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CoupledPendulums

Grade(s):MS Topic:PhysicalScienceStandards:DisciplinaryCoreIdeasMS-PS3-2Developamodeltodescribethatwhenthearrangementofobjectsinteractingatadistancechanges,differentamountsofpotentialenergyarestoredinthesystem.MS-PS3-5Construct,useandpresentargumentstosupporttheclaimthatwhenkineticenergyofanobjectchanges,energyistransferredtoorfromtheobject.ScienceandEngineeringPractices:EngaginginargumentfromevidenceAskingquestionsanddefiningproblemsPlanningandcarryingoutinvestigationConstructingexplanationsanddesigningsolutions.Objective:Understandthatenergycanbetransferredfromobjecttoobject.Todeterminewhatvariablesaffecttheperiodofacomplexpendulum.Tocreateasolutiontoknockadominooverbetween3to4secondsafterahumaninteraction(afterstartingthesystem).__________________________Materials:StringNuts(orotherweights)TapeScissorsDominos(orothertippableobject)Optionalstand:96”length1/2”PVC,2elbows,2T’s,2endcaps.AdvancedPreparation:Pre-cutPVCforstandsifyouareusingstands,orpre-builtstands.

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ActivityPart1Showstudentsasinglependulum.Askwheretheenergycomestoliftupthependulum,wherewillitgowhenyouletgoofthependulum.Askwherealongthependulumspathitwouldhurtthemostifitweretocontactyou?Wouldithurtmoreifit’sgoingfastorslow?Discuss.Takeamomenttoexperimentwiththisveryquickly.Telleveryonetomakeasimplependulumbytyingaknotaroundanut.Holdingthestringandgentlyliftingthenuttothesideandlettingitgo.TellallofthestudentsNOTtomovethenutmorethanalittlebittotheside(30degrees)–youwanttokeepthestudentsfromswingingthenutwildly.Askthestudentsforobservations.Discussasaclass.Sciencebackdrop:Atthebottomofthependulum’sswing(pictured)thegravitationalpotentialenergyhasbecomekineticenergy–soit’sgoingthefastest.Becauseit’sgoingfastestatthebottom,that’salsoprobablythepointatwhichitcouldhurtyouthemost.Part2Showthecoupledpendulumsetup(pictured).Thisconsistsofonestringstrungbetweentwopoints,withtwopendulumsattachedtothestring.Askthestudentswhattheythinkwillhappenifyouliftonependulumtowardsyouandletitswing.Tellthemtomakepredictions.Asaclass,observeanddiscuss.Askthestudentswhatpushedonthestringtomakethesecondpendulummove.Whenyouholdasinglependulum,bythestringandswingit,youcanfeelthependulumgentlytugyourfingerswhenitgoesfromsidetoside.This“tugging”iswhattransfersenergytothesecondpendulum.Asktheclasswhatvariablescouldtheyalterthatmightchangethetimeittakesforenergytotransferfromonependulumtoanother.Listtheseontheboard.Ifstudentssay“changethelength,tellthemtobemorespecific.Therearemultiplelengthsineachpendulum.Thereisthelengthfromtherigidpoletothefirstpendulum,thedistancebetweenpendulums,thelengthofeachbottompendulum.Istheheightchangeimportant?Orthelengthofthestring?Thestudentscouldexperimentwithanyofthelengths.Theycouldalsoexperimentwiththesameordifferentweights.Letthestudentsknowthatrightnow,thelengtha=bandc=dandtheweightsarethesame.Tell

HighGravitationalPotentialEnergy,NoKineticenergy(notmoving),Beginsswinging

HighestKineticEnergy(mostmotion),lowestgravitationalpotentialenergy,swingsthroughthispoint(doesn’tstop)

HighGravitationalPotentialEnergy,NoKineticenergy(notmoving),reversesdirectiontoswingback

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thestudentstheyhavetwogoals(oryoucandividethisintotwoseparatelessons–eachwithonegoal):

1) Determinewhataffectstheamountoftimeittakesfortheenergytotransferfromonependulumtoanotherpendulum.Whatvariable(s)doyouneedtomodifytochangethetimeframe?Bespecific–recordwhatyoualtered,andhowitchangedthetime.Haveatleastthreepointsofdatapriortodrawinganyconclusion.

2) Designapendulumapparatusthatknocksoveradominoinbetween3-4secondsaftertheyreleasethependulum.Theywillnotbegiventhedominoinadvance,buttheywillbegivenitsdimensions.Theywillbeabletopositionthedominopriortobeginning,buttheycannottouchthedominooncethetimingbegins.Constraints:Theapparatuscan’tbewiderthan2’.Theycannottouchthependulumapparatusoncethetimingbegins.Iftheyfail,theycantrytomodifytheirapparatusinordertoachievesuccessduringasecondtrial.

Hints:Usethetapetoeasilychangethelength(s)ofthestring,andtoensurethatthetwopendulumsdon’tsliptogetherandstarthittingeachother.Usethedimensionsofthedominotocreateamodelandfigureoutwheretoputthedomino.Modifyingthisactivity:Youcansetupadditionalconstraintsontheapparatussuchasfixingthelengthofcto5”,orstatingthatthedominohastobestandingdirectlyonthefloorordesk.Youcanmakethisactivityeasierbygivingthestudentsadominotoexperimentwith.Discussion:Askthestudentswhatvariablestheychangedinordertochangethelengthoftimeittakesfortheenergytotransferfromonependulumtotheotherpendulum.Therewillprobablybedifferentanswers,sincechangingthedistancebetweenthetwopendulums(e)resultsinlengths(a)and(b)changingaswell(unlessthelengthofthefullstringismodified).BuildingtheoptionalstandThePVCstandisonewaytohangthependulum.Thiseliminatestheneedforsomeonetoholdasticktohangthependulumsoff,anditeliminatestheneedtohangthestringbetweenobjectslikechairsanddesks–althoughthatisafinewaytocarryoutthisactivity.TheoptionalstandconsistsoftwouprightpiecesofPVC,connectedacrossthetopusinganadditionalpieceofPVCalongwithtwoelbows.Atthebottomweusetwoteesaswellastwoendcaps(orconnectors)sothatitdoesn’twobble.

FurtherChallenge

aa

b

c d

e

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You’venowexperimentedwithcoupledmotions–whereonependulumpullsonanothertocreatemotion.Nowtrytodecouplethemotion.Decouplingmeansthatwhenonependulumswingsitwon’taffectthesecondpendulum.Howcanyouminimizeoreveneliminatethetransferofenergyfromonependulumtotheother?UnderlyingScience:Theenergytransferisabeatfrequency–causedbytheinteractionbetweenthetwoswings.EverytimependulumAswings,itpushesonthestring,whichinturnpushesonPendulumB–transmittingenergytopendulumB.AccordingtoNewton’s3rdlaw(equalandoppositeforce),pendulumBpushesbackonPendulumA,slowingitdownandtakingenergyfromit!Thishappensveryregularly–soregularlyitcanbecalculated–sinceitisessentiallytwopendulums–onethefulllength,andonefromwheretheindividualpendulumscontactthestring.AppendixAincludesanactivitysheettakingpeoplethroughthecalculationfortwoidenticallengthpendulumsattachedtothesamestring.Theperiodandthefrequencyofthependulumsaredeterminedbythelengthofthependulum(downwards,nottheoveralllengthofthestring).Althoughenergytransferbetweencoupledobjectscanbesignificantlyshiftedbychangingthelengthsandweightsofthependulums,inordertotrulygetridofthecoupling,youneedtostartattachingthependulumstodifferentobjects.LIGOConnection:LIGO’smirrorsarependulumshanginginsideachamber.Becausetheyarependulumstheyaresomewhatde-coupledfromthehigherfrequencygroundvibrations,buttheycanstillvibratewhenlowerfrequencyvibrationshighthechamber.LIGOworkstodecoupleallgroundmotionfromitsinstruments.LIGO’ssensitiveopticsareonafoundationthatisseparatefromthefoundationofthebuilding–butit’sconnectedbycaulking.Thishelpstodecoupletheoptics,butitdoesn’tcompletelyeliminateit.MuchofLIGO’sengineeringeffortsgointodecouplingthesensitiveinstrumentfromthesurroundingenvironment,sothatitcandetectgravitationalwaves.Resources:https://www.ligo.caltech.edu/LA/system/media_files/binaries/301/original/Coupled_pendulums_.pdf?1472486583https://www.exploratorium.edu/snacks/coupled-resonant-pendulumsSafetyConsiderations:Nutsdroppedontoesorthrowncouldbeahazard.Cutting/stabbinghazardduetoscissors.

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AppendixA

CoupledResonanceBeatFrequency

ALIGO-SECinteractiveversion3.14

GentlypushpendulumA(perpendiculartothehorizontalstring),andwatchwhathappens.EverytimependulumAswings,itpushesonthestring,whichinturnpushesonPendulumB–transmittingenergytopendulumB.AccordingtoNewton’s3rdlaw(EqualandOppositeForce),pendulumBpushesbackonPendulumA,slowingitdownandtakingenergyfromit!Thishappensveryregularly–soregularlyitcanbecalculated.PullPendulumAandPendulumBtothesamesideandletthemswing–theyshouldswingwithoutchangingtheirswing(much)–thisisastableswing.Dosteps1-3.1)Time10swings.2)You’vegottheperiodof10swings,sodivideby10togettheperiodofoneswing(thenumberofsecondsperswing).3)Invertthistogetthefrequency(thenumberofswingspersecond). #Swings TotalTime Periodof1

SwingFrequency

SameSide OppositeSide PullPendulumAandPendulumBtooppositesidesandletthemswing–theyshouldswingwithoutchangingtheirswing(much)–thisisastableswing.Dosteps1-3againforthissituation.SubtractthefrequencyforOppositesideswingingfromthefrequencyforsamesideswinging.That’sthebeatfrequency.Invert(1/frequency=period)thistofindthebeatperiod(howlongittakestomakeonecompletetransferofenergyfrompendulumAtopendulumBandbacktopendulumAagain.Checktoseeisyouwereaccuratebyswingingjustonependulum.

A) Time10(orifyou’reimpatient,just5)completeenergytransfersifyoucan.B) Dividethetimeby10(or5ifyouwereimpatient)toseeiftheactualbeatperiodmatches

yourcalculatedbeatperiod!