll be cnseed. The nal sae he sysem s he cnguan wh m n he gund and all bjecs a es. The nal sae he sysem has m jus eachng he gund, and all bjecs n mn. Call he ze leel gaanal penal enegy be he gund leel. Bh masses wll hae he same speed snce hey ae cnneced by he pe. Assumng ha he pe des n slp n he pulley, he angula speed he pulley s elaed he speed he masses by. All bjecs hae an nal speed. E E m m m gy m gy m m m gy m gy m gh m m M m gh m M m h m m gh m m M 6.5 kg 8. kg 9.8 m s. m 6.5 kg 8. kg 7.5 kg. m s 5. Snce he lwe end he ple des n slp n he gund, he cn des n wk, and s mechancal enegy s cnseed. The nal enegy s he penal enegy, eang all he mass as wee a he CM. The nal enegy s anal KE, an abu he pn cnac wh he gund. The lnea elcy he allng p he d s s angula elcy dded by he lengh. PE KE mgh mg L ml L end end gl 9.8 m s. m 8. m s 5. The angula mmenum s gen by Eq L M. kg. m.4 ad s.64 kg m s 5. (a) The angula mmenum s gen by Eq L M.8 kg.8 m 5 e ad mn mn e 6 s 7.kg m s (b) The que equed s he change n angula mmenum pe un me. The nal angula mmenum s ze. L L 7.kg m s. m N 6. s The negae sgn ndcaes ha he que s used ppse he nal angula mmenum. 5. (a) Cnsde he pesn and plam a sysem angula mmenum analyss. Snce he ce and que ase and/ lwe he ams s nenal he sysem, he asng lweng he ams wll cause n change n he al angula mmenum he sysem. Hwee, he anal nea nceases when he ams ae ased. Snce angula mmenum s cnseed, an ncease n anal nea mus be accmpaned by a decease n angula elcy. 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 6

16 Gancl Physcs: Pncples wh Applcans, 6 h Edn (b). e s L L e s The anal nea has nceased by a ac Thee s n ne que n he de because he nly exenal ce (gay) passes hugh he cene mass he de. Thus he angula mmenum he de s cnseed. Subscp ees he uck psn, and subscp ees he sagh psn. e L L.8 e s.5 sec The skae s angula mmenum s cnsan, snce n exenal ques ae appled he..5 e s L L 4.6 kg m.77 kg m. e s She accmplshes hs by sang wh he ams exended (nal angula elcy) and hen pullng he ams n he cene he bdy (nal angula elcy). 56. Because hee s n exenal que appled he wheel-clay sysem, he angula mmenum wll be cnseed. We assume ha he clay s hwn wh n angula mmenum s ha s nal angula mmenum s. Ths suan s a ally nelasc cllsn, n whch he nal angula elcy s he same bh he clay and he wheel. Subscp epesens bee he clay s hwn, and subscp epesens ae he clay s hwn. L L wheel wheel wheel M M wheel clay wheel wheel clay clay M 5. kg. m.5e s.6 e s.4 e s 5. kg. m. kg 8. m e ad 57. (a) L M 55 kg.5 m.5 4 kg m s s e (b) he anal nea des n change, hen he change n angula mmenum s scly due a change n angula elcy. L 4 kg m s.7m N 5. s The negae sgn ndcaes ha he que s n he ppse decn as he nal angula mmenum. 58. (a) F he daly an abu s axs, ea he Eah as a unm sphee, wh an angula equency ne elun pe day. L M daly daly 5 Eah daly ad day 6. kg 6.4 m 7. kg m s day 86,4 s 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 7

17 Chape 8 anal Mn (b) F he yealy elun abu he Sun, ea he Eah as a pacle, wh an angula equency ne elun pe yea. L M daly daly Sun- daly Eah 4 ad day 4 6. kg.5 m.7 kg m s 65 day 86,4 s 59. Snce hee ae n exenal ques n he sysem, he angula mmenum he -dsk sysem s cnseed. The w dsks hae he same nal angula elcy. L L 6. The angula mmenum he dsk d cmbnan wll be cnseed because hee ae n exenal ques n he cmbnan. Ths suan s a ally nelasc cllsn, n whch he nal angula elcy s he same bh he dsk and he d. Subscp epesens bee he cllsn, and subscp epesens ae he cllsn. The d has n nal angula mmenum. L L M dsk dsk d M M.4 e s.4 e s 5 6. Snce he pesn s walkng adally, n ques wll be exeed n he pesn-plam sysem, and s angula mmenum wll be cnseed. The pesn wll be eaed as a pn mass. Snce he pesn s nally a he cene, hey hae n nal anal nea. (a) L L plam plam pesn plam plam m 9 kg m 9 kg m 75 kg. m. ad s.54 ad s. ad s (b) KE plam 9 kg m. ad s.8 J KE m plam pesn plam pesn pesn 9 kg m 75 kg. m.54 ad s 6 J. J 6. The angula mmenum he mey-g-und and peple cmbnan wll be cnseed because hee ae n exenal ques n he cmbnan. Ths suan s a ally nelasc cllsn, n whch he nal angula elcy s he same bh he mey-g-und and he peple. Subscp epesens bee he cllsn, and subscp epesens ae he cllsn. The peple hae n nal angula mmenum. L L m-g- m-g- 4M m-g- peple m-g- pesn 76 kg m.8 ad s.48 ad s 76 kg m 4 65 kg. m 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 8

18 Gancl Physcs: Pncples wh Applcans, 6 h Edn he peple jump he mey-g-und adally, hen hey exe n que n he mey-g-und, and hus cann change he angula mmenum he mey-g-und. The mey-g-und wuld cnnue ae a.8 ad s. 6. Snce he ls mass caes away n angula mmenum, he angula mmenum he emanng mass wll be he same as he nal angula mmenum. M M 5 4 L L. M.5M. 5 ad d ad s 5 ad s day 864 s The ped wuld be a ac, smalle, whch wuld make abu secnds. The a angula knec eneges he spnnng mass wuld be KE 4.5M KE 5 KE M KE 64. F u cude esmae, we mdel he hucane as a gd cylnde a. Snce he cylnde s gd, each pa has he same angula elcy. The mass he a s he pduc he densy a mes he lume he a cylnde. M V h 5 4. kg m. m 4. m.64 kg (a) (b) edge 4 edge KE M M 4 m s kg km h 4.59 J 5 J L M M edge edge.6 km h kg. m km h.7 kg m s m s.6 km h kg m s 65. Angula mmenum wll be cnseed n he Eah ased sysem, snce all ces and ques ae nenal he sysem. The nal angula elcy he saelle, jus bee cllsn, can be und m ased ased Eah. Assumng he ased becmes mbedded n he Eah a he suace, he Eah and he ased wll hae he same angula elcy ae he cllsn. We mdel he Eah as a unm sphee, and he ased as a pn mass. L L Eah Eah ased ased Eah ased The mmen nea he saelle can be gned elae ha he Eah n he gh sde he abe equan, and s he pecen change n Eah s angula elcy s und as llws. Eah Eah ased ased Eah Eah ased ased Eah Eah Eah 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 9

19 Chape 8 anal Mn ased Eah m m ased Eah Eah ased ased M M Eah 5 Eah Eah Eah 5 Eah Eah Eah % change 5 4. kg. m s ad kg s m 6.7 % 66. When he pesn and he plam ae, hey d s abu he ecal axs. nally hee s n angula mmenum pnng alng he ecal axs, and s any change ha he pesn wheel plam undeges mus esul n n ne angula mmenum alng he ecal axs. (a) he wheel s med s ha s angula mmenum pns upwads, hen he pesn and plam mus ge an equal bu ppse angula mmenum, whch wll pn dwnwads. We he angula mmenum cnsean cndn he ecal decn sle he angula elcy he plam. L L W W W P P P W The negae sgn means ha he plam s ang n he ppse decn he wheel. he wheel s spnnng cuneclckwse when ewed m abe, he plam s spnnng clckwse. (b) he wheel s pnng a a 6 angle he ecal, hen he cmpnen s angula mmenum ha s alng he ecal decn s cs 6. See he dagam. We he angula mmenum W W cnsean cndn he ecal decn sle he angula elcy he plam. L L cs 6 W W W P P P W P P 6 L W = W W Agan, he negae sgn means ha he plam s ang n he ppse decn he wheel. (c) he wheel s med s ha s angula mmenum pns dwnwads, hen he pesn and plam mus ge an equal bu ppse angula mmenum, whch wll pn upwads. We he angula mmenum cnsean cndn he ecal decn sle he angula elcy he plam. L L W W P P P W W P The plam s ang n he same decn as he wheel. he wheel s spnnng cuneclckwse when ewed m abe, he plam s als spnnng cuneclckwse. (d) Snce he al angula mmenum s, he wheel s spped m ang, he plam wll als sp. Thus P. 67. The angula mmenum he pesn unable sysem wll be cnseed. Call he decn he pesn s mn he pse an decn. elae he gund, he pesn s speed wll be T, whee s he pesn s speed elae he unable, and s he speed he m he T unable wh espec he gund. The unable s angula speed s T T, and he pesn s 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he

20 Gancl Physcs: Pncples wh Applcans, 6 h Edn T angula speed elae he gund s P pacle calculan he mmen nea. L L m T T P P T T T m 55 kg.5 m.8 m s T m T 7 kg m 55 kg.5 m T. The pesn s eaed as a pn. ad s 68. Snce he spl lls whu slppng, each pn n he edge he spl mes wh a speed CM elae he cene he spl, whee s he speed he cene he spl CM elae he gund. Snce he spl s mng he gh elae he gund, and he p he spl s mng he gh elae he cene he spl, he p he spl s mng wh a speed elae he gund. Ths s he speed he pe, assumng s unllng CM whu slppng and s a he ue edge he spl. The speed he pe s he same as he speed he pesn, snce he pesn s hldng he pe. S he pesn s walkng wh a speed wce ha he cene he spl. Thus he pesn mes wad a dsance L, n he same me he cene he spl, aelng wh hal he speed, mes wad a dsance L. The pe, say cnneced bh he pesn and he spl, mus heee unwnd by an amun L als. 69. The spn angula mmenum he Mn can be calculaed by bal angula mmenum can be calculaed by sde he Mn always aces he Eah, spn b. L M. The spn spn spn 5 Mn spn L M. Because he same b b b b b 6 spn 5 Mn spn.74 m Mn m b b b b L M L M 7. As dscussed n secn 8-, m he eeence ame he axle he wheel, he pns n he wheel ae all mng wh he same speed, whee s he speed he axle he wheel elae he gund. The p he e has a elcy he gh elae he axle, s has a elcy he gh elae he gund. p el p el cene el gund cene gund a a he gh he gh he gh. m s p el. s 6. m s gund 7. The que s und m. The angula accelean can be und m, wh an nal angula elcy. The anal nea s ha a cylnde. 8 e s ad e M.5.4 kg. m 5 m N 6. s 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he

21 Chape 8 anal Mn 7. (a) Thee ae w ces n he y-y: gay and he sng ensn. we assume ha he p he sng s held n a xed psn, hen he ensn des n wk, and s mechancal enegy s cnseed. The nal gaanal PE s cneed n anal and anslanal KE. Snce he y-y lls whu slppng a he pn cnac he sng, he elcy he CM s smply elaed he angula elcy he y-y:, whee s he CM adus he nne hub. Le m be he mass he nne hub, and M and be he mass and adus each ue dsk. Calculae he anal nea he y-y abu s CM, and hen use cnsean enegy nd he lnea speed he CM. We ake he gaanal PE be a he bm s all. m M m M CM mg F T 5 5. kg 5. m 5. kg.75 m 7.8 kg m m m M al PE KE 5. kg 5. kg.5 kg m gh m m m CM CM al al CM CM al CM CM al CM CM m m al gh CM al (b) Calculae he a KE KE..5 kg 9.8 m s. m.5 kg 7.8 kg m 5 KE KE CM CM CM CM CM al al al KE PE m gh m gh m gh kg m.895 m s 5. m.5 kg 9.8 m s. m 5. m.96 96% m s 7. (a) The lnea speed he chan mus be he same as passes e bh spckes. The lnea speed s elaed he angula speed by, and s. F F he spacng he eeh n he spckes s a dsance d, hen he numbe eeh n a spcke mes he spacng dsance mus ge he ccumeence he spcke. Nd and s (b) 5 4. F (c) F Nd. Thus N d N d N F F F F N 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he

22 Gancl Physcs: Pncples wh Applcans, 6 h Edn 74. Snce he ls mass caes away n angula mmenum, he angula mmenum he emanng mass wll be he same as he nal angula mmenum. L L m M M 5 Sun.6 4 M M. m Sun e e day day 9 4. e day.5 e s 75. (a) The nal enegy he lywheel s used w pupses ge he ca anslanal knec enegy mes, and eplace he enegy ls due cn, m a essance and m bakng. The saemen he pblem leads us gne any gaanal penal enegy changes. W KE KE F x cs8 M KE nal nal ca ca lywheel KE F x M lywheel ca ca 45 N.5 m 4 kg 95 km h J.7 J m s.6 km h (b) KE lywheel KE KE M lywheel lywheel 8.67 J 4 kg.75 m. ad s Wk (c) T nd he me, use he elanshp ha Pwe, whee he wk dne by he m wll be equal he knec enegy he lywheel. W P W 8.67 J P 5 hp 746 W hp.494 s 5 mn 76. The mass a hydgen am s. amc mass uns. The amc mass un s 7.66 kg. Snce he axs passes hugh he xygen am, wll hae n hydgen L anal nea. (a) he axs s pependcula he plane he mlecule, hen each hydgen am s a dsance L m he axs an. m L kg 9.96 m xygen pep H Ly. kg m 45 (b) he axs s n he plane he mlecule, bsecng he H-O-H bnds, each hydgen am s a dsance Ly m. Thus he mmen nea s L sn 9.6 m sn 5 hydgen 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he

23 Chape 8 anal Mn 7 45 y m L..66 kg m.9 kg m plane H 77. (a) Assumng ha hee ae n dsspae ces dng wk, cnsean enegy may be used nd he nal hegh h he hp. Take he bm he nclne be he ze leel gaanal penal enegy. We assume ha he hp s h llng whu sldng, s ha. elae he cndns a he bm he nclne he cndns a he p by cnsean enegy. The hp has bh anslanal and anal knec enegy a he bm, and he anal nea he hp s gen by m. E E m mgh m m mgh bm p h g.m s 9.8 m s. m h. m The dsance alng he plane s gen by d 4.9 m 4. m sn sn5 (b) The me can be und m he cnsan acceleaed lnea mn. Use he elanshp x 4.9 m x.6 s..m s Ths s he me g up he plane. The me cme back dwn he plane s he same, and s he al me s 5. s. 78. (a) The ce gay acng hugh he CM wll cause a clckwse que whch pduces an angula accelean. A he mmen elease, he ce gay s pependcula he lee am m he hnge he CM. gay Mg L g ML L d abu end (b) A he end he d, hee s a angenal accelean equal he angula accelean mes he dsance m he hnge. Thee s n adal accelean snce a he mmen elease, he speed he end he d s. Thus he angenal accelean s he ene lnea accelean. a a L g lnea an 79. The wheel s llng abu he pn cnac wh he sep, and s all ques ae be aken abu ha pn. As sn as he wheel s he l, hee wll be nly w ces ha can exe ques n he wheel he pullng ce and he ce gay. Thee wll n be a nmal ce cnac beween he wheel and he l nce he wheel s he l, and any ce n he wheel m he pn he sep cann exe a que abu ha ey pn. Calculae he ne que n he wheel, wh clckwse ques pse. The mnmum ce ccus when he ne que s. h h mg F h 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 4

24 Gancl Physcs: Pncples wh Applcans, 6 h Edn F h mg h F Mg h Mg h h h h 8. (a) n de n all e, he ne que n he cycls abu an axs hugh he CM and paallel he gund mus be ze. Cnsde he ee-bdy dagam shwn. Sum ques abu he CM, wh cuneclckwse as pse, and se he sum equal ze. F x F x F y an N F y (b) The cycls s n acceleang ecally, s N FN mg. The cycls s acceleang hznally, because he s aelng n a ccle. Thus he cnal ce mus be supplyng he cenpeal ce, s F m. (c) Fm F m 4. m s an an an F mg g g N 6.4 m 9.8 m s F m, he smalles unng adus esuls n he maxmum ce. The maxmum sac cnal ce s F F. Use hs calculae he adus. N 4. m s m F mg mn N mn g m s.6 m 8. Assume ha he angula accelean s unm. Then he que equed whl he ck s he mmen nea he ck (eaed as a pacle) mes he angula accelean..5 kg.5 m e ad mn m.8 m N 5. s mn e 6 s Tha que cmes m he am swngng he slng, and s cmes m he am muscles. mg x F F N y 8. Assume a mass 5 kg, cespndng a wegh abu lb. Fm Table 7-, we nd ha he al am and hand mass s abu.5% he al mass, and s he es he bdy s abu 87.5% he al mass. Mdel he skae as a cylnde mass 44 kg, and mdel each am as a hn d mass kg. Esmae he bdy as 5 cm all wh a adus 5 cm. Esmae he am dmensn as 5 cm lng. Wh he ams held ghly, we appxmae ha he ams ae pa he bdy cylnde. A skech he skae n hs cnguan s hen as shwn n he s dagam. n hs cnguan, he anal nea s M. n cylnde al bdy bdy 5 cm 5 cm 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 5

25 Chape 8 anal Mn Wh he ams exended, he cnguan changes he secnd dagam. n hs cnguan, he anal nea s M M L u bdy ams bdy bdy am am 5 cm The ces and ques nled n changng he cnguan he skae ae nenal he skae, and s he skae s angula mmenum s cnseed dung a cnguan change. Thus L L n u n n u u al bdy u n bdy 5 kg.5 m M M L n u bdy bdy am am 44 kg.5 m kg.5 m (a) The angula mmenum M wll be A M A A A L M 6. kg.6 m 7. ad s 7.8 kg m s. (b) The que equed acceleae M wll be A L 7.8 kg m s.9 m N. s (c) Snce hee ae n ques exenal he w plaes, he angula mmenum he w plaes wll be cnseed. Snce he w plaes sck gehe, hey wll hae a cmmn nal angula elcy. Ths s a ally nelasc cllsn. L L A A B A A A M M M M A B A B A B.9 ad s M M 6. kg 5. kg 5 cm 7. ad s 5 cm 84. Snce cnal lsses can be gned, enegy wll be cnseed he mable. Dene he psn PE be he bm he ack, s ha he bm he ball s nally a hegh h abe he psn PE. Snce, he mable s CM s ey clse he suace he ack. Whle he mable s n he lp, we hen appxmae ha wll be mng n a ccle adus. When he mable s a he p he lp, we appxmae ha s CM s a dsance abe he psn PE. F he mable jus be n he ege leang he ack means he nmal ce beween he mable and he ack s ze, and s he cenpeal ce a he p mus be equal he gaanal ce n he mable. m p lp mg g p lp We assume ha he mable s llng whu slppng, and s, and ha he mable s eleased m es. Use enegy cnsean elae he elease pn he pn a he p he lp. Ne ha he mable has bh anslanal and anal knec enegy. 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 6

26 Gancl Physcs: Pncples wh Applcans, 6 h Edn E E KE PE KE PE elease p elease elease p p lp lp lp p lp p p p 5 lp lp lp mgh m mg m m mg mgh m mg mg mg.7 mg h p lp 85. Snce cnal lsses can be gned, enegy wll be cnseed he mable. Dene he psn PE be bm he ack, s ha he bm he ball s nally a hegh h abe he psn PE. Snce we ae n assume ha, hen whle he mable s n he lp pn he ack, s mng n a ccle adus, and when a he p he lp, he bm he mable s a hegh abe he psn PE (see he dagam). F he mable jus be n he ege leang he ack means he nmal ce beween he mable and he ack s ze, and s he cenpeal ce a he p mus be equal he gaanal ce n he mable. m p lp mg g p lp y We assume ha he mable s llng whu slppng and s, and ha he mable s eleased m es. Use enegy cnsean elae he elease pn he pn a he p he lp. Ne ha he mable has bh anslanal and anal knec enegy. E E KE PE KE PE elease p elease elease p p lp lp lp p lp p p p 5 lp lp lp mgh m mg m m mg mgh m mg mg mg.7mg 7 7 p lp h (a) The angula accelean can be und m gen by. 6. km h 9. km h.45 m 85 e ad e.65 ad s.6ad s, wh he angula elces beng m s.6 km h (b) The me sp can be und m, wh a nal angula elcy. m s 6. km h.6 km h.45 m.65ad s s 5 Peasn Educan, nc., Uppe Saddle e, NJ. All ghs eseed. Ths maeal s peced unde all cpygh laws as hey cuenly exs. N pn hs maeal may be epduced, n any m by any means, whu pemssn n wng m he 7