Unt 5 Work and Energy 5. Work and knetc energy 5. Work - energy theore 5.3 Potenta energy 5.4 Tota energy 5.5 Energy dagra o a ass-sprng syste 5.6 A genera study o the potenta energy curve 5. Work and knetc energy () Work: unt n joue (J) y Work done by the orce F W = F d = ( F cosθ ) d = F d F : the orce aong the dspaceent Reark: Postve work s done on an object when the pont o appcaton o the orce oves n the drecton o the orce. Matheatca stateent: Work done s the dot product o the orce and the dspaceent. Dot product: A B = AB cosθ = A B + A B + A B y y z z Eape A 75.0-kg person sdes a dstance o 5.00 on a straght water sde, droppng through a vertca heght o.50. How uch work does gravty do on the person? h=.5 θ g θ d=5.00
Answer The gravty aong the dspaceent has a agntude g cos θ. By the denton o work done, the work done o the gravty on the person s gven by W = (g cos θ ) d = g (h / d ) d = gh = (75.0)(9.8)(.50) = 837.5 J Rearks: () When F or d s not a constant (see the et gure, (h, h) the oveent o the partce s aways varyng). F w = F s, W = s where s s etreey sa. The work done o (0, 0) C y the gravty on the ba s ang down s gven by 0 0 W = F ds = Fˆj (d ˆ + d y ˆj ) = F dy = g dy = gh C C h h () Power: work done per unt te Snce v = W dw F ds Power = = =. t 0 t dt dt ds dt, we have the power equas to the dot product o orce and veocty. e.g. Power = F v. I the orce and the veocty are n the sae drecton, P = Fv. (3) Knetc energy: K v. 5. Work - energy theore I an object s ovng wth an acceeraton a and a dstance d s oved, then accordng to the orua v = v + ad, we have F v = v + ( ) d. The work done on the object W s gven by v v W = Fd = ( v v ) = v Or, we can rewrte t as W = K K = K. v = K K
The work-energy theore states that the tota work done on the object by the orces equas the change n knetc energy. Reark: A proo by usng ntegraton. Suppose a partce oves n a straght ne (-D) W = F( ) d F = a = dv, v dt = d dt W = dv d = dv d ( ) = dt dt = v v = K K v v vdv Eape A 47.-kg bock o ce sdes down an ncne.6 ong and 0.90 hgh. A worker pushes up on the ce parae to the ncne so that t sdes down at constant speed. The coecent o knetc rcton between the ce and the ncne s 0.0. Fnd (a) the orce eerted by the worker, (b) the work done by the worker on the bock o ce, and (c) the work done by gravty on the ce. Answer (a) Ange o ncnaton = θ 0.90 0.90 snθ =.6 θ = 33.8 The nora reacton, N = 47. 9.8 cos33.8 = 384. N Frctona orce, k = 0.0 384. = 4.3 N F.6 k N g θ The orces whch acts on the ce: Force eerted by the worker + rctona orce = down pane orce = o Force eerted by the worker = 47. 9.8sn 33.8 4.3 = 5.0 N g snθ 3
(b) Work done by the worker = 5.0.6 = 348.33 J (c) Work done by the gravty = 47. 9.8 0.90 = 47. J 5.3 Potenta energy (a) The gravtatona potenta energy U = gh The derence o the potenta energy: U = U U = W, where W s the work done by gravty the bock s reeased. In ths case, W s postve. Reerence pont (b) The eastc potenta energy Hooke s aw states that the restorng orce s proportona to the eongaton o the sprng ro ts natura ength, e.g. F = k where F s the restorng orce and k s caed sprng constant. F F I the eongaton goes ro zero to, the work done by the apped orce F s gven by W F' d = ( F) d kd 0 = = 0 0 k = 0 k. = Suppose the sprng s stretched a dstance ntay. Then the work we have to stretch t to a greater eongaton s W = Fd = kd = k k. When a sprng s copressed by an aount, the work done o the apped orce k Fd = kd = (F: apped orce) 0 0 = U = k (.e. the eastc potenta energy stored n the sprng) 4
5.4 Tota energy E = K + U, where E: echanca energy, K: knetc energy, U: potenta energy. For an soated syste (no rctona orce), E = const. or t s conserved. K + U = const. Ths s the reerred to as the conservaton o echanca energy. Eape A 63-g bock s dropped wth an nta speed such that t hts onto a vertca ght sprng o orce constant k =.5 N/c. The bock stcks to the sprng, and the sprng copresses.8 c beore cong oentary to rest. Whe the sprng s beng copressed, how uch work s done (a) by the orce o gravty and (b) by the sprng? (c) What s the speed o the bock just beore t hts the sprng? Answer Method I: By the work-energy theor (a) Let the copresson o the sprng be. The work done by the bock s weght s (b) The work done by the sprng s 3 ( ) ( ) ( ) W = g = 63 0 9.8.8 0 = 0.304 J. (c) W.5 0.8 0 k.75 J. ( ) ( ) = = = The speed v o the bock just beore t hts the sprng s gven by K = 0 v = W + W, whch yeds v ( )( W + W ) ( )( 0.304.75) = = = 3 63 0 3.3 s Method II: By the conservaton o echanca energy Let the gravtatona potenta energy o the bock be zero when t s at ts owest pont. We have K. E. + G. P. E. + E. P. E. = constant. Thereore, vv + + 0 = 0 + 0 + kk. Ths equaton gves v + g = k, and thus the resut or v. 5
Eape (Chaengng) A partce o ass sdes down the sooth ncned ace o a wedge o ass, and ncnaton α, whch s ree to ove n a sooth horzonta tabe. Use equatons o oentu and energy to obtan an epresson or the veocty o the partce reatve to the wedge when the partce has oved a reatve dstance s ro rest down the ncned ace o the α wedge. Answer Let the veocty o the wedge be V and that o the partce reatve to the wedge be u. By conservaton o horzonta oentu, snce there s no horzonta orce actng on the syste, V (u cos α V) = 0. V = u cos α. 3 By conservaton o energy, gs sn α = ( ) V + ( u cos α V ) + ( u sn α) { } gssnα = u cos α + u cos α + u sn α, 9 9 6gs snα u = + sn α. u α V V 6
5.5 Energy dagra o a ass-sprng syste A sprng s ed at one end and the other end s attached to ass whch oves back and orth wth an aptude A Eastc potenta energy U ( ) = k. I rctona orce s negected, we have the energy o the syste beng conserved, e.g. E = constant. When =0, U = 0, K = E and K = v s the greatest veocty n the whoe process o oscaton. v At = A, E = a(u), K = 0 v = 0., F F du = = The negatve sope o the curve U() aganst, d At the orgn, du = 0, t s the equbru d poston. The resutant orce s zero. When > 0, du > 0, hence F < 0 represents d that the restorng orce s pontng to the et. When < 0, du < 0, hence F > 0 represents d that the restorng orce s pontng to the rght. When = 0, t s caed equbru poston. 7
5.6 A genera study o the potenta energy curve The oowng curve shows a proe o an arbtrary potenta energy wth the poston. Regon I Unstabe equbru Regon II Stabe equbru poston Conservaton o echanca energy E = K + U() = const. K = v 0 a b c d I E = E 0, the partce w stay at pont 0, otherwse K has to be negatve. When E = E, the oton o the partce s conned between, and are caed turnng ponts. When E = E, dependng on the nta condton o, the partce w ether ove n regon I (a b) or regon II (c d). When E = E 3, the partce w ove between = 3 and. Unstabe equbru poston: Any au pont n a potenta energy curve s an unstabe equbru poston. Stabe equbru poston: Any nu pont n a potenta energy curve s a stabe equbru poston. 8
Eape (Chaengng) A partce o ass s attached to one end o a ght eastc strng whose other end s ed to a pont O on an ncned pane (ncnaton ange = 45 o wth the horzonta). The natura ength o the strng s and ts orce constant s g/. The partce s hed on the ncned pane so that the strng es just unstretched aong a ne o greatest sope and then reeased ro rest. O 45 o (a) Suppose that the ncned pane s sooth, deterne the owest poston that the partce can reach. Deterne aso the equbru poston. (b) Suppose that the ncned pane s rough enough and the rctona coecent s µ. Show that the partce stops ater ts rst descendng a dstance d aong the greatest sope, then the dstance s gven by d = ( µ ). Answer s O 45 o (a) Note that the partce s oentary stopped at ts owest poston. By the conservaton o echanca energy, the oss n gravtatona energy becoes the gan n the eastc potenta energy. g ( ) s g ( s sn 45 o ) =, where s s the etenson o the strng when the partce s at ts owest poston. 9
The above equaton gves s ( ) =. Hence, we obtan s =. The owest poston s ocated at + s = + = ( + ) ro O. For equbru o the partce, the down pane orce o the partce baances wth the eastc orce, we can wrte o g g sn 45 = ( ) seq. s eq O 45 o Hence, we obtan s eq =. That s, the equbru poston s ocated at + seq = + = ( + ) ro O. (b) Rough surace d O 45 o Ths part ders ro part (a) by the rctona orce. In the vew o energy, we have g ( ) d ( cos 45 o ) ( sn 45 o + g d = g d ). Ater spcaton, we have ( d ) + µ =, whch gves d = ( µ ). 0