73FE7 OSCILLATIONS OBJECTIVESAimsBy studying this chapter you can expect to understand the nature and causes of oscillations.Youwill need to learn a fair number of new terms, but some care and effortindoingthatwillbewellrewarded later because the ideas and principles introduced here can beusedtounderstandawiderange of natural phenomena.In particular,theconceptsandlanguageofoscillationsformthebasisfor understanding waves and optics, which come later in this course.MinimumlearninggoalsWhen you have finished studying this chapter you should be able to do all of the following.1. Explain, interpretand use the termsoscillation,simpleharmonicmotion[SHM],amplitude,period,frequency,angularfrequency,phase,initialphase,freeoscillation,dampedoscillation,forcedoscillation,natural frequency,restoringforce,forceconstant[springconstant],drivingforce,dampingforce,transientmotion,steadystateoscillation,resonance,resonancepeak,resonantfrequency.2. Explain free, damped and forced oscillations in terms of forces and energy transfers.3. State and apply the equations and formulae of one-dimensional SHM for(a)displacement, velocity and acceleration in terms of time;(b)restoring force and acceleration in terms of displacement;(c)frequency and angular frequency in terms of force constantandmassforanobjectonaspring.4. Describe the important properties of SHM and explain how SHMs can be combined to givemore general oscillations.5. Describe and explain the phenomenon of resonance and recognise examples of resonance. PRE-LECTURE7-1 SIMPLE HARMONIC MOTION(SHM)Oscillations(periodictoandfromotions)ofobjects occur frequently both in nature and technology.Forexample,soundwaves,waterwaves,elastic waves in stretched strings etc. are made up oflargenumbersofobjectsoscillatinginacoordinated manner.Thischapterwilldescribethevarioustypesofoscillationsintermsoftheforces and energy transfers necessary for their existence.Consider a particle moving in one dimension so that its displacement from a fixed point attimetis given byx= A cos(! t)where Aand !are constants. Thisequationdescribesasimpleharmonicmotion.Thefollowingquestions are designed to lead you through some important characteristics of SHM.FE7:Oscillations 74Q7.1i) Where is the particle at t= 0 ?ii) Sketch a displacement-time graph of the motion.Note that the graph would lookthesameifyoushiftedit either to the right or to the left by a time interval T= 2"!.iii) What is the largest value of |x | ?At what times is the distance from the centre, i.e. |x|, largest?iv) Verify that the x-component of velocity, vx=dx d t , at time t isequalto- A!sin(! t) .v) Henceverifythatthespeedissmallestwhen|x|islargestandlargestwhen|x| issmallest.(Dotheseresults agree with those obtained directly from the slopes of the displacement-time graph?)vi) Verify that the acceleration component, ax=dvxd t , at time t is -#!2cos(!t). The acceleration is proportional to the displacementxandisalwaysdirectedtowardsthecentre, x= 0.The proportionality constant is-!2; that is, the acceleration component is given by:ax=-!2x ...(7.1)vii) Show that the displacement, velocity and acceleration will all have the samevaluesagainafterafurthertimeT= 2"! . The motion is cyclic, continually repeating itself.As you will have seen in (i), at time t=0,thedisplacementisA;theparticleisatoneoftheextreme points of the oscillation.It is oftenusefultostartthedescriptionoftheoscillationatsomeother part of the cycle.Mathematically this corresponds to writingx = A cos(!t+ $ ) ... (7.2)= A cos[! (t+ $! )]orshiftingthedisplacement-timegraphadistance %t = $ /! totheleft.Sincethevelocitycomponentatanytimet isjusttheslopeofthedisplacement-timegraph,andtheaccelerationcomponent is the slope of the velocity-time graph,thevelocity-timeandacceleration-timegraphsarealso shifted to the left by the same amount, $ /!.The relationships between thesethreequantities,x, vx, and axare unaltered by the shift.(The same initial phase $appears in each.)Equation 7.2 is the most general expression for the displacement of a particleundergoingsimple harmonic motion.As the name suggests, it is the simplest form of oscillatorymotion;itsimportance lies in the fact that more complex oscillations can be studied in terms of it.TerminologySymbol Name SI unitA amplitude m!angular frequencys-l(or rad.s-l)! t+ $ phase no unit (or rad)$initial phase (i.e. phase at t = 0) no unit (or rad)T period sf frequency HzFE7:Oscillations 75Frequency, angular frequency and period for any kind of oscillation are related as follows:f =1T =!2"... (7.3). The SI unit for frequency is the hertz (Hz); 1 Hz= 1cyclepersecond.Phaseisadimensionlessquantity, so it needs no unit.However, since it is often helpful to visualise phases asangles,theyareoften given the unit radian. Note that there are 2" radians in each cycle, so a frequencyof1Hzcorresponds to an angular frequency of 2" s-l or 2" rad.s-l.Two oscillations of the same frequency are said tobeinphaseatanygiventimeiftheyhavethe same phase at that time.Otherwisetheyareoutofphasebythedifferenceoftheirphases($2 - $1).Out of phase by !/2 In phaseFigure 7.1 Oscillations in phase and out of phase LECTURE7-2 FREE OSCILLATIONSIftheoscillatingsystemisisolated(i.e.ifnoenergyisbeingaddedtoortakenawayfromthesystem)theoscillationsarecalledfreeoscillations. Oscillations (free or otherwise)canoccurwhenever a restoring force capable of transforming potential energy (PE) to kineticenergy(KE)andvice versa is present.Inafreeoscillation,sincethesumofthePEandKEcannotincrease,thePEmust be largest at the extreme points of the oscillation where the KE is zero.Examples Liquid sloshing mode - the restoring forces are due to gravity. A vibrating metal plate - elastic restoring forces. Stretched string - the restoring force is provided by tension in the string.In each of these threeexamplesalltheoscillatingparticlestogetherformedastandingwavepattern.FE7:Oscillations 76Example of a single oscillating objectA ball rolls in a curved track.When viewedfromabove,theballisexecutingaone-dimensionaloscillation.The restoring force for the horizontal oscillation is the horizontal component of theforceexerted on the ball by the track.(The ball also oscillates vertically.)WTorqueCWCVCHContact forceTorqueWTorqueCWCVCHContact forceTorqueWeightFigure 7.2 A ball oscillating on a curved trackThe ball is acted on by two forces (left), a contact force C and the gravitational force W.The force Cproduces a torque about the centre of the ball because its line of action does not pass through the centre.An equivalent representation (right ) shows the single contact force replaced by a vertical contact forceand a horizontal contact force.Since thetrackcanhaveanyshape(providedthateverypointinthemiddleislowerthantheends) there are infinitely many possible restoring force functions and thereforethereareinfinitelymany different one-dimensional oscillations.SHM - the simplest free oscillationAny oscillation can be represented as a combination of SHMs for the following reasons. Any one-dimensional oscillation (freeorotherwise)canberepresentedmathematicallyasacombination of SHMs with appropriate frequencies, amplitudes and initial phases.Anexampleisthe synthesis of the displacement-time graph of a sawtooth oscillation from SHMs. Any two- or three-dimensional oscillation can betreatedastwoorthreeone-dimensionaloscillations at right anglestooneanother,sincethemotionsinthosedirectionsareindependent.(Some examples, Lissajous figures in two dimensions, are shown in 7-7, figure 7.10.)When an objectisundergoingSHM,itsaccelerationandthetotalrestoringforceatanytimeare both proportional to thedisplacementfromthecentreandtheyarebothdirectedtowardsthecentre.Example: Body on a springConsider a body on the end of a spring, moving on a frictionless surface.FE7:Oscillations 77F Fixedpointx Figure 7.3A system which does SHMFor an ideal spring,thex-componentFoftherestoringforce isequalto-kx ,wherekisaconstant; i.e. Fis proportional to displacement and is directed towards the equilibrium pointwherex= 0.The equation of motion is

max = !kx or ax = !kmx .Comparison with equation 7.1 gives the natural frequency of oscillation:

!N =kmor fN =12"km

. ... (7.4).EnergyIn a free oscillation, since the only force doing work is conservative; the total mechanicalenergy(KE+ PE) of the system is a constant.KEPEKE = 0PE(x)Totalmechanicalenergy-AA0xFigure 7.4Energy of a free oscillationThe curve shows the potential energy (PE).The line across the top represents the level of the constantmechanical energy which is also the maximum value of PE.As PE increases, KE decreases.Whendisplacement is an extreme value, x = A or x = -A, KE is zero and PE is at its maximum.For SHMthe curve is a parabola; for other oscillations the shape is different but the energy relations shown hereare the same.At the extreme pointsoftheoscillation,|x|=A,theKEiszeroandthetotalmechanicalenergy is equal to the PE.In the specialcaseofSHMtherestoringforceisproportionaltothedisplacement so the force-displacement curve is a straight line(see5-6,figure5.6).Fromtheareaunder the curve, a triangle, the PE when x= Ais:FE7:Oscillations 78maximum PE =12 A (kA) =12 kA2 =12m!N2A2 .The energy of the oscillation depends on the square of the amplitude A. Thisresultistrueforall SHMs.7-3 DAMPED OSCILLATIONSIf the total mechanical energy of an oscillating system were conserved,thesystemwouldoscillateindefinitelywiththesameamplitude.Inanyrealsituation,however,therearealwaysnon-conservative forces such as friction or dragforcespresent.Thesedissipativeforcesdecreasethesystem's total mechanical energy and can be either internal or externaltothesystem.Theamplitudewill gradually decrease and the oscillations will die out. Anoscillationwhichdiesawayisanexample of a transient motion.Examples include pendulums with damping forces.DisplacementTimeFigure 7.5 A damped oscillationThe amplitude decreases with time and the system loses mechanical energy.The frequency, fD, of a damped system is always less than fN,thenaturalfrequencythatthesystem would have if the damping forcescouldberemoved,sincethedampingforcesalwaysacttoretard the motion.7-4 FORCED OSCILLATIONSWhen non-conservative forces are present, anoscillationcandieoutunlessenergyiscontinuallysupplied to the system by a driving force.An oscillation which iskeptgoingbyaperiodicdrivingforce is called a forced oscillation.An example is a car engine.If thedrivingforceissinusoidal(i.e.ifagraphofthedrivingforceagainsttimelookslikeasine curve, perhaps shifted along the time axis), there will be a steady state oscillation setupatthefrequency,fF, of the driving force.In general,fF will be quite different from fD.For a given amplitude of driving force, the amplitude of the steadystateoscillationdependsonthe driving frequency fF.An example is a vibrating metal rule driven by an electromagnet.Each time the frequency of the driving force is changed, the new oscillation takesawhiletobecome stable, while the transient oscillations die away.Asthedrivingfrequencyisincreasedfromzero, the amplitude of the steady state oscillation gradually increasesuntilitreachesapeakandthendies away again (figure 7.6). Thisphenomenoniscalledresonance.Thefrequency(fF)ofthedriving force atwhichtheamplitudeoftheoscillationisamaximumiscalledtheresonancefrequency, fR, and the region of the graph near fR is called the resonance peak.FE7:Oscillations 79Amplitude ofthe steady stateoscillationResonance peakffRFFigure 7.6Response curve for a resonanceFor lightly damped systems, fN,fD andfR are all approximately equal. Atresonancethedriving force adds to the restoring force in thedampedsystem,producinglargeaccelerationsandoscillations.Examples of forced oscillations Shivering to heat the body in response to cold. Expansion of the chest in breathing. Beating of the heart and dilation and contraction of the arteries.In each case the driving force is provided by the muscles.Whenthesesystemsrelaxfromhigh to low potential energystates,theirmechanicalenergyisdecreasedbytheworkdonebynon-conservative forces. The systems are highly damped and require the driving force tocausethemotion. POST-LECTURE7-5 QUESTIONSQ7. 2 List the forces that are acting on the following oscillating objects and in each case statewhethertheforcesare restoring, damping or driving forces:i) a tree swaying in the breeze,ii)the earth vibrating after an earthquake,iii) a child on a swing,iv)a vibrating eardrum,v)a marker buoy bobbing in the water,vi)the atoms in a solid when a sound wave passes through.Simple harmonic motionIngeneralonecanobtainanorderofmagnitudeestimateforthesquareofthenaturalangularfrequency (!) of an oscillation by dividing an order of magnitude estimateoftherestoringforceperFE7:Oscillations 80displacement (k)by the mass (m) of theobjectinmotion.ThisrelationisexactforSHM(seeequation 7.4) but it also gives results which are roughly OK for other systems.Q7. 3 When a spring is stretched by a pair of 2.4 N forces its length increasesby62mm.Predicttheperiodofoscillation when an object of mass 0.32 kg is hung from the springandsetvibrating.Whatwillhappento the period if the object is replaced by one with 10 times the mass?Q7. 4Awoodenrectangularpontoonofdensity& =0.50 ' 1 03kg.m-3 floatsinwater(density &W=1. 00'103kg. m-3) sothattwofaces, eachofareaA=20m2,arehorizontal.The vertical sides each have a length L = l.0 m.Figure 7.7i) How far is the bottom face below the water at equilibrium?(You may neglect the weightofthedisplacedair in this problem.)ii) Whatisthetotalforceactingonthepontoonwhenitisasmalldistancexbelowtheequilibriumposition with the same two faces horizontal?iii) What would this force be if the pontoon were a distance x above the equilibrium position?iv) Hence show that if the pontoon were released from a position other than its equilibriumpositionitwouldexecute SHM in the vertical direction if no non-conservative forces were present.(Whatnon-conservativeforces would you expect to be present and what effect would they have?)v) What is the natural frequency of the above SHM ?Q7. 5 Use energy considerations to findthemaximumspeedofanobjectundergoingSHMintermsoftheamplitude and natural (angular) frequency of the motion.Q7. 6Asomewhatbizarreproposalforarapidtransit system betweenBostonandWashington(a distance of 630 km)madeuseoftheEarth'sgravitationalattraction.(ScientificAmerican,August1965,pp30-40.) Strippedofitstechnicalcomplications,theproposalhadacapsuletravellingthroughanevacuatedtunneldrilled in a straight line between those cities.Forthefirsthalfofitsjourney,thecapsulewould gradually be getting closer to thecentreoftheearth(i.e.slidingdownanincline);forthesecond half it would betravellingawayfromtheEarth's centre.BostonWashingtonCentre ofthe EarthFigure 7.8In the idealised case where non-conservative forces are neglected, the potential energies at thebeginningand the end of the journey would be equal.(Intheactualproposal,thecapsulewashelpedalongbycompressed air to overcome the non-conservative forces.) In this idealisedcase,theaccelerationofthecapsule would bea= -C xwherex isthedisplacementfromthemidpointofthejourneyandCisaconstantequalto1.55 '10-6s-2.The capsule would therefore be undergoing part of a SHM.i) How long would the one way trip from Boston to Washington take ?ii) What would be the maximum speed of the capsule on the journey ?7-6 FURTHER DISCUSSION OF RESONANCEIn the video lecture it was stated that the Tacoma Narrows Bridge collapse wascausedbyresonance.The driving force in that case, however, was notsinusoidal nor did it oscillate at thefrequencyofthesteady state oscillations of thebridge.Infactthedrivingforcewasprovidedbyawindblowinginonedirectionfor a time long compared to the period of the bridge's oscillation.Howthendoesresonance occur ?FE7:Oscillations 81Anydrivingforcewhatevermaybeconsideredmathematicallytobeacombinationofsinusoidal driving forces with appropriate frequencies, amplitudes and initialphases.TheprocedureforsplittingupaforceinthiswayiscalledFourieranalysis(aftertheFrenchmathematicalphysicist whodeveloped it to study the flow of heat through metals). Each of these sinusoidalforces can be thought of as acting on the damped oscillating system independently of the othersinusoidal components setting up a sinusoidal steady state oscillation at its own frequency.Theseoscillationsadduptogivetheactualoscillationcausedbythefulloriginaldrivingforce.Schematically, this procedure can be summarised as shown in figure 7.9.+ ++ + + + . . . . . . Fulldrivingforce Fulloscillation ! ! Sum of sinusoidaldriving forces Sum of sinusoidalsteady state oscillations is equivalent toFigure 7.9Resonant response to a complex driving forceThe ratio of the amplitude of a steady state oscillation to that of thesinusoidaldrivingforcethat caused it is largest when the driving force has afrequencyequaltotheresonancefrequency.Quite oftentherefore,thecombinedoscillationisdominatedbythesteadystateoscillationswithfrequency approximately equal to fR, particularly if the resonance curve is very sharply peaked.These ideas are the basis of radioandTVreceivers,forexample.Thedrivingforceisthecomplicatedelectricalsignal(thesumofallthesignalssentoutbythevariousstationsplusinterference from fluorescent lights, spark plugsetc.)fromtheaerialorantenna.Thetuningcircuitof a radio or TVsetisanoscillatingsystemwhoseresonancefrequencycanbevariedtoselectthesteady state oscillations at the frequency of the transmitting station.The resonance peak forthesetuning circuits has to be sufficiently narrow that only one station is transmitting atafrequencywithin the peak at any given setting.Inthebridgecollapse,thedrivingforceprovidedbythewindhadalargesinusoidalcomponent at one of the natural frequencies of oscillation of the bridge. Thiscomponentcausedthebridge to resonate and ultimately break apart.Q7. 7 Can you think of any other situations in which resonance plays an important role ?FE7:Oscillations 827-7 APPENDIX: LISSAJOUS FIGURESWhen two SHMs atrightanglesarecombined,thepathtracedoutwillbeaLissajousfigureiftheratio of the two frequencies is equal to a ratio of two integers. Theshapeisalsoaffectedbythephase difference between the oscillations.11/21/32/33/43/54/55/6Figure 7.10 Some Lissajous figuresThe number at the left of each row is the ratio of the frequency of the vertical oscillation to that of thehorizontal oscillation for the figures in that row.The figures in each row correspond to different initialphases of these two oscillations.