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CH 15
SimpleHarmonicMotion
I. Oscillatorymotion
A. Motionwhichisperiodicintime,thatis,motionthatrepeatsitselfintime.
B. Examples:
1. Powerlineoscillateswhenthewindblowspastit
2. Earthquakeoscillationsmovebuildings
C. Sometimestheoscillationsaresosevere,thatthesystemexhibitingoscillationsbreakapart.
1. TacomaNarrowsBridgeCollapse"Gallopin'Gertie"
a) http://www.youtube.com/watch?v=j‐zczJXSxnw
II. SimpleHarmonicMotion
A. http://www.youtube.com/watch?v=__2YND93ofEWatch the video in your spare time. This professor is my teaching Idol.
B. Inthefigurebelowsnapshotsofasimpleoscillatorysystemisshown.Aparticlerepeatedlymovesbackandforthaboutthepointx=0.
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C. Thetimetakenforonecompleteoscillationistheperiod,T.InthetimeofoneT,thesystemtravelsfromx=+x
m,to–x
m,andthenbackto
itsoriginalpositionxm.
D. Thevelocityvectorarrowsarescaledtoindicatethemagnitudeofthespeedofthesystematdifferenttimes.Atx=±x
m,thevelocityis
zero.
E. Frequencyofoscillationisthenumberofoscillationsthatarecompletedineachsecond.
1. Thesymbolforfrequencyisf,andtheSIunitisthehertz(abbreviatedasHz).
2. Itfollowsthat
F. Anymotionthatrepeatsitselfisperiodicorharmonic.
G. Ifthemotionisasinusoidalfunctionoftime,itiscalledsimpleharmonicmotion(SHM).
1. MathematicallySHMcanbeexpressedas:
2. Here,
a) xmistheamplitude(maximumdisplacementofthesystem)
b) tisthetime
c) wistheangularfrequency,and
d) fisthephaseconstantorphaseangle
3. Figure(a)belowplotsthedisplacementoftwoSHMsystemsthataredifferentinamplitudes,buthavethesameperiod.
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4. Figure(b)belowplotsthedisplacementoftwoSHMsystemswhicharedifferentinperiodsbuthavethesameamplitude.
a) Thevalueofthephaseconstantterm,,dependsonthevalueofthedisplacementandthevelocityofthesystemattimet=0.
5. Figure(c)belowplotsthedisplacementoftwoSHMsystemshavingthesameperiodandamplitude,butdifferentphaseconstants.
6. ForanoscillatorymotionwithperiodT,
a) Thecosinefunctionalsorepeatsitselfwhentheargumentincreasesby2.Therefore,
b) Here,istheangularfrequency,andmeasurestheangleperunittime.ItsSIunitisradians/second.Tobeconsistent,thenmustbeinradians.
)()( Ttxtx
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III. SimpleHarmonicMotionGraphs
A. ThedisplacementequationandgraphofSHM:
1.
2.
B. ThevelocityequationandgraphofSHM:
1. ( )v t
2. Themaximumvalue(amplitude)ofvelocityisxm.Thephaseshiftof
thevelocityis/2,makingthecosinetoasinefunction.
C. TheaccelerationequationandgraphofSHM:
1. ( )a t
2.
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3. Theaccelerationamplitudeis.
4. InSHMa(t)isproportionaltothedisplacementbutoppositeinsign.
IV. Newton’s2ndlawforSHM
A. SHMisthemotionexecutedbyasystemsubjecttoaforcethatisproportionaltothedisplacementofthesystembutoppositeinsign.
1. ForexampleaspringsubjecttoHooke’slaw:
2.
3. Theblock‐springsystemshownaboveformsalinearSHMoscillator.
a) Thespringconstantofthespring,k,isrelatedtotheangularfrequency,,oftheoscillator:
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B. SampleProblemforForcelaw:
1. Anoscillatorconsistsofablockofmass0.500kgconnectedtoaspring.Whensetintooscillationwithamplitude35.0cm,theoscillatorrepeatsitsmotionevery0.500s.Findthe(a)period,(b)frequency,(c)angularfrequency,(d)springconstant,(e)maximumspeed,and(f)magnitudeofthemaximumforceontheblockfromthespring.
a) Solution:
(a) T = (b) f = (c) = (d) k = (e) vm = (f) Fm =
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V. EnergyinSHM
A. Thepotentialenergyofalinearoscillatorisassociatedentirelywiththespring.
B. Thekineticenergyofthesystemisassociatedentirelywiththespeedoftheblock.
1.
C. Diagram
D. Thetotalmechanicalenergyofthesystem:
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E. SampleProblem:
1. Anoscillatingblock–springsystemhasamechanicalenergyof1.00J,anamplitudeof10.0cm,andamaximumspeedof1.20m/s.Find(a)thespringconstant,(b)themassoftheblock,and(c)thefrequencyofoscillation.
a) Solution:
(a)
(b) (c)
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I. Pendulums
A. Inasimplependulum,aparticleofmassmissuspendedfromoneendofanunstretchablemasslessstringoflengthLthatisfixedattheotherend.
B. First,let’sprovethatforasimplependulumthemotionisALSOsimpleharmonicmotionhavingthesamegeneralsolution.
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C. Second,let’sproveforasimplependulumtheAngularfrequencyandPeriodequations.
a) Thisistrueforsmallangulardisplacements,.
b) Ifwekeep<10°wemakelessthan1%error.
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D. Physicalpendulum
1. Aphysicalpendulumcanhaveacomplicateddistributionofmass.Ifthecenterofmass,C,isatadistanceofhfromthepivotpoint(figure),thenforsmallangularamplitudes,themotionissimpleharmonic.
2. Theperiod,T,is: 2I
Tmgh
a) Here,IistherotationalinertiaofthependulumaboutO.
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E. Sampleproblem:
1. InFig.below,astickoflengthL=1.85moscillatesasaphysicalpendulum.(a)Whatvalueofdistancexbetweenthestick'scenterofmassanditspivotpointOgivestheleastperiod?(b)Whatisthatleastperiod?
a) Solution:
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II. Dampedoscillation
A. Inadampedoscillation,themotionoftheoscillatorisreducedbyanexternalforce.
1. Example:Ablockofmassmoscillatesverticallyonaspringonaspring,withspringconstant,k.Fromtheblockarodextendstoavanewhichissubmergedinaliquid.Theliquidprovidestheexternaldampingforce,F
d.
2. Anotherexample,dashpot:
B. Oftenthedampingforce,Fd,isproportionaltothe1
stpowerofthe
velocityv(remembercoffeefilterdemoinDemoDay#1).Thatis,
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C. FromNewton’s2ndlaw,thefollowingDiffEqresults:
Or
D. Let’sthinkaboutthesolution:
1. Weknowthatthesolutionto 0mx kx istheSHMgeneralequation:
2. WeknowthatthedashpotordragisrobbingsomeenergysotheremustbeadecayofAmplitudeovertimetoo,yes?Thisleadsustotheform:
a) NoticethattheAmplitudeXmismultipliedbyafactor, 2( )bt
mmx e
.
Thisfactoristhedecayenvelope 2( )bt
me
(thedottedlinesonthegraph
below).
b) NoticetoothattheAngularfrequencyisnotthesame.ItisnowOmegaPrime.Wewilldiscussthismoreshortly.
3. LookatthegraphoftheCosinefunctiondecayingboundbytheasymptotesof
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a) Thefigureshowsthedisplacementfunctionx(t)forthedamped
oscillatordescribedbefore.Theamplitudedecreasesas 2( )bt
mmx e
with
time.
4. ’(dampedangularfrequency)isthenewresultantangularfrequency,andisgivenby:
22 2
2' ( )
2 4
b k b
m m m =
5. Total Energy decreases with time similar to the amplitude (but with a
different multiplier), so:
a) Since 21
2 mE U K kX
b) Then2
2 221 1( ) ( ) ( )
2 2
bt bt
m mm mE t kX e kX e
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E. DampedOscillatorsampleproblem:
1. Foradampedsimpleharmonicoscillator,theblockhasamassof1.2kgandthespringconstantis5.5N/m.Thedampingforceisgivenby–b(dx/dt),whereb=260g/s.Theblockispulleddown10.3cmandreleased.(a)Calculatethetimerequiredfortheamplitudeoftheresultingoscillationstofallto1/5ofitsinitialvalue.(b)Howmanyoscillationsaremadebytheblockinthistime?
Solution part a: Solution part b:
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III. Forcedoscillationsandresonance
A. Whentheoscillatorissubjectedtoanexternalforcethatisperiodic,theoscillatorwillexhibitforced/drivenoscillations.
1. Example:Aswinginmotionispushedwithaperiodicforceofangularfrequency,
d.
B. Therearetwofrequenciesinvolvedinaforceddrivenoscillator:
1. w,thenaturalangularfrequencyoftheoscillator,withoutthepresenceofanyexternalforce,and
2. wd,(Driveangularfrequency)theangularfrequencyoftheappliedexternalforce.
C. Resonancewilloccurintheforcedoscillationifthenaturalangularfrequency,,isequalto
d.
1. Thisistheconditionwhenthevelocityamplitudeisthelargest,andtosomeextent,alsowhenthedisplacementamplitudeisthelargest;thebelowfigureplotsdisplacementamplitudeasafunctionoftheratioofthetwofrequencies.
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D. ForcedOscillationandResonance,sampleproblem:
1. For ( ) cos( )mx t x t supposetheamplitudexmisgivenby
whereFmisthe(constant)amplitudeoftheexternaloscillatingforceexertedonthespringbytherigidsupportinFig.below.Atresonance,whatarethe(a)amplitudeand(b)velocityamplitudeoftheoscillatingobject?