1 !ℎ= ’(∆*) 2 - bungee journal – for a community of bungee...
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LukeFarley November9,2017Partner:KarlensJoas PHYS113-05
ADynamicLookatourBungeeCord
Inthisexperiment,wesetouttomorerealisticallyinvestigatethepropertiesofourbungeecordbyconductingadynamicexperimentinwhichthecordwouldbehaveinamannersimilartothatofanactualbungeejump.Hence,wedesignedanexperimentinwhichwecouldusetheConservationofEnergytodetermineanew,dynamicbungeeconstant,whichwehypothesizedwouldbeslightlydifferentfromthestaticconstantweanalyzedinourpreviousexperiment.OurexperimentwasbuiltonthefollowingequationwhichfollowsfromtheConservationofEnergy,claimingthattheoriginalpotentialenergyduetogravityshouldbeequaltothefinalspringpotentialenergy.
𝑚𝑔ℎ =12𝑘(∆𝑥)
,
Here,histheverticaldistancebetweenthedroppointandtheheightofthemassatthebottomofthejump,and∆xisthebungeecord’sdisplacementfromequilibrium.Wedroppedamassattachedtothebungeecordandrecordedtheoriginalheighth,aswellasthecord’sdisplacement,∆x,tofindadynamicbungeeconstantk.Forthreedifferentequilibriumlengths,wedroppedfivedifferentmassesandcomputedtheaveragek-valueateachequilibriumlength.Wethenplottedthek-valuesagainsttheequilibriumlengths,andfoundamodelforktobek=47.232L2–48.402L+14.772+/-.0936N/m,wheretheuncertaintywascalculatedwithalinearregressionanalysis.However,weunderstandthatthismodelcannotbeaccurate,asitisquadratic,andthuspredicts thatkwilleventuallybegin to increaseas theequilibrium lengthbecomes longer.We believe our datamay have been skewed by the difficulty of collectingaccuratedatawithaslow-motioncameraandtape-measure.Figure1:DiagramofExperimentSet-up
Withamountclampedtothetable,wehungourbungeecordandtiedthehangingmasstoitwithasmallslipknot.Wealsohungatapemeasuredownfromthemountinordertoaccuratelymeasurethechangeinheightandthecord’sdisplacement.WeusedaniPhonecamera,specificallytheCoachMyVideoapplication,inordertocapturethepositionofthemassatthebottomofitsfall.Table1:Summaryofk-valuesbasedonequilibriumlength.Uncertaintywascalculatedbytakingtheaveragestandarderrorinalinearregressionanalysisforeachofourk-values.
EquilibriumLength(m)+/-.001m k(N/m)+/-.109115N/m
0.351 3.602323
0.448 2.567988
0.557 2.466327Chart1:TheplotofPEgrav(mgh)vs.1/2∆x^2forallthreeequilibriumlengthsL.
AsvisualizedinChart1,therelationshipbetweenthepotentialenergyduetogravityand½∆x2wasindeedfairlylinearforallthreeofourequilibriumlengths.Notice,however,thattheslopeofthelinesforequilibriumlengths.448mand.557marealmostidentical.Althoughwewouldexpecttoseethek-valuecontinuetodecreaseataconsiderablerateastheequilibriumlength
PEgrav =3.6023*1/2∆x2
PEgrav =2.568*1/2∆x2
PEgrav =2.4663*1/2∆x2
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PEgrav(J)
1/2∆x2 (m2)
PEgrav vs.1/2∆x2
L=.351m
L=.448m
L=.557m
Linear(L=.351m)
Linear(L=.448m)
Linear(L=.557m)
increases,thiswasnotthecaseforourdata,whichledtofurtherunexpectedresultsinattemptingtomodeltherelationshipbetweenkandequilibriumlength.Ourunexpectedresultsarelikelyduetohumanerrorinmeasuringdisplacementwiththeslow-motioncameraandtape-measure.Asinourpreviousstaticexperiment,weplottedtheseaveragek-valuesagainsttheirrespectiveequilibriumlengths.Chart2:PlotofkagainstL,whereListheequilibriumlength,foreachofouraveragek-values.
Afterfittingseveraldifferentmodelsonourdataandconductinglinearregressionanalyses,wefoundaquadraticequationtofitourdatabest,asseeninChart2,whichisdescribedbytheequationk=47.232L2–48.402L+14.772+/-.0936N/m.Othermodelshadhigherstandarderrors,between.25and.35N/m,anditwasalsovisuallyclearthattheydidnotfitthedataaswell.Intuitively,however,thisquadraticcannotbeanaccuratemodelfork,asthiswouldsuggestthatthek-valueswouldbegintoincreaseagainastheequilibriumlengthbecomeslonger,whichweknowshouldnothappen.Insummary,weconductedadynamicexperimentwithourbungeecordbydroppingamassattachedtothecordandrecordingthedistanceitfellaswellasthedistancethecordstretched.WeusedthisinformationalongwiththeCWEtheoreminordertofindanexpectedk-valueforourbungeecordatthreedifferentequilibriumlengths.Furthermore,inanefforttodevelopamodelforkgivenanyequilibriumlength,weplottedkagainstequilibriumlength.Sincewebelieveourquadraticmodeltobeinaccurateduetopotentialerrorindatacollectionwiththeslow-motionvideo,wewilllikelyconsiderthefindingsofourfellowclassmatesintheBungeeJournalwhenplanningourfinalbungeejump.Onmyhonor,Ihaveneithergivennorreceivedanyunacknowledgedaid.
k=47.232L2 - 48.402L+14.772
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k(N/m
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Eq(m)
kvs.L
kvs.Eq
Poly.(kvs.Eq)