Chapter 7 Kinetic energy, potential energy, work
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1 Chpter 7 netc energy, potentl energy, work I. netc energy. II. ork. III. ork - netc energy theorem. IV. ork done by constnt orce: Grttonl orce V. ork done by rble orce. VI. Power - Sprng orce. - Generl: 1D, 3D, ork-netc Energy Theorem VII. Potentl energy Energy o congurton VIII. ork nd potentl energy IX. Conserte / Non-conserte orces X. Determnng potentl energy lues: grttonl potentl energy, elstc potentl energy
2 Energy: sclr quntty ssocted wth stte (or condton o one or more objects. I. netc energy 1 Energy ssocted wth the stte o moton o n object. m (7.1 Unts: 1 Joule = 1J = 1 kgm /s = N m II. ork Energy trnserred to or rom n object by mens o orce ctng on the object. To + rom - - Constnt orce: m d d 1 1 m m( md m( d 1 m( d d 1 ork done by the orce = Energy trnser due to the orce.
3 - To clculte the work done on n object by orce durng dsplcement, we use only the orce component long the object s dsplcement. The orce component perpendculr to the dsplcement does zero work. d cos d d (7.3 cos φ d - Assumptons: 1 =cte, Object prtcle-lke. Unts: 1 Joule = 1J = 1 kgm /s A orce does + when t hs ector component n the sme drecton s the dsplcement, nd when t hs ector component n the opposte drecton. = when t hs no such ector component. Net work done by seerl orces = Sum o works done by nddul orces. Clculton: 1 net = net net = net d
4 II. ork-netc Energy Theorem (7.4 Chnge n the knetc energy o the prtcle = Net work done on the prtcle III. ork done by constnt orce - Grttonl orce: d mgd cos (7.5 Rsng object: = mgd cos18º = -mgd g trnsers mgd energy rom the object s knetc energy. llng object: = mgd cos º = +mgd g trnsers mgd energy to the object s knetc energy.
5 - Eternl ppled orce + Grttonl orce: g (7.6 Object sttonry beore nd ter the lt: + g = The ppled orce trnsers the sme mount o energy to the object s the grttonl orce trnsers rom the object. IV. ork done by rble orce - Sprng orce: kd (7.7 Hooke s lw k = sprng constnt mesures sprng s stness. Unts: N/m
6 Hooke s lw 1D k ork done by sprng orce: - Assumptons: Sprng s mssless m sprng << m block Idel sprng obeys Hooke s lw ectly. Contct between the block nd loor s rctonless. Block s prtcle-lke. - Clculton: 1 The block dsplcement must be dded nto mny segments o nntesml wdth,. j ( cte wthn ech short segment.
7 ork done by n ppled orce + sprng orce: d k d s j s ( 1 1 s k k s = I Block ends up t =. 1 s k s Block sttonry beore nd ter the dsplcement: = =- s The work done by the ppled orce dsplcng the block s the negte o the work done by the sprng orce. ( 1 1 S k k d k
8 ork done by generl rble orce: 1D-Anlyss (7.1 (,,,, lm g j g j j g j j d more better ppromton Geometrclly: ork s the re between the cure ( nd the -s.
9 ork-netc Energy Theorem - Vrble orce d m d ( 3D-Anlyss (, (, ( ; ˆ ˆ ˆ z y k j z y z y k dz j dy d r d ˆ ˆ ˆ z z z y y y r r z y dz dy d d dz dy d dr d d dt d m d m m m d m d m 1 1 d d dt d d d dt d md d d d m
10 V. Power Tme rte t whch the ppled orce does work. - Aerge power: mount o work done n n mount o tme t by orce. P g t (7.1 - Instntneous power: nstntneous tme rte o dong work. P d dt (7.13 φ P d dt cos d dt cos d dt cos (7.14 Unts: 1tt=1=1J/s 1 klowtt-hour = 1 k h = J=3.6MJ
11 54. In the gure ( below N orce s ppled to 4kg block t downwrd ngle θ s the block moes rghtwrd through 1m cross rctonless loor. nd n epresson or the speed t the end o tht dstnce the block s ntl elocty s: ( nd (b 1m/s to the rght. (c The stuton n (b s smlr n tht the block s ntlly mong t 1m/s to the rght, but now the N orce s drected downwrd to the let. nd n epresson or the speed o the block t the end o the 1m dstnce. N N d ( cos d.5m( y mg y mg (.5m (Ncos.5(4kg cos m / s ( b 1m / s.5m.5(4kg (1m / s ( c 1m / s.5m J (Ncos.5(4kg J (Ncos.5(4kg J 1 cos m / s 1 cos m / s
12 18. In the gure below horzontl orce o mgntude N s ppled to 3kg psychology book, s the book sldes dstnce o d=.5m up rctonless rmp. ( Durng the dsplcement, wht s the net work done on the book by,the grttonl orce on the book nd the norml orce on the book? (b I the book hs zero knetc energy t the strt o the dsplcement, wht s the speed t the end o the dsplcement? y g N mg gy N d Only g, do ( net net g g work or cos3 net mg sn 3 (17.3N 14.7N.5m 1.31J net d ( b 1.31J.5m.93m / s
13 55. A kg lunchbo s sent sldng oer rctonless surce, n the poste drecton o n s long the surce. Begnnng t t=, stedy wnd pushes on the lunchbo n the negte drecton o, g. below. Estmte the knetc energy o the lunchbo t ( t=1s, (b t=5s. (c How much work does the orce rom the wnd do on the lunch bo rom t=1s to t=5s? Moton conce 1 1 t(1 t downwrd prbol d dt d dt t.m / s cte m (kg(.m / s (.4N( t.1t.4n ( b t 5s J ( t 1s.8m / s.5(kg(.8m / s.64j ( c (5s.64.64J (1s
14 74. ( nd the work done on the prtcle by the orce represented n the grph below s the prtcle moes rom =1 to =3m. (b The cure s gen by =/, wth =9Nm. Clculte the work usng ntegrton ( Are under cure (11.5squres(.5m(1N 5.75J ( b d 9 9( 1 6J An eletor hs mss o 45kg nd cn crry mmum lod o 18kg. I the cb s mong upwrd t ull lod t 3.8m/s, wht power s requred o the orce mong the cb to mntn tht speed? mg m 45kg 18kg 63kg mg totl net mg (63kg(9.8m / s g 61.74kN P P (61.74kN(3.8m / s 34.61k
15 A sngle orce cts on body tht moes long n -s. The gure below shows the elocty component ersus tme or the body. or ech o the nterls AB, BC, CD, nd DE, ge the sgn (plus or mnus o the work done by the orce, or stte tht the work s zero. t A B C D 1 m BC B C CD C D, DE D E AB A B E
16 5. A 5g block s dropped onto reled ertcl sprng tht hs sprng constnt o k=.5n/cm. The block becomes ttched to the sprng nd compresses the sprng 1 cm beore momentrly stoppng. hle the sprng s beng compressed, wht work s done on the block by ( the grttonl orce on t nd (b the sprng orce? (c ht s the speed o the block just beore t hts the sprng? (rcton neglgble (d I the speed t mpct s doubled, wht s the mmum compresson o the sprng? mg d s mg ( g g d mgd (.5kg(9.8m / s (.1m. 9J 1 ( b s kd.5(5n / m(.1m 1. 8J ( c.9j 1.8J net 3.47m / s.5m.5m.5(.5kg.5m g s ( d I ' 6.95m / s Mmum sprng compresso n? net d ' mgd '.5kd '.3m.5m '
17 6. In the gure below, cord runs round two mssless, rctonless pulleys; cnster wth mss m=kg hngs rom one pulley; nd you eert orce on the ree end o the cord. ( ht must be the mgntude o you re to lt the cnster t constnt speed? (b To lt the cnster by cm, how r must you pull the ree end o the cord? Durng tht lt, wht s the work done on the cnster by (c your orce ( the cord nd (d the grttonl orce on the cnster? P ( Pulley Hnd 1: cte cord : T net T mg mg 98N T 98N T T T (b To rse m.m, two segments o the cord must be shorten by tht mount. Thus, the mount o the strng pulled down t the let end s:.4m P1 mg ( c d (98N (.4m 3. 9J ( d g mgd (.m(kg(9.8m / s 3. 9J + g = There s no chnge n knetc energy.
18 I. Potentl energy Energy ssocted wth the rrngement o system o objects tht ee orces on one nother. Unts: J Emples: - Grttonl potentl energy: ssocted wth the stte o seprton between objects whch cn ttrct one nother the grttonl or - Elstc potentl energy: ssocted wth the stte o compresson/etenson o n elstc object. II. ork nd potentl energy I tomto rses grttonl orce trnsers energy rom tomto s knetc energy to the grttonl potentl energy o the tomto-erth system. I tomto lls down grttonl orce trnsers energy rom the grttonl potentl energy to the tomto s knetc energy.
19 U Also ld or elstc potentl energy Sprng compresson s Sprng orce does on block energy trnser rom knetc energy o the block to potentl elstc energy o the sprng. Sprng etenson s Sprng orce does + on block energy trnser rom potentl energy o the sprng to knetc energy o the block. Generl: - System o two or more objects. - A orce cts between prtcle n the system nd the rest o the system.
20 - hen system congurton chnges orce does work on the object ( 1 trnserrng energy between E o the object nd some other orm o energy o the system. - hen the congurton chnge s reersed orce reerses the energy trnser, dong. III. Conserte / Nonconserte orces - I 1 = lwys conserte orce. Emples: Grttonl orce nd sprng orce ssocted potentl energes. - I 1 nonconserte orce. Emples: Drg orce, rctonl orce E trnserred nto therml energy. Non-reersble process. - Therml energy: Energy ssocted wth the rndom moement o toms nd molecules. Ths s not potentl energy.
21 - Conserte orce: The net work t does on prtcle mong round eery closed pth, rom n ntl pont nd then bck to tht pont s zero. - The net work t does on prtcle mong between two ponts does not depend on the prtcle s pth. Conserte orce b,1 = b, Proo: b,1 + b, = b,1 = - b, b, = - b, b, = b,1 IV. Determnng potentl energy lues ( d U orce s conserte Grttonl potentl energy: y y U ( mg dy mg mg( y y y y y mgy Chnge n the grttonl potentl energy o the prtcle-erth system.
22 U, y U ( y mgy Reerence congurton The grttonl potentl energy ssocted wth prtcle-erth system depends only on prtcle s ertcl poston y relte to the reerence poston y=, not on the horzontl poston. Elstc potentl energy: U k 1 1 ( k d k k Chnge n the elstc potentl energy o the sprng-block system. Reerence congurton when the sprng s t ts reled length nd th block s t =. 1 U, U ( k Remember! Potentl energy s lwys ssocted wth system. V. Conserton o mechncl energy Mechncl energy o system: Sum o ts potentl (U nd knetc ( energes.
23 E mec =U+ Assumptons: - Only conserte orces cuse energy trnser wthn the system. - The system s solted rom ts enronment No eternl orce rom n object outsde the system cuses energy chnges nsde the system. U ( 1 ( U U1 U 1 U U E mec = + U = - In n solted system where only conserte orces cuse energy chnges, the knetc energy nd potentl energy cn chnge, but ther sum, the mechncl energy o the system cnnot chnge. 1 - hen the mechncl energy o system s consered, we cn relte the sum o knetc energy nd potentl energy t one nstnt to tht t nother nstnt wthout consderng the ntermedte moton nd wthout ndng the work done by the orces noled.
24 y E mec = constnt E mec U U 1 U 1 Potentl energy cures ndng the orce nlytclly: du ( U ( ( ( (1D moton d - The orce s the negte o the slope o the cure U( ersus. - The prtcle s knetc energy s: ( = E mec U(
25 Turnng pont: pont t whch the prtcle reerses ts moton (=. lwys (=.5m Emples: = 1 E mec = 5J=5J+ = < 1 E mec = 5J= >5J+ < mpossble Equlbrum ponts: where the slope o the U( cure s zero (= U = -( d U/d = -(
26 U(/d = -( Slope Equlbrum ponts E mec,1 E mec, E mec,3 Emple: 5 E mec,1 = 4J=4J+ = nd lso = 5 neutrl equlbrum >> 1, 5 >> 4 E mec, = 3J= 3J+ = Turnng ponts 3 =, = prtcle sttonry Unstble equlbrum 4 E mec,3 =1J=1J+ =, =, t cnnot moe to > 4 or < 4, snce then < Stble equlbrum
27 Reew: Potentl energy =- U - The zero s rbtrry Only potentl energy derences he physcl menng. - The potentl energy s sclr uncton o the poston. - The orce (1D s gen by: = -du/d
28 P1. The orce between two toms n dtomc molecule cn be represented by the ollowng potentl energy uncton: 6 1 ( U U where U nd re constnts. Clculte the orce ( ( U U U d du Mnmum lue o U( mn 1 ( 1 ( ( ( U U U U d du U U s ppro. the energy necessry to dssocte the two toms.
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