Chpter 8 Potentl energy nd conservton o energy I. Potentl energy Energy o congurton II. Wor nd potentl energy III. Conservtve / Non-conservtve orces IV. Determnng potentl energy vlues: - Grvttonl potentl energy - Elstc potentl energy I. V. Conservton o hncl energy VI. Eternl wor nd erml energy VII. Eternl orces nd nternl energy chnges VIII. Power I. Potentl energy Energy ssocted w e rrngement o system o objects t eert orces on one noer. nts: J Emples: - Grvttonl potentl energy: ssocted w e stte o seprton between objects whch cn ttrct one noer v e grvttonl orce. - Elstc potentl energy: ssocted w e stte o compresson/etenson o n elstc object. II. Wor nd potentl energy I tomto rses grvttonl orce trnsers energy rom tomto s netc energy to e grvttonl potentl energy o e tomto-er system. I tomto lls down grvttonl orce trnsers energy rom e grvttonl potentl energy to e tomto s netc energy.
W lso vld or elstc potentl energy Sprng compresson s Sprng orce does W on bloc energy trnser rom netc energy o e bloc to potentl elstc energy o e sprng. Sprng etenson s Sprng orce does +W on bloc energy trnser rom potentl energy o e sprng to netc energy o e bloc. Generl: - System o two or more objects. - orce cts between prtcle n e system nd e rest o e system. - When system congurton chnges orce does wor on e object W trnserrng energy between KE o e object nd some oer orm o energy o e system. - When e congurton chnge s reversed orce reverses e energy trnser, dong W. III. Conservtve / Nonconservtve orces - I W W lwys conservtve orce. Emples: Grvttonl orce nd sprng orce ssocted potentl energes. - I W W nonconservtve orce. Emples: Drg orce, rctonl orce KE trnserred nto erml energy. Non-reversble process. - Therml energy: Energy ssocted w e rndom movement o toms nd molecules. Ths s not potentl energy.
- Conservtve orce: The net wor t does on prtcle movng round every closed p, rom n ntl pont nd en bc to t pont s zero. - The net wor t does on prtcle movng between two ponts does not depend on e prtcle s p. Conservtve orce W b, W b, Proo: W b, + W b, W b, -W b, W b, - W b, W b, W b, IV. Determnng potentl energy vlues W F d Force F s conservtve Grvttonl potentl energy: y y mg dy mg[ y] y mg y y mg y y Chnge n e grvttonl potentl energy o e prtcle-er system., y y mgy Reerence congurton The grvttonl potentl energy ssocted w prtcle-er system depends only on prtcle s vertcl poston y reltve to e reerence poston y, not on e horzontl poston. Elstc potentl energy: d [ ] Chnge n e elstc potentl energy o e sprng-bloc system. Reerence congurton when e sprng s t ts reled leng nd e bloc s t., Remember! Potentl energy s lwys ssocted w system. V. Conservton o hncl energy Mechncl energy o system: Sum o ts potentl nd netc K energes. 3
E + K ssumptons: W K W - Only conservtve orces cuse energy trnser wn e system. - The system s solted rom ts envronment No eternl orce rom n object outsde e system cuses energy chnges nsde e system. K + K K + K + K + K + - In n solted system where only conservtve orces cuse energy chnges, e netc energy nd potentl energy cn chnge, but er sum, e hncl energy o e system cnnot chnge. - When e hncl energy o system s conserved, we cn relte e sum o netc energy nd potentl energy t one nstnt to t t noer nstnt wout consderng e ntermedte moton nd wout ndng e wor done by e orces nvolved. y E constnt E K + K + K + Potentl energy curves Fndng e orce nlytclly: d W F F D moton d - The orce s e negtve o e slope o e curve versus. - The prtcle s netc energy s: K E 4
Turnng pont: pont t whch e prtcle reverses ts moton K. K lwys K.5mv Emples: E 5J5J+K K < E 5J >5J+K K< mpossble Equlbrum ponts: where e slope o e curve s zero F -F d /d -F /d -F Slope Equlbrum ponts E, E, E,3 Emple: 5 E, 4J4J+K K nd lso F 5 neutrl equlbrum >>, 5 >> 4 E, 3J 3J+K K Turnng ponts 3 K, F prtcle sttonry nstble equlbrum 4 E,3 JJ+K K, F, t cnnot move to > 4 or < 4, snce en K< Stble equlbrum 5
6 Revew: Potentl energy W - - The zero s rbtrry Only potentl energy derences hve physcl menng. - The orce D s gven by: F -d/d - The potentl energy s sclr uncton o e poston. P. The orce between two toms n dtomc molecule cn be represented by e ollowng potentl energy uncton: 6 where nd re constnts. Clculte e orce F [ ] + 7 3 7 6 3 5 6 d d F Mnmum vlue o. [ ] 7 3 mn F d d s ppro. e energy necessry to dssocte e two toms.
VI. Wor done on system by n eternl orce Wor s energy trnser to or rom system by mens o n eternl orce ctng on t system. When more n one orce cts on system er net wor s e energy trnserred to or rom e system. No Frcton: Remember! Frcton: W K+ Et. orce K+ only when: - System solted. - No et. orces ct on system. - ll nternl orces re conservtve. F v v m + d.5 v v / d m F v v Fd d m v d W Fd K + d v Fd mv mv + d Generl: W Fd + d Emple: loc sldng up rmp. Therml energy: d Frcton due to cold weldng between two surces. s e bloc sldes over e loor, e sldng cuses terng nd reormng o e welds between e bloc nd e loor, whch mes e bloc-loor wrmer. Wor done on system by n eternl orce, rcton nvolved W Fd + 7
VI. Conservton o energy Totl energy o system E hncl + E erml + E nternl - The totl energy o system cn only chnge by mounts o energy trnserred rom or to e system. W + + Epermentl lw E nt -The totl energy o n solted system cnnot chnge. There cnnot be energy trnsers to or rom t. Isolted system: + + nt In n solted system we cn relte e totl energy t one nstnt to e totl energy t noer nstnt wout consderng e energes t ntermedte sttes. Emple: Trolley pole jumper Run Internl energy muscles gets trnserred nto netc energy. Jump/scent Knetc energy trnserred to potentl elstc energy trolley pole deormton nd to grvttonl potentl energy 3 Descent Grvttonl potentl energy gets trnserred nto netc energy. 8
VII. Eternl orces nd nternl energy chnges Emple: ster pushes hersel wy rom rlng. There s orce F on her rom e rlng t ncreses her netc energy. One prt o n object ster s rm does not move le e rest o body. Internl energy trnser rom one prt o e system to noer v e eternl orce F. ochemcl energy rom muscles trnserred to netc energy o e body. W F, et K Fcosϕ d Non solted system K + W F, et Fd cosϕ Fd cosϕ Chnge n system s hncl energy by n eternl orce Proo: v v + d.5m Mv Mv Md K F cosϕ d VIII. Power verge power: P vg t Instntneous power: de P dt 9
6. In e gure below, bloc sldes long p t s wout rcton untl e bloc reches e secton o leng L.75m, whch begns t heght hm. In t secton, e coecent o netc rcton s.4. The bloc psses rough pont w speed o 8m/s. Does t rech pont where e secton o rcton ends? I so, wht s e speed ere nd not, wht gretest heght bove pont does t rech? N mg cos3 8.5m mg N µ N.48.5m 3.4m C Only conservtve orces K mv + K C + C mvc + mghc vc 5m / s The netc energy n C turns nto erml nd potentl energy loc stops. K.5mv.4m c K mgy + d.4m mg d sn 3 + 3.4md d.49 meters c C c d > L.75m loc reches Isolted system.4m.5mv + 3.67m +.5m v 3.5m / s + + K + K + + L.4m.5mv + mg y y + µ mglcos3.5mv + mglsn 3 + µ mglcos3 c C C 9. mssless rgd rod o leng L hs bll o mss m ttched to one end. The oer end s pvoted n such wy t e bll wll move n vertcl crcle. Frst, ssume t ere s no rcton t e pvot. The system s lunched downwrd rom e horzontl poston w ntl speed v. The bll just brely reches pont D nd en stops. Derve n epresson or v n terms o L, m nd g. b Wht s e tenson n e rod when e bll psses rough? c lttle grl s plced on e pvot to ncrese e rcton ere. Then e bll just brely reches C when lunched rom w e sme speed s beore. Wht s e decrese n hncl energy durng s moton? d Wht s e decrese n hncl energy by e tme e bll nlly comes to rest t ter severl osclltons? K ; D K + K + mgl mv v c v W d c + gl b F m T mg v m T mg T m v L L + K + K cent mv mgl + mv gl + gl v v c + g gl T 5mg y v D L T F c mg C The derence n heghts or n grvttonl potentl energes between e postons C reched by e bll when ere s rcton nd D durng e rctonless movement Is gong to be e loss o hncl energy whch goes nto erml energy. c mgl d The derence n heght between nd D s L. The totl loss o hncl energy whch ll goes nto erml energy s: mgl
. 3g slo hngs 3m bove e ground. Wht s e grvttonl potentl energy o e slo-er system we te e reerence pont y to be t e ground? I e slo drops to e ground nd r drg on t s ssumed to be neglgble, wht re b e netc energy nd c e speed o e slo just beore t reches e ground? K + K + b K 94.J ground ; K mgh 3.g9.8m / s 3m 94.J c K mv v K m 7.67m / s 3. metl tool s shrpen by beng held gnst e rm o wheel on grndng mchne by orce o 8N. The rctonl orces between e rm nd e tool grnd smll peces o e tool. The wheel hs rdus o cm nd rottes t.5 rev/s. The coecent o netc rcton between e wheel nd e tool s.3. t wht rte s energy beng trnserred rom e motor drvng e wheel nd e tool to e netc energy o e mterl rown rom e tool? v F8N rev π.m v.5 3.4m / s P v 57.6N 3.4m / s 8W s rev Pmotor 8W µ N µ F.38 N 57.6N Power dsspted by rcton Power sup pled motor 8. bloc w netc energy o 3J s bout to collde w sprng t ts reled leng. s e bloc compresses e sprng, rctonl orce between e bloc nd loor cts on e bloc. The gure below gves e netc energy o e bloc K nd e potentl energy o e sprng s uncton o e poston o e bloc, s e sprng s compressed. Wht s e ncrese n erml energy o e bloc nd e loor when e bloc reches poston. m nd b e sprng reches ts mmum compresson? Isolted system.m E, E b K 3J Grph : E K, K 3J 3J 7J, m K 3J E J, 4J 3J 6J + 7J + v K.m, 3J 3J 4J 6J mg N
. g bloc s pushed gnst sprng w sprng constnt 5 N/m compressng t cm. ter e bloc s relesed, t trvels long rctonless horzontl surce nd 45º nclne plne. Wht s e mmum heght reched by s bloc?