CH 15 Simple Harmonic Motion

[SHIVOK SP211] October 30, 2015

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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?

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CH 15 Simple Harmonic Motion