5.Mastering Physics Mechanics 2 - assessed PDF

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09/04/2017

MasteringPhysics: Mechanics 2 - assessed

Question5 LearningGoal: Tounderstandthatcontactbetweenrollingobjectsandwhattheyrollagainstimposesconstraintsonthechangein position(velocity)andangle(angularvelocity). Thewayinwhichabodymakescontactwiththeworldoftenimposesaconstraintrelationshipbetweenitspossiblerotation andtranslationalmotion.Aballrollingonaroad,ayoyounwindingasitfalls,andabaseballleavingthepitcher'shandare allexamplesofconstrainedrotationandtranslation.Inasimilarmanner,therotationofonebodyandthetranslationof anothermaybeconstrained,ashappenswhenafiremanunrollsahosefromitsstoragedrum.

Situationslikethesecanbemodeledbyconstraintequations,relatingthecoupledangularandlinearmotions.Although theseequationsfundamentallyinvolveposition(theangleofthewheelataparticulardistancedowntheroad),itisusually therelationshipofvelocitiesandaccelerationsthatarerelevantinsolvingaprobleminvolvingsuchconstraints.The velocitiesareneededintheconservationequationsformomentumandangularmomentum,andtheaccelerationsare neededforthedynamicalequations. Itisimportanttousethestandardsignconventions:positiveforcounterclockwiserotationandpositiveformotiontowardthe right.Otherwise,yourdynamicalequationswillhavetobemodified.Unfortunately,afrequentresultwillbetheappearanceo negativesignsintheconstraintequations. Considerameasuringtapeunwindingfromadrumofradius . Thecenterofthedrumisnotmoving;thetapeunwindsasitsfree endispulledawayfromthedrum.Neglectthethicknessofthe tape,sothattheradiusofthedrumcanbeassumednottochange asthetapeunwinds.Inthiscase,thestandardconventionsforthe angularvelocity andforthe(translational)velocity oftheend ofthetaperesultinaconstraintequationwithapositivesign(e.g., if ,thatis,thetapeisunwinding,then also).

PartA Assumethatthefunction representsthelengthoftapethathasunwoundasafunctionoftime.Find throughwhichthedrumwillhaverotated,asafunctionoftime. Expressyouranswer(inradians)intermsof

,theangle

andanyothergivenquantities.

Hint1.Findtheamountoftapethatunrollsinonecompleterevolutionofthedrum Ifthemeasuringtapeunwindsonecompleterevolution(

),howmuchtape,

,willhaveunwound?

ANSWER: =

ANSWER: =

radians

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Correct

PartB Thetapeisnowwoundbackintothedrumatangularrate .Withwhatvelocitywilltheendofthetapemove?(Note thatourdrawingspecifiesthatapositivederivativeof impliesmotionawayfromthedrum.Becarefulwithyour signs!Thefactthatthetapeisbeingwoundbackintothedrumimpliesthat ,andfortheendofthetapeto moveclosertothedrum,itmustbethecasethat . Answerintermsof

andothergivenquantitiesfromtheproblemintroduction.

Hint1.Howtoapproachtheprobelm Thefunction expressionfor

isgivenbythederivativeof

withrespecttotime.Computethisderivativeusingthe

foundinPartAandthefactthat

Expressyouranswerintermsof

.

and .

ANSWER: =

ANSWER: =

Correct

PartC Since isapositivequanitity,theansweryoujustobtainedimpliesthat willalwayshavethesamesignas thetapeisunwinding,bothquanititeswillbepositive.Ifthetapeisbeingwoundbackup,bothquantitieswillbe negative.Nowfind ,thelinearaccelerationoftheendofthetape. Expressyouranswerintermsof

,theangularaccelerationofthedrum:

.If

.

ANSWER: =

Correct

PartD Perhapsthetrickiestaspectofworkingwithconstraintequationsforrotationalmotionisdeterminingthecorrectsignfor thekinematicquantities.Consideratireofradius rollingtotheright,withoutslipping,withconstantxvelocity .Find ,the(constant)angularvelocityofthetire.Becarefulofthesignsinyouranswer;recallthatpositiveangularvelocity correspondstorotationinthecounterclockwisedirection. Expressyouranswerintermsof

and .

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ANSWER: =

Correct Thisisanexampleoftheappearanceofnegativesignsinconstraintequationsatirerollinginthepositivedirection translationallyexhibitsnegativeangularvelocity,sincerotationisclockwise.

PartE Assumenowthattheangularvelocityofthetire,whichcontinuestorollwithoutslipping,isnotconstant,butratherthat thetireaccelerateswithconstantangularacceleration .Find ,thelinearaccelerationofthetire. Expressyouranswerintermsof and . ANSWER: =

Correct

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