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# 465 Note 11 for PHYS 250 at PSU

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This 47 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 23 views.

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Date Created: 02/06/15

Chapter 11 Vibrations and Waves Units of Chapter 11 Simple Harmonic Motion Energy in the Simple Harmonic Oscillator The Period and Sinusoidal Nature of SHM The Simple Pendulum Damped Harmonic Motion Forced Vibrations Resonance Wave Motion Types of Waves Transverse and Longitudinal Units of Chapter 11 Reflection and Transmission of Waves nterference Principle of Superposition Standing Waves Resonance Refraction Diffraction Mathematica Representation of a Traveling Wave Simple Harmonic Motion Copyright 2005 Pearson Prentice Hall Inc If an object vibrates or oscillates back and forth over the same path each cycle taking the same amount of time the motion is called periodic The mass and spring system is a useful model for a periodic system Simple Harmonic Motion We assume that the surface is frictionless There is a point where the spring is neither stretched nor compressed this is the equilibrium position We measure displacement from that point X 0 on the previous figure The force exerted by the spring depends on the displacement F kx 111 Simple Harmonic Motion The minus sign on the force indicates that it is a restoring force it is directed to restore the mass to its equilibrium position k is the spring constant The force is not constant so the acceleration is not constant either F Simple Harmonic Motion a x A x0 Displacement is measured from i0 pmax the equilibrium point ma sln luve I I I d Amplitude Is the maxrmum x0 0 displacement F4 17 A cycle is a full toandfro x0 FA motion this figure shows half a cycle C Period is the time required to complete one cycle Frequency is the number of cycles completed per second Copyright 2005 Pearson Prentice Hall Inc Simple Harmonic Motion If the spring is hung 2 4060 vertically the only change i is in the equilibrium l x W position which is at the x measured point where the spring from here force equals the gravitational force L a b Copyright 2005 Pearson Prentice Hall Inc 111 Simple Harmonic Motion Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion SHM and is often called a simple harmonic oscillator Energy in the Simple Harmonic Oscillator We already know that the potential energy of a spring is given by PE kx2 The total mechanical energy is then The total mechanical energy will be conserved as we are assuming the system is frictionless 112 Energy in the Simple Harmonic Oscillator If the mass is at the limits of its motion the energy is all potential If the mass is at the equilibrium point the El VVVVV energy is all kinetic We know what the c x A x0 xA ltv0 potential energy Is at the E mv2gkx2 turning points E kA2 114a x Copyright 2005 Pearson Prentice Hall Inc 112 Energy in the Simple Harmonic Oscillator The total energy is therefore 142 And we can write mv2 kx2 kA2 114 This can be solved for the velocity as a function of position v vmax 1 115 113 The Period and Sinusoidal Nature of SHM If we look at the projection onto the X axis of an object moving in a circle of radius A at a constant speed vmax we find that the X component of its velocity varies as This is identical to SHM lt A b Copyright 2005 Pearson Prentice Hall Inc 113 The Period and Sinusoidal Nature of SHM Therefore we can use the period and frequency of a particle moving in a circle to find the period and frequency T 277391 1173 f 1 1 k 117b T 277 m 113 The Period and Sinusoidal Nature of SHM We can similarly find the position as a function of time x A cos mt 118a A 00827Tft 118b A cos2m T 118c 113 The Period and Sinusoidal Nature of SHM Paper motion E The top curve is a graph of the previous equa on The bottom curve is the same but shifted 14 period so that it is a sine function rather than a cosine Displacement x Velocity V Acceleration a T I Copyright 2005 Pearson Prentice Hall Inc 113 The Period and r Sinusoidal Nature of SHM The velocity and acceleration can be calculated as functions of time the results are below and tare plotted at left 390 vmax sin wt 119 k vmax A m a amaXCOS2wtT 1110 imam kA m 114 The Simple Pendulum A simple pendulum consists of a mass at the end of a lightweight cord We assume that the cord does not stretch and that its mass is negligible 114 The Simple Pendulum In order to be in SHM the restoring force must be proportional to the negative of the displacement Here we have F mg sin 6 which is proportional to sin 0 mgcos 6 and not to 0 itself mg sine mg TABLE 11 1 Copyrighl 2005 Pearson Prentice Hall Inc Sin 0 at Angles 0 0 degrees radians sino Difference However ifthe 0 0 o 0 1 001745 001745 0005 angle ls small 5 008727 008716 01 sin 0 z 10 017453 017365 05 15 026180 025882 11 20 034907 034202 20 30 052360 050000 47 Copyright c 2005 Faarsan Prenllne Hall Int 114 The Simple Pendulum Therefore for small angles we have mg Fz L 2c where x L9 The period and frequency are T 2wZ 1111a g 1 8 1111b 277 Ma The S pe Pemdwum Copyright 2005 Pearson Prentice Hall Inc Sea ag Hemg 31 the WM cam be mme dered maee eeg and the amp tude e ema g the per m deeg mm depemd mm the maee 115 Damped Harmonic Motion Damped harmonic motion is harmonic motion with a frictional or drag force If the damping is small we can treat it as an envelope that modifies the undamped oscillation Copyright 2005 Pearson Prentice Hall Inc 115 Damped Harmonic Motion However if the damping is large it no longer resembles B SHM at all t A underdamping there are a few small oscillations before the oscillator comes to rest C B critical clamping this is the fastest way to get to equilibrium C overdamping the system is slowed so much that it takes a long time to get to equilibrium 115 Damped Harmonic Motion There are systems where damping is unwanted such as clocks and watches Then there are systems in which it is wanted and often needs to be as close to critical damping as possible such as automobile shock absorbers and earthquake protection for buildings Attached to car frame Piston Viscous fl uid Attached to gt car axle Prenlme Hall In 116 Forced Vibrations Resonance Forced vibrations occur when there is a periodic driving force This force may or may not have the same period as the natural frequency of the system If the frequency is the same as the natural frequency the amplitude becomes quite large This is called resonance Amplitude of 116 Forced Vibrations Resonance A g A The sharpness of the EL resonant peak depends go on the damping If the E damping is small A it 39Q B can be quite sharp if C f the damping IS larger flo External frequency f Copyright 2005 Pearson Prentice Hall Inc B it is less sharp Like damping resonance can be wanted or unwanted Musical instruments and TVIradio receivers depend on it 117 Wave Motion 39 f quot 3 334 My Copyright 2005 Pearson Prentice Hall Inc Velocity of rope particle A wave travels Velociwwave along its medium 1 but the individual particles just move up and down 117 Wave Motion All types of traveling waves transport energy Study of a single wave pulse shows that it is begun with a vibration and transmitted through internal forces in the b medium Continuous waves start with vibrations too If the vibration is SHM then the wave will be sinusoidal Copyright 2005 Pearson Prentice Hall lnc 117 Wave Motion Wave characteristics Amplitude A Wavelength l Frequency fand period T Wave velocity 1112 Copyright 2005 Pearson Prentice Hall Inc 118 Types of Waves Transverse and Longitudinal F Wavelength gtl Compression Expansion b lg Wavelength at oupyrigmo zuas Pearson Frenllce Hle Inc The motion of particles in a wave can either be perpendicular to the wave direction transverse or parallel to it longitudinal 118 Types of Waves Transverse and Longitudinal Sound waves are longitudinal waves Drum 39 membrane Compress10n Expansmn 118 Types of Waves Transverse and Longitudinal Earthquakes produce both longitudinal and transverse waves Both types can travel through solid material but only longitudinal waves can propagate through a fluid in the transverse direction a fluid has no restoring force Surface waves are waves that travel along the boundary between two media g V IXr V l gt S quot f quot K Hr139 39 k gt 1111 Reflection and Transmission of Waves U l W M a b Copyright 2005 Pearson Prentice Hall Inc A wave hitting an obstacle will be A wave reaching the end of its medium but where the medium is still free to move will be reflected b and its reflection will be upright reflected a and its reflection will be inverted 1111 Reflection and Transmission of Waves Light section Heavy section Transmitted pulse Re ected pulse b Copyrighl 2005 Pearson Prentice Hall Inc A wave encountering a denser medium will be partly reflected and partly transmitted if the wave speed is less in the denser medium the wavelength will be shorter 1111 Reflection and Transmission of Waves Two or threedimensional waves can be represented by wave fronts which are curves of surfaces where all the waves have the same phase R ay Lines perpendicular to the wave fronts are called rays they point in Ra the direction of propagation of the wave 311101 QABM a b Copyright 2005 Pearson Prentice Hall Inc 1111 Reflection and Transmission of Waves The law of reflection the angle of incidence equals the angle of reflection Re ected Copyright 2005 Pearson Prentice Hall Inc 1112 Interference Principle of Superposition The superposition principle says that when two waves pass through the same point the displacement is the arithmetic sum of the individual displacements In the figure below a exhibits destructive interference and b exhibits constructive interference Pulses far apart approaching A V Time Pulses overlap precisely Pulses far apart receding V Copyright 2005 Pearson Prentice Hall Inc 1112 Interference Principle of Superposition These figures show the sum of two waves In a they add constructively in b they add destructively and in c they add partially destructively r V V VV I V VVV IVV VV 2 Vm a b c Copyright 2005 Pearson Prentice Hall Inc 1113 Standing Waves Resonance a The frequencies of the standing waves on a Lizi particular string are called resonant frequencies Fundamental 0139 rst harmonic fl HZ They are also referred to as the fundamental and First overtone or second harmonic f2 2f ha rm 0 n iCS FgM Second overtone or third harmonic f3 3f 1 13 Copyright 2005 Pearson Prentice Hall Inci 1113 Standing Waves Resonance Antinode Node b Node Antinode Copyright 2005 Pearson Prentice Hall Inc Standing waves occur when both ends of a string are fixed In that case only waves which are motionless at the ends of the string can persist There are nodes where the amplitude is always zero and antinodes where the amplitude varies from zero to the maximum value 1116 Mathematical Representation of a Traveling Wave To the left we have a snapshot of a traveling amplitude k A A wave at a single point in time Below left the same wave is shown y wave at wave at timer traveling Mathematical Representation of a Traveling Standing from Traveling Wave A full mathematical description of the wave describes the displacement of any point as a function of both distance and time yx t A 427 x 14 1122 yxt A sin27 x 14 Add them 39 39 39 39 yx t 2A sin27 x c0s27 t Standing Waves Resonance The wavelengths and frequencies of standing waves are 2L A v n z123 H fN v n v nf19 n132939 1116 Mathematical Representation of a Traveling Wave A full mathematical description of the wave describes the displacement of any point as a function of both distance and time 2 y Asin77 x mg 1122 Summary of Chapter 11 For SHM the restoring force is proportional to the displacement The period is the time required for one cycle and the frequency is the number of cycles per second Period for a mass on a spring T 271 l SHM is sinusoidal During SHM the total energy is continually changing from kinetic to potential and back Summary of Chapter 11 A simple pendulum approximates SHM if its amplitude is not large Its period in that case is T27rZ g When friction is present the motion is damped If an oscillating force is applied to a SHO its amplitude depends on how close to the natural frequency the driving frequency is If it is close the amplitude becomes quite large This is called resonance Summary of Chapter 11 Vibrating objects are sources of waves which may be either a pulse or continuous Wavelength distance between successive crests Frequency number of crests that pass a given point per unit time Amplitude maximum height of crest Wave velocity v Af

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