EXAMIIIPhysics218NameSectionNumberUSEFULINFORMATIONIff(x)==nkx’’Iff(x)=f(x)dx=1—A1)J•d=mv2()—mv2(i)IfPisconservative:J.df=—[U()—U(1)JandF=-r=rxFI=m1r?DONOTWASTETIMEDOINGARITHMETICI3.1.(25points)Derivetheexpressionsfortherandgcomponentsofthevelocityandacceleration.2.(25points)Amanofmassm1sitsonasled,massm2onthetopofa frictionlessbillofheightH.Thesledstartsdownthehillwithaninitialvelocityvodirectedtowardsthe North.a.Whatistheman’svelocityatthebottomofthehill?Call itvB.b.Atthebottomofthehillthesurfaceissolidicesothatthereisnofriction.Themanjumpsoffthesledontoanothersled,massm3whichisatrest. Theemptysledgoesoffatan angleOwithvelocityofmagnitudevl.ObtainthenecessaryequationstodeterminethepositionofthemanTsecondsafterhejumpsfromone sledtotheother?DONOTSOLVETHEEQUATIONS!‘I’9,TopV;j3.(25points)Amasslessrodcanrotatewithoutfrictionaboutaverticalaxle.AsmallmassmlisfixedtotherodadistanceHfromtheaxle.Asecondsmallmassm2isinitiallyadistanceSfromthefirst mass,asshown.STherodandthemassesaresetintomotionrotatingabout theaxlewithangularvelocityw.Att=0m2beginstomovetowardsmisothatthedistancebetweenthemis.9—ctwherecisaknownconstant.a.Whatwillbetheangularvelocityoftherodasafunctionoftimewhilem2ismovingtowardsm1?b.Whatistheforcethattherodexertsonm2whileitismoving?c.IftherodwerenotmasslessbutinsteadhadamomentofinertiaI,jabouttheaxle,whatwouldbethe angularvelocityoftherodasafunctionoftimewhilem2ismovingtowardsml?4.(25points)Anelectronwithmassmandchargeofmagnitudeqisattractedtoaproton,whichisfixedattheorigin,byaforceofmagnitudeF=742where7isaknownconstant,qisthechargeoftheprotonandristhedistanceoftheelectronfromtheorigin.a.Iftheelectronmovesinthex,yplaneinacircleofradiusR,whatisitsangularmomentumabouttheorigin?(b.Ifinsteadofmovinginacircletheelectron’spositionwasgivenbyr(t)=r(O)+c1t.6(t)0(0)+c2twherer(0),ci,6(0),andc2areknown,whatwouldbethekineticenergyoftheelectron?c.Calculatetheworkdonebytheforceexertedbytheprotoniftheelectronmovesfromthepointr=R,&=0tothepointr=2R,0=/Physics218:MechanicsExam3,20overnber2007Printyournameneatly:Lastname:Pirstname,USignyourname:Yourinstructor:______________________________Yoursection:PleasefillinyourStudentIDnumber(UIN):—————IMPORTANTReadthesedirectionscarefully:•Thereare4problemstotalling100points.Checkyour examtomakesureyouhaveallthepages.Workeachproblemonthepagetheproblemison.Youmayusetheback.Ifyouneedextrapages,Ihaveplentyupfront.•Eachproblemwithitsassociatedfigureisselfexplanatory.Ifyoumustaskaquestion,thencometothefront,beingasdiscreteaspossiblesoasnottodisturbothers.•Indicatewhatyouaredoing!‘Wecannotgivefullcreditformerelywritingdowntheanswer.Neatnesscounts!IwillgivegenerouspartialcreditifIcantellthatvrareontherighttrack.Thismeansyoumustbeneatandorganized.•Thereispotentiallyusefulinformationonthelastpage.Donotwastetimeoncomplexalgebraorarithmetic.Grading:_______________________________________ __________________________Problem1Problem2Problem3Problem4TOTALla.(20points)Derivetheexpressionsforthe1randcomponentsofthevelocityandacceleration.lb.(5points)Anobjectofmassmismovingintheiplane.Itspositionvarieswithtimesothatavectorfromtheorigintotheobjecthasalengthgivenbykt2wherekisaknownconstantandtisthetimeinseconds.Thevectorpointsatananglewiththexaxisthatvariesaccordingtobt2wherebisaknownconstant.Whatistheforce,asafunctionoftime,actingontheobject?2.(25points)Inanexperimenta smallballwithmass7bhangsfromamasslessrodoflengthS.Itstartsatrestinthepositionshown,swingsdownandstrikesasmallobject,massm1whichisatrestonafrictionlesssurface.,mba.Iftheobjectstickstotheballhowhighwilltheballgoafterstrikingtheobject?b.Ifinsteadofstickingtotheballtheobjectbouncesoffit,sothatthecollisionisperfectlyelastic,writedowntheequationsthatcouldbesolvedforhowhighwilltheballgo.DONOTSOLVETHEEQUATIONSSS3.(25points)Amanstandsonamasslessplatformthatisfreetorotateinthehorizontalplane.Heholdsaweightineachhandthathasmassm.HehashisarmsextendedsothattheyhavelengthS.Thesystemissetintorotationsothattheangularvelocityoftheplatformisw0.a.Assumetheman’smasscanbeneglectedcomparedtotheweights.Whatforcewouldhavetoheappliedtooneoftheweightsat thedistanceSsothatint0secondstheplatform,whichisinitiallyatrest,isrotatingwithangularvelocityw0?b.Assumingtheman’smasscanbeneglectedcomparedtotheweights,whatwouldbetheangularvelocityofthesystemifhestartswithangularvelocityw0andthenbringshisarmsinsothatthedistancefromhiscenterisreducedfromStoc.Ifinsteadofbeingmasslessassumethemancanbeconsideredtobeacylinderwithmomentofinertia‘manaboutanaxisthroughhiscenter.Heisagain setspinningwithhisarms,consideredmassless,extendedadistanceSandstill holdingthemasses.m.Hisinitialangillarvelocityisagainw0andhebringshisarmsinasbeforesothatthedistanceisagainreducedto.Whatwillthenewangularvelocityhe?mm4.(25points)ArodofmassMlengthSisfreetorotateintheverticalplaneaboutafrictionlesspinthroughoneend.ThemomentofinertiaabouttheendisI.Therodisuniformsothattheforceofgravityactsatits center.FrictionlessHingenifoRod,LengthSMassMxlEPointca.Whatisthetorqueexertedontherodbygravityabout thepinwhentherodishorizontal?b.Iftherightendoftherodisreleasedfromrestfindtheaccelerationdofthepointmarkedcattherightendwhentherodishorizontal.c.Whatisthetorqueexertedontherodbygravityaboutthepiiiwhentherodisvertical?d.Iftherightendoftherodisreleasedfromrestfiudtheaccelerationofthepointmarkedcat therightendwhentherodisvertical,intermsofitsangularve1ocitvEXAMHIPhysics218NameSectionNumber......USEFULINFORMATIONIff(s)=kx=nkxIff(s)=kxff(x)dx=1—IfFisconservative:-çr2JPd=-[U(i)-UQ,1)]1°1andÔU9U=ayL=ixfl=i’xFI=>mjrDONOTWASTETIMEDOINGARITHMETIC/31/1.(25points)Derivetheexpressionsforthei7-andi9componentsofthevelocityandacce1erati.Ic2.(25points)Threeblocksareslidingonafrictionlesstable.One,withmasamhasaknown velocityofmagnitudev1inthedirectiongivenbytheknown-angleD1.Thesecond,mass2m,hasanunknownvelocityofmagnitudeu2inthedirectiongivenbytheknownangle02.Athirdblock,mass3m,hasavelocityofknownmagnitudev3andmovesalongthe+xaxis.Thethreeblockscollideattheorigin, sticktogetherandgoofftogetheralongthe+xaxiswiththeunknownvelocityU..Whatwastheoriginalvelocityofthesecondblockandwhatisthemagnitudeofthefinalvelocity?1Fb.Wháiistheforceexertedonthebug
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