Study Guide - CH. 13-14
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This 8 page Study Guide was uploaded by Stephanye Vela on Wednesday January 14, 2015. The Study Guide belongs to PHY 102 at a university taught by Dr. Neil F. Johnson in Fall. Since its upload, it has received 168 views.
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Date Created: 01/14/15
13 14 partially 19 static electricity Periodic Motion lb a motion that repeats itself over and over Period T T time reqfor 1 cycle of a periodic motion S unit secondscycle s a cycle that is an oscillation is dimensionless Frequencyf lb the frequency of an oscillation is the number of oscillations per unit of time I ftells us how frequently or rapidly an oscillation takes place higher the frequency the more rapid the oscillations 1 1 S unit cyclesecond ls 5391 if the period of an oscillation T is very small corresponding to rapid oscillations the corresponding frequency will be large as expected I 1 Hz cyclesecond Simple Harmonic Motion 39classic example is provided by the oscillations of a mass attached to a spring Consider an airtrack cart of mass m attached to a spring force constant k when the spring is at its equilibrium length neither stretched nor compressed the cart is at the position X 0 where it will remain at rest if left undisturbed F 1 the mass is at its maximum positive value of x it s velocity is O and the force on it points to the left with maximum magnitude the mass is at the equilibrium position of the spring here the speed has its maximum value and the force exerted by the spring is O I F m m the mass is at its maximum displacement in the negative x direction the velocity is 0 here and the force points to the right with maximum if 3939 magnitude F mm equilibrium the mass is at the equilibrium position of the spring with 0 force acting on it and maximum speed I I F g the mass has completed one cycle of its oscillation about x 0 ill Icimw If the cart is displaced from equilibrium by a distance x the spring exerts a restoring force given by Hooke s Law In words 39A spring exerts a restoring force whose magnitude is proportional to the distance it is displaced from the equilibrium This direct proportionality between distance from equilibrium and force is the key feature of a massspring system that leads to simple harmonic motion As for the direction of the spring 39The force exerted by a spring is opposite in direction to its displacement from equilibrium this accounts for the minus sign in F kx In general a restoring force is one that always points toward the equilibrium position Position versus Time in Simple Harmonic Motion 2n x A cos lt t A coswt SI unit m this type of dependence of time as a sine or cosine is a characteristic of simple harmonic motion Connections Between Uniform Circular Motion and Simple Harmonic Motion 39an example would be a turntable that rotates with a constant angular speed a 27tT taking the time Tto complete a revolution 39Constant Angular Speed 27139 quotT 39Angular Position 6 2 wt Position as a Function of Time 2n x A cos 6 A coswt A cos SI unit m 39when referring to simple harmonic motion or other periodic motion we have a slightly different name for a In these situations a is called the angular frequency Angular Frequency a 39Angular frequency a is 27 times the frequency SI unit rads s 1 39Velocity of an object in uniform circular motion of radius r has a magnitude equal to v no Velocity in Simple Harmonic Motion v Awsinwt SI unit ms 39Acceleration of an object in uniform circular motion has a magnitude given by acp rwz Acceleration in Simple Harmonic Motion a Aw2 coswt SI unit ms2 Acceleration and position vary with time in the same way but with opposite signs That is when the position has its maximum positive value the acceleration has its maximum negative value and so on Maximum Speed and Acceleration The maximum speed of an object in simple harmonic motion is vmax 2 Am Its maximum acceleration has a magnitude of amax sz Period of a Mass on a Spring Period of a Mass on a Spring 39the period of a mass m attached to a spring of force constant k is T 2 F 39Period increases with the mass and decreases with the spring s force constant SI unit s Factors affecting the motion of a mass on a spring the motion of a mass on a spring is determined by the force constant of the spring k the mass m and the amplitude A 0 Increasing the force constant causes the mass to oscillate with a greater frequency 0 Increasing the mass lowers the frequency of oscillation o If the force constant and the mass are both increased be the same factor the effects described above cancel resulting in no change in the motion 0 An increase in amplitude has no effect on the oscillation frequency However it will increase the maximum speed and the maximum acceleration of the mass Vertical Spring when a mass m is attached to a vertical spring it causes the spring to stretch In fact the vertical spring is in equilibrium when it exerts an upward force equal to the amount yo given by kyo mg or yo mgk Thus a mass on a vertical spring oscillates about the equilibrium point y y0 In all other respects the oscillations are the same as for a horizontal spring Energy Conservation in Oscillatory Motion In an ideal system with no friction of other nonconservative forces the total energy is conserved For example the total energy E of a mass on a horizontal spring is the sum of its kinetic energy K mvz and its potential energy U kxz Therefore E KU 1 21k2 2mv 2 x Since E remains the same throughout the motion it follows that there is a continual tradeoff between kinetic and potential energy In simple harmonic motion the total energy is proportional to the square of the amplitude of motion Total Energy U mythe total energy in simple harmonic motion is proportional to the amplitude A squared For a mass on a spring the total energy E is E 1 kA2 2 Potential Energy as a Function of Time 1 U EkA2 cos3wt Kinetic Energy as a Function of Time 1 2 2 K EkA sm wt The Pendulum The Simple Pendulum U wyconsists of a mass m suspended by a light string or rod of length L The pendulum has a stable equilibrium when the mass is directly below the suspension point and oscillates about this position if displaced from it Period of a Pendulum small amplitude L T 2n 9 SI unit s Tdepends on the length of the pendulum L and on the acceleration of gravity g It is independent however of the mass m and the amplitude A Damped Oscillations Qgtn most physical systems there is some loss of mechanical energy to friction air resistance or other nonconservative forces As the mechanical energy of a system decreases its amplitude of oscillation decreases as well This type of motion is referred to as a damped oscillation 39in a typical situation an oscillating mass may lose its mechanical energy to a force such as air resistance that is proportional the speed of the mass and opposite in direction The force in such case can be written as F bv F and v are vectors here constant b is referred to as the damping constant a measure of the strength of the damping force S unit kgs Underdamping a system of mass m and a damping constant b continues to oscillate as its amplitude steadily decreases with time The decrease in amplitude is exponential A AOe btZm Critical Damping no oscillations occur The system simply relaxes back to the equilibrium in the least possible time Overdamping an overdamped system also relaxes back to the equilibrium with no oscillations The relaxation occurs more slowly in this case than in critical damping Driven Oscillations and Resonance QgtIf an oscillating system is driven by an external force it is possible for energy to be added to the system This added energy may simply replace energy lost to friction or in the case of resonance it may result in oscillations of large amplitude and energy Natural frequency the natural frequency of an oscillating system is the frequency at which it oscillates when free from external disturbances Resonance the response of an oscillating system to a driving force of the appropriate frequency CHAPTER 14 Types of Waves Wave a disturbance that propagates from one place to another U mytransverse waves in a transverse wave the displacement of individual particles is at right angles to the direction of the propagation of the wave ex light amp radio waves Qgtlongitudinal waves in a longitudinal wave the displacement of individual particles is parallel to the direction of propagation of the wave ex sound 39a simple wave can be thought of as a regular rhythmic disturbance that propagates from one point to another repeating itself both in space and in time Crests points on a wave corresponding to maximum upward displacement Troughs points corresponding to maximum downward displacement Wavelength distance from one crest to the next or from one trough to the next is the repeat length of the wave Wavelength A A 2 distance over which a wave repeats S unit m Speed of a Wave distance A 21f time S unit ms Waves on a String 39speed of a wave is determined by the properties of the medium through which it propagates 39n the case of a string of length L there are two basic characteristics that determine the speed of a wave i tension in the string and ii the mass of the string Definition of Mass per Length 1 u 2 mass per length mL S unit kgm we expect the speed v to increase with the tension F and decrease with the mass per length u Speed of a Wave on a String v G H EI TJ S unit ms speed increases with F and decreases with 1 Reflections 39a reflected wave pulse fixed end a wave pulse on a string is inverted when it reflects from an end that is tied down 39a reflected wave pulse free end a wave pulse on a string whose end is free to move is reflected without inversion Harmonic Wave Functions 39a harmonic wave has the shape of a sine or a cosine Wave Function Qgta harmonic wave of wavelength A and period T is described by the following expression tA Zn Znt yx cos Ax T Sound Waves Speed of Sound in Air RT 20 C m 17 343 z 770 mih S unit ms speed of sound is determined in part by how stiff the material is The stiffer the material the faster the sound wave Just as having more tension in a string causes a faster wave 39the frequency of sound determines its pitch Highpitched sounds have high frequencies low pitched sounds have low frequencies Sound Intensity Qgtntensity I is a measure of the amount of energy per time that passes through a given area Since energy per time is power P the intensity of a wave is I P T A SI unit Wm2 Iloudness of a sound is determined by its intensity Intensity with Distance from a Point Source If a point source emits sound with a power P and there are no reflections the intensity a distance rfrom the source is P 47tr2 SI unit Wm2 The Doppler Effect U orchange in pitch due to the relative motion between a source of sound and the receiver Moving Observer 39an observer moving toward the source with a speed u the sound appears to have a higher speed v u As a result more compressions move past the observer in a given time than if the observer had been at rest To the observer the sound has a frequency f that is higher than the frequency of the source f f is greater than f this is the Doppler effect Doppler Effect for Moving Observer SI unit 1s s 1 in this expression u amp v are speeds and hence are always positive When an observer approaches a source the frequency heard by the observer is greater than the frequency of the source that is f gtf This means we must use the plus sign in f 1 uvf simiary we must use the minus sign when the observer moves away from the source Moving Source 39with a stationary observer and a moving source the Doppler effect is not due to the sound wave appearing to have a higher or lower speed as when the observer moves 39the speed of a wave is determined by the properties of the medium through which it propagates Thus once the source emits a sound wave it travels through the medium with its characteristic speed v regardless of what the source is doing Doppler Effect for Moving Source SI unit 1s s391 as before u amp v are positive quantities minus sign used when the source moves toward the observer plus sign when the source moves away from the observer Doppler Effect for Moving Source and Observer 39f the observer moves with a speed U0 and the source moves with a speed us the Doppler effect gives 1 amp f 55 f S 11 SI unit 1s s 1 in the numerator the plus sign corresponds to the case in which the observer moves in the direction of the source whereas the minus sign indicates motion in the opposite direction in the denominator the minus sign corresponds to the case in which the source moves in the direction of the observer whereas the plus sign indicates motion in the opposite direction Superposition amp Interference UltwgtSuperposition the combination of two or more waves to form a resultant wave Qgtlnterference constructive waves that add to give larger amplitude destructive waves that add to give a smaller amplitude
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