DiscussionQuestions447horizontalspring.NeithermnorkchangeswhentheapparatusistakentoMars,sotheperiodisunchanged.Theonlydifferenceisthatinequilibrium,thespringwillstretchashorterdistanceonMarsthanonearthduetotheweakergravity.13.5Answer:noJustasforanobjectoscillatingonaspring,attheequilibriumpositionthespeedofthependulumbobisinstantaneouslynotchanging(thisiswherethespeedismaximum,soitsderivativeatthistimeiszero).Butthedirectionofmotionischangingbecausethependulumbobfollowsacircularpath.Hencethebobmusthaveacomponentofaccelerationperpendiculartothepathandtowardthecenterofthecircle(seeSection3.4).Tocausethisaccelerationattheequilibriumpositionwhenthestringisvertical,theupwardtensionforceatthispositionmustbegreaterthantheweightofthebob.Thiscausesanetupwardforceonthebobandanupwardaccelerationtowardthecenterofthecircularpath.13.6Answer:(i)TheperiodofaphysicalpendulumisgivenbyEq.(13.39),T=2ir\/7d.Thedistanced—LfromthepivottothecenterofgravityisthesameforboththerodandthePROBLEMSsimplependulum,asisthemassm.Thismeansthatforanydisplacementangle0thesamerestoringtorqueactsonboththerodandthesimplependulum.However,therodhasagreatermomentofinertia:‘rod=m(2I)=niL2arid‘smpIe=rnL2(allthemassofthependulumisadistanceLfromthepivot).Hencetherodhasalongerperiod.13.7Answer:(ii)Theoscillationsareunderdampedwithadecreasingamplitudeoneachcycleofoscillation,likethosegraphedinFig.13.26.Iftheoscillationswereundamped.theywouldcontinueindefinitelywiththesameamplitude.Iftheywerecriticallydampedoroverdamped,thenosewouldnotbobupanddownbutwouldreturnsmoothlytotheoriginalequilibriumattitudewithoutovershooting.13.8Answer:(i)Figure13.28showsthatthecurveofamplitudeversusdrivingfrequencymovesupwardatallfrequenciesasthevalueofthedampingconstantbisdecreased.Henceforfixedvaluesofkandm,theoscillatorwiththeleastdamping(smallestvalueofb)willhavethegreatestresponseatanydrivingfrequency.Fonstrurtor-assignedhomewargotowww.memigphysksomDiscussionQuestionsQ13.1.AnobjectismovingwithSHMofamplitudeAontheendofaspring.Iftheamplitudeisdoubled,whathappenstothetotaldistancetheobjecttravelsinoneperiod?Whathappenstotheperiod?Whathappenstothemaximumspeedoftheobject?Discusshowtheseanswersarerelated.Q13.2.Thinkofseveralexamplesineverydaylifeofmotionsthatare,atleastapproximately,simpleharmonic.InwhatrespectsdoeseachdifferfromSHM?Q13.3.DoesatuningforkorsimilartuninginstrumentundergoSHM?Whyisthisacrucialquestionformusicians?Q13.4.Aboxcontainingapebbleisattachedtoanidealhorizontalspringandisoscillatingonafriction-freeairtable.Whentheboxhasreacheditsmaximumdistancefromtheequilibriumpoint,thepebbleissuddenlyliftedoutverticallywithoutdisturbingthebox.Willthefollowingcharacteristicsofthemotionincrease,decrease,orremainthesameinthesubsequentmotionofthebox?Justifyeachanswer.(a)frequency;(b)period;(c)amplitude;(d)themaximumkineticenergyofthebox;(e)themaximumspeedofthebox.Q13.5.Ifauniformspringiscutinhalf,whatistheforceconstantofeachhalf?Justifyyouranswer.HowwouldthefrequencyofSHMusingahalf-springdifferfromthefrequencyusingthesamemassandtheentirespring?Q13.6.TheanalysisofSHMinthischapterignoredthemassofthespring.Howdoesthespring’smasschangethecharacteristicsofthemotion?Q13.7.Twoidenticalglidersonanairtrackareconnectedbyanidealspring.CouldsuchasystemundergoSHM?Explain.Howwouldtheperiodcomparewiththatofasinglegliderattachedtoaspringwhoseotherendisrigidlyattachedtoastationaryobject?Explain.Q13.8.YouarecapturedbyMartians,takenintotheirship.andputtosleep.Youawakesometimelaterandfindyourselflockedinasmallroomwithnowindows.Allthemartianshaveleftyouwithisyourdigitalwatch,yourschoolring,andyourlongsilver-chainnecklace.ExplainhowyoucandeterminewhetheryouarestillonQ13.9.ThesystemshowninFig.13.17ismountedinanelevator.Whathappenstotheperiodofthemotion(doesitincrease,decrease,orremainthesame)iftheelevator(a)acceleratesupwardat5.0m/s2;(b)movesupwardatasteady5.0m/s;(c)acceleratesdownwardat5.0m/s2?Justifyyouranswers.Q13.1O.Ifapendulumhasaperiodof2.5sonearth,whatwouldbeitsperiodinaspacestationorbitingtheearth?Ifamasshungfromaverticalspringhasaperiodof5.0sonearth,whatwoulditsperiodbeinthespacestation?Justifyeachofyouranswers.Q13.11.Asimplependulumismountedinanelevator.Whathappenstotheperiodofthependulum(doesitincrease,decrease,orremainthesame)iftheelevator(a)acceleratesupwardat5.0m/s2,(b)movesupwardatasteady5.0mIs.(c)acceleratesdownwardat5.0m/s2,(d)acceleratesdownwardat9.8m/s2?Justifyyouranswers.Q13.12.Whatshouldyoudotothelengthofthestringofasimplependulumto(a)doubleitsfrequency;(b)doubleitsperiod;(c)doubleitsangularfrequency?Q13.13.Ifapendulumclockistakentoamountaintop,doesitgainorlosetime,assumingitiscorrectatalowerelevation?Explainyouranswer.Q13.14.Whentheamplitudeofasimplependulumincreases,shoulditsperiodincreaseordecrease?Giveaqualitativeargument;donotrelyonEq.(13.35).Isyourargumentalsovalidforaphysicalpendulum?Q13.15.Whydoshortdogs(likeChihuahuas)walkwithquickerstridesthandotalldogs(likeGreatDanes)?Q13.16.Atwhatpointinthemotionofasimplependulumisthestringtensiongreatest?Least?Ineachcasegivethereasoningbehindyouranswer.Q13.17.Couldastandardoftimebebasedontheperiodofacertainstandardpendulum?Whatadvantagesanddisadvantageswouldsuchastandardhavecomparedtotheactualpresent-daystandarddiscussedinSection1.3?Q13.18.Forasimplependulum.clearlydistinguishbetweenw(theangularvelocity)andw(theangularfrequency).Whichisconstantandwhichisvariable?earthorhavebeentransportedtoMars.448CHAPTER13PeriodicMOtionQ1319.Agliderisattachedtoafixedidealspringandoscillatesonahorizontal,friction-freeairtrack.Acoinisatopthegliderandoscillatingwithit.Atwhatpointsinthemotionisthefrictionforceonthecoingreatest?Atwhatpointsisitleast?Justifyyouranswers.Q13.20.Indesigningstructuresinanearthquake-proneregion,howshouldthenaturalfrequenciesofoscillationofastructurerelatetotypicalearthquakefrequencies?Why?Shouldthestructurehavealargeorsmallamountofdamping?ExercisesSection13.1DescribingOscillation13.1.ApianostringsoundsamiddleAbyvibratingprimarilyat220Hz.(a)Calculatethestring’speriodandangularfrequency.(b)CalculatetheperiodandangularfrequencyforasopranosinginganAoneoctavehigher,whichistwicethefrequencyofthepianostring.13.2.Ifanobjectonahorizontal,frictionlesssurfaceisattachedtoaspring,displaced,andthenreleased,itwilloscillate.Ifitisdisplaced0.120rnfromitsequilibriumpositionandreleasedwithzeroinitialspeed,thenafter0.800sitsdisplacementisfoundtobe0.120montheoppositeside,andithaspassedtheequilibriumpositiononceduringthisinterval.Find(a)theamplitude:(b)theperiod;(c)thefrequency.13.3.Thetipofatuningforkgoesthr
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