Introductory Physics I
Introductory Physics I PHYS 250
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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 u wWWW f a 0 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 Copyright 2005 Pearson Prentice Hall Inc 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 gtF Simple Harmonic Motion a x A x0 Displacement is measured from i0 pm the equilibrium point maxlnP051the d Amplitude is the maximum x0 b displacement F4 7 A cycle is a full toandfro 397 7 motion this figure shows half a cycle Period is the time required to complete one cycle Frequency is the number of cycles completed per second 6 x A x 0 Copyright 2005 Pearson Prentice Hall Inc Simple Harmonic Motion If the spring is hung 9 vertically the only change i is in the equilibrium position which is at the point where the spring force equals the gravitational force I a b Copyright 2005 Pearson Prentice Hall Inc T x now measured from here 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 Ez kAz a 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 H VVVVV energy is all kinetic 7 iiquot We know what the potential energy is at the turning points E kA2 114a x Copyright 2005 Pearson Prentice Hall Inc 112 Energy in the Simple Harmonic Oscillator The total energy is therefore CAB And we can write va ax2 kA2 114c This can be solved for the velocity as a function of position x2 115 A2 11 vmax 1 where v zmax kmA2 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 x2 E This is identical to SHM v vmax 1 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 277 I 117a 1 1 k f 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 cos2mT 118c 113 The Period and Sinusoidal Nature of SHM l The top curve is a graph of the previous equa on lt Paper motion gtlt 0150000000000 PT 39T T T 3 The bottom curve is the same but shifted C 14 period so that it is x 39 39 a sine function rather Am than a cosine 0 l I l Velocity V Displacement x Acceleration a 113 The Period and Sinusoidal Nature of SHM The velocity and acceleration can be calculated as functions of gt Q in alt I H Nl u Jgtm H N A gt m v lt vmax g time the results are below and M are plotted at left 0 i I r WT iT TiT v vmax sin wt 119 max I i i i I i k b k I I I I I Hi A i I 3T i i I 0 I I 4 I I ta mum COS21TtT 1110 1 1 I I 2 C Copyright 2005 Pearson Prentice Hall Inc 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 mg cos 9 114 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 19 which is proportional to sin 0 mg cos 9 and not to 0 itself TABLE 1 11 Copyrighl 2005 Pearson Prentice Hall Inc Sin 0 at Angles 0 0 degrees radians sin Difference However if the 0 0 0 0 angle is small 1 001745 001745 0005 5 008727 008716 01 sin 0 z 10 017453 017365 05 15 026180 025882 11 20 034907 034202 20 30 052360 050000 47 114 The Simple Pendulum Therefore for small angles we have N mg N L x where x L0 The period and frequency are T 2w 2 1111a g 1 g 1111b 277 Mae The Simp e Pehdtthth Copyright 2005 Pearson Prentice Hall Inc Se ee ehg ee the eetet eeh he eehe deted meee eee and the emh tudle e eme g the heated deee het eeeehd eh the meee 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 A t A underdamping there are v a few small osclllatIons 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 gt gt Piston IX Viscous 39 fluid Attached to x car axle 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 oscillating system 116 Forced Vibrations Resonance to A The sharpness of the resonant peak depends on the damping If the damping is small A it B can be quite sharp if g the damping is larger f0 B it is less sharp External frequency f Copyright 2005 Pearson Prentice Hall Inc Like damping resonance can be wanted or unwanted Musical instruments and TVIradio receivers depend on it 117 Wave Motion f v 4 39 iii Copyright 2005 Pearson Prentice Hall Inc A wave travels along its medium 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 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 f and period T Wave velocity 1112 Crest 7L 39 Amplitude Trou h g 1 Copyright 2005 Pearson Prentice Hall Inc 118 Types of Waves Transverse and Longitudinal Wavelength gtl Compression Expansion 4 Cnpyvighl 2005 Pearson Prenl39 Hall The motion of particles in a wave can either be perpendicular to the wave direction transverse or parallel to it longitudinal l Wavelength 4 b 118 Types of Waves Transverse and Longitudinal Sound waves are longitudinal waves Drum I membrane Compressmn Expansmn Copyright 2005 Pearson Prenlice Hall Inc 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 V i l al Jtt 3 mmmmmmmmmmmm a mu maan Han r 1111 Reflection and Transmission of Waves U i 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 Heavy section section 7 was a Transrmtted Y 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 Lines perpendicular to the wave fronts are called rays they point in Ra the direction of propagation of the wave 311101 QABM E o 2005 Pearson Prentice Hall inc 1111 Reflection and Transmission of Waves The law of reflection the angle of incidence equals the angle of reflection Incident 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 a b gt gt lt Pulses far apart approaching Time Pulses overlap precisely gt Pulses far apart receding 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 V VV VV V VVV VV VV z 2 AVAVA a b C Copyright 2005 Pearson Prentice Hall Inc 1113 Standing Waves Resonance a The frequencies of the standing waves on a hill particular string are called resonant frequencies Fundamental or rst harmonic fl I H They are also referred to as i 2 the fundamental and First overtone or second harmonic f2 2fl O n MW Second overtone or third harmonic f3 3f 1 b Copyright 2005 Pearson Prentice Hall Incr 1113 Standing Waves Resonance Antinode Node Antinode 3 Node Antinode e c 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 y keeeexeeeea l I x To the left we have a AC V snapshot of a traveling amplitude lt x 4 wave at a single point in time Below left the same wave is shown y w vae at meta traveling l 1 0 IJ A a n 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 a n123 n fN n v nflv n132a39 H 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 2w 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 lts period in that case is T27rZ 8 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