大學物理雙語奧本漢姆ChapWorkEnergy

會計學1大學物理雙語奧本漢姆ChapWorkEnergy7-1WorkandPower(P147-)Theworkdonebyaforceindisplacingabodyasthescalarproductofand.1.Work研究力和運動的空間過程關系。Workisascalarquantity,nodirection.positivework;質點受的力與它的位移的標積zerowork;negativework.第1頁/共37頁Explanations:(1)Iftheforceisconstantinthewholepath,ABoABWhenaparticlemovesfromAtoBalongacurvepath,thetotalworkdonebytheforceequalstheintegraloffromAtoB:(2)Workdonebyavariableforce:—displacement

fromAtoB第2頁/共37頁If(3)Workdonebyseveralforces:(4)Specialexpressioninscalarproduct:Theworkdonebyseveralforces=algebraicsumoftheworkdonebytheindividualforce.第3頁/共37頁(5)Geometricalrepresentationofwork:Fxx1x2dxTheworkdonebyaforceequalstheareaundertheFversusxcurve

——變力曲線與位移軸在兩點x1,

x2

之間所包圍的面積.(6)Unitofwork:Dimensionis:MLT–2L=ML2T–2InSIsystem,1N·m=1J(joule)=1kg·m2/s2

xFxFxFW=0W<0W>0第4頁/共37頁2.Power(功率)(P186)Instantaneouspower:TheinstantaneouspowerPisdefinedastherateatwhichworkisperformed.TheSIunitofpoweristhewatt,1W=1J/s1horsepower=1hp=746W第5頁/共37頁abAparticlemovesfromatobalonganarbitrarypath.andmayvaryfrompointtopointalongthepath.drrThatis,7-2Work-KineticEnergyTheorem(P156-)第6頁/共37頁Networkdoneonanobjectequalstothechangeinitskineticenergy.Explanations:(1)Workisthemeasurementsforthechangeinthekineticenergy(K)ofabody.——Bodygainsthekineticenergy.—Bodyloses“K”&doesworkoutside.(2)Bothworkand“K”arescalars.Theyhavesameunitsanddimensions;“K”onlydependsonthespeedofinitialandfinalstates,butworkdependsontherealprocess(動能是狀態(tài)量,功是過程量).第7頁/共37頁(4)BothWandKdependonRF.Work-kineticenergytheoremisonlyholdforinertialRF.LookatexamplesinSec.7-4afterclass.(3)Forasystemwithmoreparticles,work-kineticenergytheoremshouldbe:Thesumworkdoneonasystembyallexternalandinternalforcesisequaltothechangeinkineticenergyofthesystem(質點系動能定理系統(tǒng)外力和內(nèi)力做功總和等于系統(tǒng)動能的增量).第8頁/共37頁Example:Asimplependulum(單擺)ofmassm=1kgisreleasedfrompositiono=30o.

Itmovesinacirculararcofradius

R=1minaverticalplane.Usethework-kineticenergytheoremtodeterminethespeedofthependulumbobattheangle=10o.Solution::constant:variableFTrROdrFTrr第9頁/共37頁RABB′C〞C′OCFTr第10頁/共37頁7-3Con’servativeandNonconservativeForces1.WorkdonebyanditsfeaturesxyomgyiyfabcdheConsideringabodyundermgnearthesurfaceofEarth,fromatobalongthepathofacb.Theworkdonebythegravity(重力):(P168)第11頁/共37頁Thefeatureofthegravity:Wgonaparticledependsonlyontheinitialandfinalposition(orheight),anddoesnotdependontherealpathtakenbytheparticle.1.Wgonaparticlemovingaroundaclosedpath=0.2.xyomgyiyfabcdhe第12頁/共37頁BasedonHooke’slaw:2.Workdonebyanditsfeatures(P172)havethesamefeatures!OxiABxfx3.Conservative&nonconservativeforcesMathematically,thetwocommonfeaturesofcanbewrittenas:gsFandFvv第13頁/共37頁Thenetworkdonebyconservativeforce(保守力)

onaparticlemovingaroundanyclosedpathiszero.Definition1:Definition2:Workdonebyaconservativeforceisrecoverableorindependenceofpaths(與路徑無關).Thesetwodefinitionsareequivalentandpowerfulbecausetheyallowustosimplifydifficultproblemswhenonlyconservativeforcesareinvolved.第14頁/共37頁ababA2.0kgblockslidesalongafrictionlesstrackfrompointatob.Ittravelsthroughatotaldistanceof2.0malongthetrack,andanetverticaldistanceof0.80m.Howmuchworkisdoneonitbythegravitationalforceduringtheslide?Solution:Example:J7.150cos°v==mgdWQ第15頁/共37頁Theworkdoneby

conservativeforce

=minuschangeof“U”relatedtotheconservativeforce

(保守力作功等于該過程始末兩狀態(tài)勢能增量的負值).Ifwedefineawork-function——U

as7-4PotentialEnergy(P170)Lookatexample8-1ofP171Then,UUUWifconserD-=--=)(第16頁/共37頁Explanations:1.“U”isthefunctionofstates.2.Therelativityofpotentialenergy:Wab=Ua-Ub——Thedifferenceofpotentialenergy(“U”)hasabsolutemeaning,buttoassignaparticularUvaluetoanobjectmakessenseonlyifthereference“U”valueisgiven.Referenceof“U”isarbitrarytochoose.3.Apotentialenergyisassociatedwithasystemasawhole

(相互作用能).勢能是保守力對路徑的線積分DefinitionofU:第17頁/共37頁Ifworkdonebyforceisdependentonthepath.Anonconservativeforceisalsocalleddissipativeforce(耗散力),suchasafrictionforce.Theworkdonebyfrictionalforceisequaltotheincreaseinthermalenergy.Ingeneral,Theiscallednonconservativeforce.ò1L0.rdFrrsFEfth=D第18頁/共37頁7-5Work-EnergyPrinciple(1)1.Work-energyprincipleExternalForce:0Inphysics,energyhavedifferentforms.Oneformofenergycantransfertoanotherform,butitcannotmagicallyappearordisappear.TheyobeyalawcalledthelawofConservationofEnergy.A,B&theEarthasasystem??=+=iiWWWWWextextincicintandInternalForce:,ABFBAFrrABFBAFABrr第19頁/共37頁(2)(3)2.Thelawofconservationofenergy(P174)—mechanicalenergyofasystem.1.(4)Ifonlyconservativeforcesdowork,mechanicalenergyconserves——bothkineticenergyandpotentialenergytransfereachother.—Work-energyprincipleor第20頁/共37頁Thetotalenergyofasystemcanchangeonlybyamountsofenergythataretransferredtoorfromthesystem.2.ThetotalenergyEofanisolatedsystemcannotchange——conservationofenergy.3.Forisolatedsystem,ifasystemisisolatedfromitsenvironment,thentherecanbenoenergytransferstoorfromit.

第21頁/共37頁Acrate(木箱)ofmass180kgisreleasedthatwasbeingheldatrestatthetopofa3.7m-long-rampinclined39otothehorizontal.=0.28(a)Howfastisthecratemovingasitreachesthebottomoftheramp?(b)Howfarwillitsubsequentlyslideacrossthefloor?(c)Dotheanswerstothemchangeifwehalvethemassofthecrate?(a)Solution:Example:qmmcosmgFNFkkk==\q=cosmgFNQCFNFmgABs第22頁/共37頁Fromthealgebraicformoftheresults,itcanbeseenthattheanswersdonotdependonmass.Answer:(c)CNfmgABsqqmsin21cos2mgdmvmgdk-=-s/m5.5)cos(sin2=-=\qmqkgdv(b))cos(sin2102qmqmkkmgdmvmgs--=-=-4.5/)cos(sinmdskk=-=\mqmq第23頁/共37頁AchainhaslengthLandmassM,andwasputonafrictionlesstable.Att=0,thechainisstationaryandlengthdishangedovertheedge.Atthemomentthatthewholechainleavethetable,whatisthe(a)

Wgand(b)thevelocityofthechain?Solution:(a)Theway:1.Example:G1L-dG2FNxOx第24頁/共37頁ApplyingtheKinetic-energytheorem,onlyWg,Theway:2.(b)Earth,tableandchainasasystem,Emec=CG1Ifthefrictioncoefficientoftable’ssurfaceis,howaboutthev?netWK=D\)(221222dLLMgMvKWWGnet-==D==L-dxOG2FNU=0x第25頁/共37頁7-6PotentialEnergyDiagrams1.Relationbetweenforceandpotential(P173)CalculateUorUwhentheforceisgiven;1.hastwosidesofinformation:UUUWifD-=--=][conserwhereò=oPrrconserprFUrrdò-=-=DficonserconserrdFWUrr第26頁/共37頁Ifweconsidera1DsysteminwhichU

dependsonlyonx,U=U(x),agraphcanthenbedrawnofUversusx.ThecomponentofFinthex-direction,F(x),isthenegativeoftheslopeoftheUversusxcurve.Forgeneralcase,U=U(x,y,z)U(x)ExO

Findingforceanalytically(由勢能求保守力)2.xxFxUxxxifD-@D-=D)()(,cesin第27頁/共37頁Aconservativeforceequalsminusgradientofthepotentialenergyrelatedtotheforce

保守力沿某一給定的l

方向的分量等于與此保守力相應的勢能函數(shù)沿l方向的空間變化率(梯度).Youmayfindbothmagnitudeanddirectionofaconservativeforceatanypointinaconservative-force-fieldfrompotentialenergydiagram!Hey!第28頁/共37頁LetU=0forr=,theUofsystemforparticleinpositionrisgivenby:Aspecialcase:Gravitationalpotentialenergy(Sec.8-7)——gravitationalpotentialenergyE)(2ErrrmMG>-=第29頁/共37頁Discussions:TheU-rcurveofgravitationalpotentialenergyastheFigureshowing:1.2.Theworkdonebythegravitationalforceisindependentofthepath,sogravitationisalsoaconservativeforce.rUO勢能曲線第30頁/共37頁2.Equilibriumpointsandturningpoints(P189)

ApotentialenergyU(x)diagramisasfollows:ThetotalenergyE=U+K=constantandcanberepresentedasahorizontallineonthisgraph.第31頁/共37頁However,x2,x4arestableequilibriumpoints,x3isanunstableequilibriumpointand

x5isneutralequilibrium.(E=U,soK=0,F=0,Particlemustbestationary).Correspondstothepointx0at