SM2-B19-物理-Remedial-6

r(0PurerolHogwiHcontinuewithoutanyforce0Kineticfrictionwilldecreaselinearspeedandincreaseangularspeedandafteracertaintimepurerollingwillbeachieved.vrcooVKineticfrictionwilldecreaseangularspeedandincreaselinearspeed,andafteracertaintimepurerollingwillbeachieved.0)r26!20.AspoolofthreadconsistsofacylinderofradiusR,withendcapsofradiusR:asintheendviewshowninFigure.Themassofmespool,includingthethread,ismand\itsmomentofinertiaaboutanaxisthrough/gitscenteris/ThespoolisplacedonaroughhorizontalsurfacesomatitrollswithoutslippingwhenaforceTactingtotherightisappliedtothefreeendofmethread.Whatisthemagnitudeanddirectionofthefrictionalforceexertedbythesurfaceonmespoof?Answer:T-f~molThetorques(inclockwise)around]thecenterofmassare:JJR2~Ti?i=laForrollingwithoutslipping,a=-Haa♦ml-h,rTherestisalgebra!=——(r-f)R,nv'-Tf!i=—TRiR2m-IT-Iff(l+mJtf)=T(1+inR2)m22(1+aT(1+mR|Jf=\AlternateApproachTakingContactPointasReferenceofRotationAualiTherestisalgelgajf(R-/?,)=(/+wK;Answer:T-fThetorque(inclockwise)/aboutthecontactpoint:/T(R1~R[)={f+mR;)adma.2/?2Jr-/!(R.-R,)~(f+mR:Im/?,✓1mR;T-HITR,R,=IT+mR;T-(I+mR\)/I+mR[R,J+mR;ForrollingwithoutslippingafaR,‘Assumption:PureRollingSteph\ttasottiitgM*aa/f*aafa=r{-a)raaThedirectionofangularaccelerationvectorisposiliveThedirectionofangularaccelerationvectorispositiveplanebasedonrighthandplanebasedonrighthandrule.rule.29(1)T+f=maTR,+1a2)a=R,(-a)a\\/H'/a(3)!m\RSi/Therestisalgebra!Tr+/it(2):TR,+JR:=IIWRyR,amTRlR:+mfR:2=-IxT-lx/w/fc2+/xf=-IxT-mTR{R:I+mRiR2lT/:I+mR,fAsthefrictionisnegative,thedirectionoffrictionisactuallypointingtotheleft,andthemagnitudeisa=r(-a)/+mRxR:J.T*Thedirectionofangularaccelerationvectorispositiveforgoingoutoftheverticalplanebasedonrighthandrule./+fR:'Alsofrom(3),a=R,(-a)-atheangularvelocityisactuallyinclockwisedirection./aR,30ConstraintsonTfor/?,=0:/Weuseÿ.toapproximatefJ.Forpurerolling,'l+mR,R:'\-I/=V-TI+m%fo)ÿ/\«*I+mR2~jaiam[t;»ÿRiRiTTTIfi?1=0/VerticalDiscpulledattheCenter/\2rfJjngO+mR£)I+mxOxR,I+mR,T-f=maJRt-TRx=la...(2)...(1)i/y1+——...(3)=3fj,mg(forpurerolling)=MM\:mR,/+m«/JXItisthefrictionthatstartstherollingpossible.Butthefriction\canonlycomefromthenormal\forceofmagnitudeequaltothe\weightoftheobject.Sof≤im\g.2///=-\T0(noconflict)I+mR,1TR2I+mR{JJR,—0(noconflict)l¥vampkAvowplacedonaroughstrikewhichhasacoefficientofstackfncnoovt.ismadeofmoidenticaldiscsofradiusR,andasmallpinausedtohold(hetwodiscsandtoaccommodatethethreadmthegapIftheyoyotspulledhorizontallyatthecenterasshewninthefigurewithaforceT.whatisdielargestvalueofTtoensurethatpurerollingcanbemamtaroedfortheyoyoontheroughsurface?Assumethatthemassofthewholesystemofyoyoism,andthemomentofinertiaaboutthecenterofyoyois/,where/2jL.Rollingfrictioncanbeapproximatedbystaticfrictionifitispurerolling.Themassofthethreadaswellasitsthicknessisvervsmall.i)Answer:Lettherollingfrictionbe/,andwearbitrarilysetthedirectionoffrictiontopointtotheleft,LetthelinearaccelerationoftheyoyosystembeJ.theangularaccelerationbeor,andtheirdirectionsareasshowninthefollowingfigure.Wehave:T-f=maSoa=—0mfxR-TxO=IaSofxR=Ia...(2)'3**'T-faFrom(2),fxR=IT-fRmxRfxttixR'=IxT-lxffxmxR'+/x/=IxTfx(mR2+/)=/xT*0)«fxR=Ict-W*«a%***RA/ispositive,mR2+rthedirectionof/iscorrect.IxTSo/*—-/aTaFrom(2),since/ispositive,awillalsobepositive.Thedirectionofangularaccelerationisalsocorrect.TBut/fismgSo≤MM3IxTf≤p.mgThereforeT3//,mg.OtherwiserheyoyowillrubontheroughsurfaceifT3fJfmg,i.e.,purerollingwillnotbepossibleifyoupulltheyoyothreadtoohardontheroughhorizontalsurface.mR:+/mR'xTr≤w'mRmR'+WhatifR*R,?*(1):T-f=ma**aT-T=maam♦\a=0/VrSoitdoesnothavelinearmotioninitially.Thismeansthattheangulardisplacementinclockwise(positive)directionduetorotationisnotundonebythelineardisplacementfromthecenterofmassofthediscsoworkwillbedonebythefrictiontodecreasetheclockwiseangularvelocityofthedisc.SoourinitialassumptionisnotvalidthuspurerollingcannotoccurinitiallyT-f=ma_fR,-77J,=la...(2)a=R.a(3)i/+nifyRl/+mR{'*/TT**l--flif=R,frffifcr’ATWhatistheeffectduetotheroughnessofthesurface?(i)IfRrO,JR,-TxO=Ia*ik«a=JR2\/,«»\mThustherotationispositive(clockwise)asRVlongas//O.Butitwilldegeneratetopurelinearmotion(withoutrotation)if=0.Tt//Vv-A,A.-vlL(if)IfRrR2From_Fornearlysmoothsurface,jus-»0.SoT»fTherotationwillstartincounter¬clockwise.anditwillberubbingthesurfaceifm*0.Butforveryroughsurface,psoc.WewillneedalargeTtoovercomethestaticfrictiontostartthemotion(orstartacceleration).From(1)Tfsotherotationwillsoonerstartwithrubbingincounter-clockwise.Basedontheconventionsindicatedonthediagram,wehavethefollowingtwo/la.equations-7aisnegative.T-f=maJR2-rftJLja(1)(2)Wealsohavethefollowingconstraint:/≤Msm§21.AspoolofwireofmassMandradiusRisunwoundunderaconstantforceFasinFigure.Assumingthespoolisauniformsolidcylinderthatdoesn’tslip,showthat(a)theaccelerationofthecenterofmassis4F/3A/and(b)theforceoffrictionistotherightandequalinmagnitudeto/73.(c)Ifthecylinderstartsfromrestandrollswithoutslipping,whatisthespeedofitscenterofmassafterithasrottedthroughadistanced?Answer:Stepbystepreasoning...(a)F-f-Ma...(1)1FR+jR-Ia...(2)1