Oscillatorymotion2.pptx

Asimplependulumisdefinedtohaveasmallmass,alsoknownasthependulumbob,whichissuspendedfromalightwireorstring.Exploringthesimplependulumabitfurther,wecandiscovertheconditionsunderwhichitperformssimpleharmonicmotions,andwecanderiveaninterestingexpressionforitsperiod.2.3 The Simple Pendulum LetP1representthepositionofthebobwhenitisdisplacedthroughasmalldistance.Theweightmgisbrokenintocomponents.Tensioninthestringexactcancelsthecomponentparalleltothestring.Thisleavesanetrestoringforcebacktowardtheequilibriumposition.ThecomponentofmgtangentialtothearcPP1providestherestoringforce.P1P2.3 The Simple Pendulum Forsmallangles,.Fore.g.,when(about6o),adifferenceofonly0.2percent.For angle less than 15o,the difference between(in radians)and is less than 1 percent.Thus,to be a very good approximation for small angles.ApplyingtheformulaforthearcPP1,weget2.3 The Simple Pendulum Forsmallangles,then,theexpressionfortherestoringforceisThemotionisapproximatelysimpleharmonic.ComparingthisequationwiththestandardequationfromSHM,namely F=-kx,wegetThus,forangleslessthanabout15o,therestoringforceisdirectlyproportionaltothedisplacement,andthesimplependulumisasimpleharmonicoscillator.2.3 The Simple Pendulum The period of a simple pendulumisdirectlyproportionaltothesquarerootofthelengthisinverselyproportionaltotheaccelerationduetothegravityisindependentofthemassofthebobisindependentoftheamplitudeofoscillation,providedtheamplitudeissmallTheperiodinSHMisgivenby2.3 The Simple Pendulum Example 2-7 The period of a pendulum is uesd to measure the acceleration of gravityWhatistheaccelerationofgravityinaregionwhereasimplependulumhavingalength75.000cmhasaperiodof1.7357s?2.3 The Simple Pendulum SolutionSquaringEquationSolvingforg,2.3 The Simple Pendulum Question1.(a)Whatistheeffectontheperiodofapendulumofdoublingitslength?(b)Ofdecreasingitslengthtoofpreviouslength?2.3 The Simple Pendulum 2.4 Damped Harmonic Motion,Forced Oscillations And Resonance1.Damped Harmonic MotionDampingisanyeffect,eitherdeliberatelyengenderedorinherenttoasystem,thattendstoreducetheamplitudeofoscillationsofanoscillatorysystem.Anoscillator,inactualpractice,almostalwaysliesinaresistingmedium,likeair,oiletc.,wherepartofitsenergyisdissipatedinovercomingtheopposingfrictionalorviscousforcesanditsamplitude,therefore,goesondecreasingprogressively.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceIntheabsenceofanysuchforces,theoscillationswillcontinuewithoutanychangeinamplitude.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceForasystemwithasmallamountofdamping,theperiodandfrequencyarenearlythesameasforasimpleharmonicmotion,buttheamplitudegraduallydecreases.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceSincenatureoscillatingsystemaredampedingeneral,whydoweeventalkabout(undamped)simpleharmonicmotion?Theansweristhatmucheasiertodealwithmathematically.Andifthedampingisnotlarge,theoscillationscanbethoughtofassimpleharmonicmotion.Althoughfrictionaldampingdoesalterthefrequencyofvibration,theeffectisusuallysmallifthedampingissmall.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceForasystemwithdamping,thedampingforceremovesenergyfromthesystem,usuallyintheformofthermalenergy.EquationingeneralfordampingforcesasWnc=KE+PE,OrWnc=(KE+PE)WhereWnciswokdonebythedampingforce.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceIfyougraduallyincreasetheamountofdampingonasystem,theperiodandfrequencybegintobeeffected,becausedampingopposesandhenceslowsthebackandforthmotion.Ifverylargedamping,thesystemdoesnotevenoscillate-itslowlymovestowardequilibrium.2.4 Damped Harmonic Motion,Forced Oscillations And ResonancextBADisplacement versus timeC(A)The critically damped harmonic oscillator(B)The overdamped harmonic oscillator(C)The underdamped harmonic oscillatorCriticaldampingisdefinedtobetheconditioninwhichthedampingofanoscillatorreturningasquicklyaspossibletoitsequilibriumpositionwithoutbackandforththroughit.CurveBrepresentsanoverdampedsystem.Itmovesmoreslowingtowardequilibriumthanthecriticallydampingsystem.Withlessthancriticaldamping,thesystemwillreturntoequilibriumfasterbutwillovershootandcrossoverafewtimes,suchasystemisunderdamped.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceCritical damping underdampedoverdampedInmanysystems,theoscillatorymotioniswhatcounts,asinclocksandwatches,anddampingneedstobeminimized.Inothersystems,oscillationsaretheproblem,suchasacarsspring,soaproperamountofdampingisdesired.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceExample 2-8Supposea0.200kgmassisconnectedtoaspring,butwithsimplefrictionbetweenitandthesurfacehavingacoefficientoffriction=0.0800.Whattotaldistancedoesthemasstravelifitisreleased0.100mfromequilibriumstartingatv=0?Theforceconstantofthespringisk=50.0N/m.2.4 Damped Harmonic Motion,Forced Oscillations And Resonance SolutionTheworkdoneequalstheinitialstoredelasticpotentialenergy.Thatis,ButweknowWnc=fd,andweenterthefrictionasf=mg;thusCombiningthesetwoequations,weseethat2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceSolvingfordandenteringknownvaluesyields2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceQuestion1.Whatdistancewouldthemassinexample 2-8travelifthissystemhadadampingforce20timesgreater,itwerereleased0.100mfromequilibriumwithzeroinitialvelocity?Andwhichtypeofdampedharmonicoscillatoritis?2.4 Damped Harmonic Motion,Forced Oscillations And Resonance 2.Forced OscillationsIfanoscillatorisdisplacedandthenreleaseditwillbegintovibrate,itvibratesatnaturalfrequency.Ifnomoreexternalforcesareappliedtothesystemitisafree oscillator.Ifaforceiscontinuallyorrepeatedlyappliedtokeeptheoscillationgoing,itisaforced oscillator.Thenaturalfrequencyisthefrequencyatwhichasystemwouldoscillateiftherewerenodrivingandnodampingforce.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceDrag forceThe motion dies out eventually!Provide energy(force it vibrate)Add an additional force called as a driving force to a harmonic oscillator.This motion is a forced vibration.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceInaforcedoscillation,hencetheenergytransferredtotheoscillationsystem,theamplitudeofoscillationisfoundtodependonthedifferencebetweenwandw0aswellasontheamountofdamping.Theamplitudewillreachamaximumwhenthefrequencyoftheexternalforceequalsthenaturalfrequencyofthesystem.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceInthecasewithsmalldamping,theincreaseinamplitudenearf=f0isverylarge.Thiseffectisknownasresonance.The natural oscillating frequency f0 of a system is called its resonant frequency.2.4 Damped Harmonic Motion,Forced Oscillations And Resonance 3.ResonanceThephenomenonofdrivingasystemwithafrequencyequaltoitsnaturalfrequencyiscalledresonance.Asystembeingdrivenatitsnaturalfrequencyissaidtoresonate.2.4 Damped Harmonic Motion,Forced Oscillations And Resonance The natural oscillating frequency f0 of a system is called its resonant frequency.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceMagneticresonanceimagingisawidelyusedmedicaldiagnostictoolinwhichatomicnuclei,mostlyofhydrogen,aremadetoresonatebyincomingmicrowaves.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceTheresonanceisaninterestingfeatureofoscillation.Thisphenomenonattractsinterestasitmakespossibletoachieveextraordinaryresult(materialfailureoflargestructure)withsmallforce!Resonancealsoexplainswhyearthquakecausesdifferentiatingresulttodifferentstructuresmostdevastatingwhereresonanceoccurs!2.4 Damped Harmonic Motion,Forced Oscillations And Resonance The great tenor Enrico Caruso was said to be able to shatter a wineglass by singing a note of just the right frequency at full voice.2.4 Damped Harmonic Motion,Forced Oscillations And ResonanceTacomabridgewasdestroyedin1940justafterfourmonthsofitsopening.Therearedifferenttheoriesexplainingthisincident.Thecentraltothesetheoriesisresonancewhichcouldassimilateenoughforcefromthegentlebreezeandultimatelycausethematerialfailure.Windisnttheonlypotentialthreat,however.Whenanarmymarchesacrossabridge,thesoldiersoftenbreakstepsothattheirrhythmicmarchingwillstartresonatingthroughoutthebridge.Asufficientlylargearmymarchingatjusttherightcadencecouldsetthedeadlyvibrationintomotion.2.4 Damped Harmonic Motion,Forced Oscillations And Resonance