Forceisgreaterwherethecurveissteeper力在曲線陡處較大-ckw課件

7.ConservationofEnergy 能量守恆Conservative&Non-conservativeForces 守恆和非守恆力PotentialEnergy 位能ConservationofMechanicalEnergy 機械能的守恆PotentialEnergyCurves 位能曲線7.ConservationofEnergy 能量守恆HowmanydifferentenergyconversionstakeplaceastheYellowstoneRiverplungesoverYellowstoneFalls?當黃石河在黃石瀑布下墮時，經(jīng)歷了幾種能量轉換。P.E.K.E.,sound,&heat位能

動能，聲音，和熱HowmanydifferentenergyconvWorkagainstconservativeforce(pathindependent),對守恆力作功(與路徑無關)，e.g.,gravity,&staticE

如重力和靜電場:Workcanbestored

potentialenergy.

功可存

位能Workagainstnon-conservativeforce(pathdependent),對非守恆力作功(隨路徑而定)，e.g.,friction&E+B(t)如摩擦和E+B(t):Forfriction,workisdissipated

heat

&entropy.摩擦:功有消耗

熱和熵Workagainstconservativeforc7.1.Conservative&Non-conservativeForces

守恆和非守恆力Fisconservativeif如果每條密閉路徑C都是foreveryclosedpathC.F便是守恆的。ispath-independent與路徑無關i.e.,ispath-dependent由路徑?jīng)Q定

WABWBA

+

WAB=0

WAB=

WBA=WAB

Fisnon-conservativeifthereisaclosedpathCsuchthat如果有一條密閉路徑C是F便是非守恆的。WBAWABMathematica7.1.Conservative&Non-conserExample:Workdoneonclimberbygravity

重力對爬山者所作功Goingup上山時:W1=(

mg)h=

mghGoingdown下山時:W2=(

mg)(

h)=mghRoundtrip來回:W=W1+W2=0Horizontaldisplacementrequiresnowork.水平方向的位移不需作功。Gravityisconservative重力是守恆的.Climberdoes(+)work.爬山者作(正)功。(+)workdoneonclimber.爬山者被作(正)功。Example:WorkdoneonclimberExample:Workdoneontrunkbyfriction

摩擦對櫃子所作功Goingright往右時: W1=(

mg)L=

mgLGoingleft往左時: W2=(

mg)(

L)=

mgLRoundtrip來回:W=W1+W2=2mgL

0Frictionisnon-conservative摩擦是非守恆的.Moveralwaysdoes(+)workagainstfriction.搬櫃子的人永遠都要對摩擦作(正)功。Example:WorkdoneontrunkbGOTIT懂嗎?7.1.Ifittakesthesameamountofworktopushatrunkacrossaroughfloorasitdoestoliftaweighttothesamedistancestraightupward.如果把一個櫃子推過一段粗糙樓板地所需的功，和把一個砝碼朝上舉起到同樣距離所需的功一樣。Howdotheamountsofworkcompareifthetrunk&weightaremovedalongcurvedpathsbetweenthesamestarting&endpoints?如果起和終點不變，但櫃子和砝碼的路徑都改成曲線，兩者所需的功比較起來會怎樣?Ans.Workisgreaterforthetrunk.答：櫃子需要的功較大。GOTIT懂嗎?7.1.Ifittakest7.2.PotentialEnergy 位能Conservativeforce守恆力:Potentialenergy=storedwork 位能

=所存的功

=(workdonebyforce) =(力作的功)Note:onlydifferenceofpotentialenergymatters.注

:祇有位能差算數(shù)。1-Dcase:一維時：ConstantF:不變的力F:7.2.PotentialEnergy 位能ConserGravitationalPotentialEnergy 重力位能Horizontalcomponentofpathdoesnotcontribute. 路徑的水平分量無貢獻Verticallift:垂直上提

mgGravitationalPotentialEnergyExample7.1.RidingtheElevatorA55kgengineertakesanelevatorfromherofficeonthe33rdfloortothe59thfloor.一個55kg的工程師乘電梯從她辦公室的33樓到59樓。Later,shedescendstostreetlevel. 稍後，她下去一樓。Ifshetakesherofficeasthezeroofpotentialenergy,如果她把她的辦公室當成位能的原點，andifthedistancebetweenfloorsis3.5m,而且樓層的高度是3.5m，what’sherpotentialenergy 在inheroffice. 她的辦公室，onthe59thfloor. 59樓，和atstreetlevel? 一樓時，

她的位能為何？(a)(b)(c)33591Example7.1.RidingtheElevaApplication:PumpedStorage

應用:抽水式存能NorthfieldMountainPumpedStorageProject,MA,USA美國麻省北田山抽水式儲能計劃Excesselectricenergystoredbypumpingwatertohigherground.(seeProb.29)過剩的電能以抽水到高地的方式儲存(參閱習題29)Application:PumpedStorage

ElasticPotentialEnergy 彈性位能Idealspring:理想彈簧

parabolic拋物線x

x0 x=x0 x

x0壓縮

平衡

伸長UisalwayspositiveU永遠都是正的x0=equilibriumposition

平衡位置LetSettingx0=0:ElasticPotentialEnergy 彈性位能IExample7.2.SpringsvsGasoline 彈簧對汽油

Acar’ssuspensionconsistsofspringswithoveralleffectivek=120kN/m.一輛汽車的避震器內(nèi)的彈簧組合的有効k=120kN/m.Howmuchthesespringsneedbecompressedtostorethesameamountofenergyasin1gramofgasoline?這些彈簧要壓縮多少，才能儲存與1公克汽油相同的能量？FromAppendixC:Energycontentsofgasoline=44MJ/kg.從附錄C：汽油的能含量=44MJ/kg。Springscan’tcompetewithgasolineasenergysource.彈簧在能源上無法與汽油相比。Example7.2.SpringsvsGasolExample7.3.ClimbingRope 爬山索Climbingropesarespringytocushionfalls. 爬山索都有彈性，以減輕下墜時的損傷。ConsiderropewithF=

kx+bx2

,where

k=223N/m,b=4.10N/m2.茲有一索，其F=

kx+bx2，且k=223N/m,b=4.10N/m2.Findthepotentialenergywhenit’sstretched2.62m,takingU=0atx=0.求把它拉長2.62m時的位能，設

x=0時U=0。whichis

3%lessthanU=?kx2=765J.比U=?kx2=765J少

3%Example7.3.ClimbingRope 爬山7.3.ConservationofMechanicalEnergy

機械能的守恆Mechanicalenergy機械能:LawofConservationofMechanicalEnergy機械能守恆定律:if若(nonon-conservativeforces)無非守恆力7.3.ConservationofMechanicaExample7.4.TranquilizinganElephant 麻醉大象Abiologistusesaspring-loadedguntoshoottranquilizerdartsintoanelephant.一生物學家用彈簧槍把麻醉鏢射入一頭大象。Thegun’sspringhask=940N/m,&iscompressed

x=25cmbeforefiringa38-gdart.槍的彈簧k=940N/m，壓縮了

x=25cm後射出38-g的鏢。Assumingthegunpointshorizontally,atwhatspeeddoesthedartleavethegun?設槍朝水平方向，鏢離槍口時的速率為何？Initialstate初態(tài):Finalstate終態(tài):

Problemishardertosolveby2ndlaw.此題以第二定律來解會比較難。Example7.4.TranquilizinganExample7.5.Spring&Gravity 彈簧比重力A50-gblockisplacedagainstaspringatthebottomofafrictionlessslope.在一個無摩擦力的斜坡底，一50-g的方塊緊靠著一個彈簧。Thespringhask=140N/mandiscompressed11cm.彈簧的k=140N/m，且已壓縮了11cm。Whentheblockisreleased,howhighuptheslopedoesitrise?放開方塊後，它能沿斜坡升到多高？Initialstate初態(tài):Finalstate終態(tài):

Example7.5.Spring&GravityGOTIT?7.3.BowlingBall 保齡球Abowlingballistiedtotheendofalongropeandsuspendedfromtheceiling.一個保齡球由一根長繩子吊在天花板下。Astudentholdstheballtohernose,thenreleasesitfromrest.一個學生把球捧在她的鼻子前，然後放手。Shouldsheduckasitswingsback? 它擺回來時她要不要往後退？GOTIT?7.3.BowlingBall 保齡球Example7.6.SlidingBlock 滑動的質塊Ablockofmassmislaunchedfromaspringofconstantkthatiscompressedadistancex0.一質量為m的質塊，由一條長度壓縮了x0

，常數(shù)為k的彈簧彈出。Theblockthenslidesonahorizontalsurfaceoffrictionalcoefficient

.質塊然後在一摩擦系數(shù)為的水平面上滑動。Howfardoestheblockslidebeforecomingtorest? 質塊要滑多遠才能停下來？Initialstate始態(tài)：Workdoneagainstfriction對摩擦作的功:

Finalstate終態(tài):Launch發(fā)射時:Conservationofenergy:能量守恆：Example7.6.SlidingBlock 滑動7.4.PotentialEnergyCurves 位能曲線

Frictionlessroller-coastertrack

無摩擦的雲(yún)霄飛車軌道HowfastmustacarbecoastingatpointAifit’storeachpointD?車在A點要走多快才能達到D點？Criterion條件:turningpoints轉折點能量高，車可到任何地方車限於此兩點之間車限於此potentialbarrier位能屏障potentialwell位能井7.4.PotentialEnergyCurves 位Example7.7.H2 氫分子NearthebottomofthepotentialwellofH2

在H2的位能井底附近，U=U0+a(x

x0)2,where其中U0

=0.760aJ,a=286aJ/nm2,x0=0.0741nm.(1aJ=1018J)Whatrangeofatomicseparationisallowedifthetotalenergyis0.717aJ?

如果總能量是0.717aJ，原子間的距離在甚麼範圍？Turningpoints轉折點：

Example7.7.H2 氫分子NeartheForce&PotentialEnergy 力與位能Force~slopeofpotentialcurve

力~位能曲線的坡度(xalongdirectionofF)x沿F的方向Whenthecurveisrisingtotheright,forceistotheleft.曲線向右爬升時，力朝左