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Chapter 15Oscillations and Waves

Oscillations and Waves Simple Harmonic Motion Energy in SHM Some Oscillating Systems Damped Oscillations Driven Oscillations ResonanceMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/20122

Simple Harmonic MotionSimple harmonic motion (SHM)occurs when the restoring force(the force directed toward a stableequilibrium point) is proportionalto the displacement fromequilibrium.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/20123

Characteristics of SHM Repetitive motion through a central equilibrium point. Symmetry of maximum displacement. Period of each cycle is constant. Force causing the motion is directed toward theequilibrium point (minus sign). F directly proportional to the displacement fromequilibrium.Acceleration - ω2 x DisplacementMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/20124

A Simple Harmonic Oscillator (SHO)Frictionless surfaceThe restoring force is F kx.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/20125

Two Springs with Different AmplitudesFrictionless surfaceMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/20126

SHO Period is Independent of the AmplitudeMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/20127

The Period and the Angular FrequencyThe period of oscillation isT where ω is the angular frequency ofthe oscillations, k is the springconstant and m is the mass of theblock.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/20122πω.ω km8

Simple Harmonic MotionAt the equilibrium point x 0 so, a 0 also.When the stretch is a maximum, a will be a maximum too.The velocity at the end points will be zero, andit is a maximum at the equilibrium point.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/20129

Representing Simple Harmonic MotionMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201210

Representing Simple Harmonic MotionMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201211

Representing Simple Harmonic MotionPosition - xmax AVelocity - vmax ωAAcceleration - amax ω2AMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201212

A simple harmonic oscillator can be describedmathematically by:x ( t ) Acosωtdxv (t ) -Aωsinωtdtdva (t ) -Aω 2 cosωtdtOr by:x ( t ) Asinωtdxv (t ) Aωcosωtdtdva (t ) -Aω 2 sinωtdtMFMcGraw-PHY 2425where A is the amplitude ofthe motion, the maximumdisplacement fromequilibrium, Aω vmax, andAω2 amax.Chap 15Ha-Oscillations-Revised 10/13/201213

Linear Motion - Circular FunctionsMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201214

Projection of Circular MotionMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201215

Circular Motion is the superposition of two linear SHO that are900 out of phase with each othery A sin(ωt )x A cos(ωt )MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201216

Shifting Trig Functions sin x A ωt -φ cos x AThe minus sign means that thephase is shifted to the right.{ }sin t 2πϕ cos TA plus sign indicated the phaseis shifted to the leftx Asin ωt - π2 x A ( sinωt cos π2 - sin π2 cosωt )x A ( sinωt (0) - (1)cosωt )x -AcosωtMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201217

Shifting Trig Functions π sin ωt - 02 Shifted Trig Functions1.50πωt - 02πωt 2π 11Tt ; ω 2π2ωπ TTt 2 2π 4MFMcGraw-PHY 0.001.002.00-0.50-1.00-1.50TimeChap 15Ha-Oscillations-Revised 10/13/2012183.00

EnergyMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201219

Equation of Motion & EnergyAssuming the table is frictionless: FClassic form for SHMAlso,x - kx ma xka x ( t ) - x ( t ) - ω2 x ( t )m1 21 2E K ( t ) U ( t ) mv ( t ) kx ( t )22MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201220

Spring Potential EnergyMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201221

Spring Total EnergyMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201222

Approximating Simple Harmonic MotionMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201223

Approximating Simple Harmonic MotionMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201224

Potential and Kinetic EnergyMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201225

The period of oscillation of an object in an ideal mass-springsystem is 0.50 sec and the amplitude is 5.0 cm.What is the speed at the equilibrium point?At equilibrium x 0:1 2 1 2 1 2E K U mv kx mv222Since E constant, at equilibrium (x 0) theKE must be a maximum. Here v vmax Aω.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201226

Example continued:The amplitude A is given, but ω is not.2π2π 12.6 rads/secω T0.50 sand v Aω (5.0 cm )(12.6 rads/sec ) 62.8 cm/secMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201227

The diaphragm of a speaker has a mass of 50.0 g and responds to asignal of 2.0 kHz by moving back and forth with an amplitude of1.8 10 4 m at that frequency.(a) What is the maximum force acting on thediaphragm?()22()F F ma mAω mA2πf 4πmAf maxmax22The value is Fmax 1400 N.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201228

Example continued:(b) What is the mechanical energy of the diaphragm?Since mechanical energy is conserved, E Kmax Umax.U maxK max1 2 kA21 2 mvmax2The value of k is unknown so use Kmax.K max 1 21122mvmax m( Aω ) mA2 (2πf )222The value is Kmax 0.13 J.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201229

Example: The displacement of an object in SHM is given by:y (t ) (8.00 cm )sin [(1.57 rads/sec ) t ]What is the frequency of the oscillations?Comparing to y(t) A sinωt gives A 8.00 cmand ω 1.57 rads/sec. The frequency is:ω 1.57 rads/secf 0.250 Hz2π2πMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201230

Example continued:Other quantities can also be determined:The period of the motion is2π2πT 4.00 secω 1.57 rads/secxmax A 8.00 cmvmax Aω (8.00 cm )(1.57 rads/sec ) 12.6 cm/secamax Aω 2 (8.00 cm )(1.57 rads/sec ) 19.7 cm/sec22MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201231

What About Gravity?When a mass-spring system is oriented vertically,it will exhibit SHM with the same period andfrequency as a horizontally placed system.The effect of gravity is canceled out.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201232

Why We Ignore Gravity with Vertical SpringsMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201233

The Simple PendulumA simple pendulum is constructed by attaching amass to a thin rod or a light string. We will alsoassume that the amplitude of the oscillations issmall.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201234

The Simple PendulumThe pendulum is bestdescribed using polarcoordinates.The origin is at the pivotpoint. The coordinates are(r, φ). The r-coordinatepoints from the originalong the rod. The φcoordinate is perpendicularto the rod and is positive inthe counter clockwisedirection.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201235

Fφ mg sin φ maφndApply Newton’s 2Law to the pendulumbob.2v Fr T mg cos φ m rIf we assume that φ 1 rad, then sin φ φ and cos φ 1, the angularfrequency of oscillations is then: Fφ mg sin φ maφ mLα mg sin φ mLαα ( g / L) sin φα ( g / L)φThe period of oscillations isMFMcGraw-PHY 2425ω T gL2πωChap 15Ha-Oscillations-Revised 10/13/2012 2πLg36

Example: A clock has a pendulum that performs one full swingevery 1.0 sec. The object at the end of the string weighs 10.0 N.What is the length of the pendulum?LT 2πggT 2 (9.8 m/s 2 )(1.0 s )L 0.25 m224π4π2Solving for L:MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201237

The gravitational potential energy of a pendulum isU mgy.Taking y 0 at the lowest point of the swing, show that y L(1-cosθ).θLcosθLLy L(1 cos θ )y 0MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201238

The Physical PendulumA physical pendulum is any rigid object that is free tooscillate about some fixed axis. The period ofoscillation of a physical pendulum is not necessarily thesame as that of a simple pendulum.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201239

The Physical PendulumIT 2πMgDI is the moment of inertia aboutthe given axis. The Icm from thetable will need to be modifiedusing the parallel axis theorem.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201240

Compound PendulumIT 2πMgDI Irod IdiskM mrod MdiskD distance from the axis tothe center of mass of the rodand disk.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201241

Damped OscillationsWhen dissipative forces such as friction are notnegligible, the amplitude of oscillations will decreasewith time. The oscillations are damped.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201242

Damped Oscillations Equationsd 2xdxm 2 b mω0 x 0dtdt-t x(t) A0 exp cos( ω ' t δ) 2τ 'ω ω0 b 1- 2mω0 2ω0 kmτ m; bc 2mω0bFor b bc the system is overdamped. For b bc the system iscritically damped. The object doesn’t oscillate and returns to itsequilibrium posion very rapidly.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201243

Damped OscillationsMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201244

Graphical representations of damped oscillations:MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201245

Damped Oscillations Overdamped: The system returns to equilibrium withoutoscillating. Larger values of the damping the return toequilibrium slower. Critically damped : The system returns to equilibrium asquickly as possible without oscillating. This is oftendesired for the damping of systems such as doors. Underdamped : The system oscillates (with a slightlydifferent frequency than the undamped case) with theamplitude gradually decreasing to zero.Source: Damping @ WikipediaMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201246

Damped OscillationsThe larger the damping the more difficult it is to assigna frequency to the oscillation.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201247

Damped Oscillations2EαAMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201248

Forced OscillationsMFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201249

Forced Oscillations and ResonanceA force can be applied periodically to a damped oscillator(a forced oscillation).When the force is applied at the natural frequency of thesystem, the amplitude of the oscillations will be amaximum. This condition is called resonance.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201250

Forced Oscillations Equationsd 2xdxm 2 b mω0 x F0 cosωtdtdtmafrictionspringapplied forcex Acos( ωt - δ )A F0m 2 (ω02 - ω 2 )2 b 2 ω 2MFMcGraw-PHY 2425bωtanδ m(ω02 - ω 2 )Chap 15Ha-Oscillations-Revised 10/13/201251

Energy and Resonanced 2xdxm 2 b mω0 x F0 cosωtdtdtAt resonance v and Fo are in phasevx dx -ωAsin(ωt - δ)dtv x -ωAsin(ωt --tEnergy α A2 A02 exp τ -tE 21 mω 2 A2 E0 exp τ E0 21 mω 2 A02 ; τ m bMFMcGraw-PHY 2425π2) ω Acosωtω0 mQ ω0 τ bChap 15Ha-Oscillations-Revised 10/13/201252

Power Transferω0Q ωThe dissipation in the system, represented by “b” keeps theamplitude from going to infinity.MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201253

Tacoma Narrows BridgeNov. 7, 1940MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201254

Tacoma Narrows BridgeNov. 7, 1940MFMcGraw-PHY 2425Chap 15Ha-Oscillations-Revised 10/13/201255