大学物理双语奥本汉姆ChapWorkEnergy

会计学1大学物理双语奥本汉姆ChapWorkEnergy7-1WorkandPower(P147-)Theworkdonebyaforceindisplacingabodyasthescalarproductofand.1.Work研究力和运动的空间过程关系。Workisascalarquantity,nodirection.positivework;质点受的力与它的位移的标积zerowork;negativework.第1页/共37页Explanations:(1)Iftheforceisconstantinthewholepath,ABoABWhenaparticlemovesfromAtoBalongacurvepath,thetotalworkdonebytheforceequalstheintegraloffromAtoB:(2)Workdonebyavariableforce:—displacement

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

——变力曲线与位移轴在两点x1,

x2

之间所包围的面积.(6)Unitofwork:Dimensionis:MLT–2L=ML2T–2InSIsystem,1N·m=1J(joule)=1kg·m2/s2

xFxFxFW=0W<0W>0第4页/共37页2.Power(功率)(P186)Instantaneouspower:TheinstantaneouspowerPisdefinedastherateatwhichworkisperformed.TheSIunitofpoweristhewatt,1W=1J/s1horsepower=1hp=746W第5页/共37页abAparticlemovesfromatobalonganarbitrarypath.andmayvaryfrompointtopointalongthepath.drrThatis,7-2Work-KineticEnergyTheorem(P156-)第6页/共37页Networkdoneonanobjectequalstothechangeinitskineticenergy.Explanations:(1)Workisthemeasurementsforthechangeinthekineticenergy(K)ofabody.——Bodygainsthekineticenergy.—Bodyloses“K”&doesworkoutside.(2)Bothworkand“K”arescalars.Theyhavesameunitsanddimensions;“K”onlydependsonthespeedofinitialandfinalstates,butworkdependsontherealprocess(动能是状态量,功是过程量).第7页/共37页(4)BothWandKdependonRF.Work-kineticenergytheoremisonlyholdforinertialRF.LookatexamplesinSec.7-4afterclass.(3)Forasystemwithmoreparticles,work-kineticenergytheoremshouldbe:Thesumworkdoneonasystembyallexternalandinternalforcesisequaltothechangeinkineticenergyofthesystem(质点系动能定理系统外力和内力做功总和等于系统动能的增量).第8页/共37页Example:Asimplependulum(单摆)ofmassm=1kgisreleasedfrompositiono=30o.

Itmovesinacirculararcofradius

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

onaparticlemovingaroundanyclosedpathiszero.Definition1:Definition2:Workdonebyaconservativeforceisrecoverableorindependenceofpaths(与路径无关).Thesetwodefinitionsareequivalentandpowerfulbecausetheyallowustosimplifydifficultproblemswhenonlyconservativeforcesareinvolved.第14页/共37页ababA2.0kgblockslidesalongafrictionlesstrackfrompointatob.Ittravelsthroughatotaldistanceof2.0malongthetrack,andanetverticaldistanceof0.80m.Howmuchworkisdoneonitbythegravitationalforceduringtheslide?Solution:Example:J7.150cos°v==mgdWQ第15页/共37页Theworkdoneby

conservativeforce

=minuschangeof“U”relatedtotheconservativeforce

(保守力作功等于该过程始末两状态势能增量的负值).Ifwedefineawork-function——U

as7-4PotentialEnergy(P170)Lookatexample8-1ofP171Then,UUUWifconserD-=--=)(第16页/共37页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).势能是保守力对路径的线积分DefinitionofU:第17页/共37页Ifworkdonebyforceisdependentonthepath.Anonconservativeforceisalsocalleddissipativeforce(耗散力),suchasafrictionforce.Theworkdonebyfrictionalforceisequaltotheincreaseinthermalenergy.Ingeneral,Theiscallednonconservativeforce.ò¹L0.rdFrrsFEfth=D第18页/共37页7-5Work-EnergyPrinciple(1)1.Work-energyprincipleExternalForce:0Inphysics,energyhavedifferentforms.Oneformofenergycantransfertoanotherform,butitcannotmagicallyappearordisappear.TheyobeyalawcalledthelawofConservationofEnergy.A,B&theEarthasasystemåå=+=iiWWWWWextextincicintandInternalForce:,ABFBAFrrABFBAFABrr第19页/共37页(2)(3)2.Thelawofconservationofenergy(P174)—mechanicalenergyofasystem.1.(4)Ifonlyconservativeforcesdowork,mechanicalenergyconserves——bothkineticenergyandpotentialenergytransfereachother.—Work-energyprincipleor第20页/共37页Thetotalenergyofasystemcanchangeonlybyamountsofenergythataretransferredtoorfromthesystem.2.ThetotalenergyEofanisolatedsystemcannotchange——conservationofenergy.3.Forisolatedsystem,ifasystemisisolatedfromitsenvironment,thentherecanbenoenergytransferstoorfromit.

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

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

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

Findingforceanalytically(由势能求保守力)2.xxFxUxxxifD-@D-=D)()(,cesin第27页/共37页Aconservativeforceequalsminusgradientofthepotentialenergyrelatedtotheforce

保守力沿某一给定的l

方向的分量等于与此保守力相应的势能函数沿l方向的空间变化率(梯度).Youmayfindbothmagnitudeanddirectionofaconservativeforceatanypointinaconservative-force-fieldfrompotentialenergydiagram!Hey!第28页/共37页LetU=0forr=,theUofsystemforparticleinpositionrisgivenby:Aspecialcase:Gravitationalpotentialenergy(Sec.8-7)——gravitationalpotentialenergyE)(2ErrrmMG>-=第29页/共37页Discussions:TheU-rcurveofgravitationalpotentialenergyastheFigureshowing:1.2.Theworkdonebythegravitationalforceisindependentofthepath,sogravitationisalsoaconservativeforce.rUO势能曲线第30页/共37页2.Equilibriumpointsandturningpoints(P189)

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

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