AP Physics Vibrations and Waves Chapter 13 Vibrations

AP Physics Vibrations and Waves

Chapter 13: Vibrations and Waves 13. 1 13. 2 13. 3 13. 4 13. 5 Simple Harmonic Motion Equations of Motion Wave Properties Standing Waves and Resonance

Homework for Chapter 13 • Read Chapter 13 • HW 13. A : pp. 447 -448: 8, 9, 11, 12, 15, 16, 17, 20, 31 -38, 41, 42. • HW 13. B: pp 449 -450: 51, 52, 54, 55, 58, 60, 62, 64, 71, 76 – 80.

Simple Harmonic Motion

13. 1: Simple Harmonic Motion • The motion of an oscillating object depends on the restoring forces that make it go back and forth. • The simplest type of restoring force is a spring force. Hooke’s Law: Fs = -kx where k is the spring constant and x is the displacement • The negative sign indicates that the force is opposite to the displacement from the springs relaxed position. • Motion under the influence of the type of force described by Hooke’s Law is called: simple harmonic motion (SHM) • It is called harmonic because the motion can be described by sines and cosines.

13. 1: Simple Harmonic Motion Fig 13. 1, p. 420. A block on a spring undergoes simple harmonic motion. a) The block is at the equilibrium position, x = 0. a) The force of a hand, Fh, pulls the block for a displacement of x = A. The force of the spring is Fs. • At the time of the release, t = 0. • The time it takes to complete one period of oscillation is T. c) At t = T/4, the block is back at the equilibrium position. d) at t = T/2, the block is at x = -A. e) During the next half of the cycle, the motion is to the right. f) At t = T, the object is back at its starting position.

13. 1: Simple Harmonic Motion displacement - the distance of an object, including direction ( x), from its equilibrium position. amplitude (A) - the magnitude of the maximum displacement of a mass from its equilibrium position. period (T) - the time needed to complete one cycle of oscillation. frequency (f) - the number of cycles per second. • frequency and period are related by: f= 1 T • The SI unit of frequency is 1/s, or hertz (Hz). This is also known as cycles per second.

13. 1: Simple Harmonic Motion The Energy and Speed of a Spring-Mass System in SHM Recall - the total potential energy stored in a spring is: U = ½ kx 2 The total kinetic and potential energies of a spring-mass system is equal to its total mechanical energy. E = K + U = ½ mv 2 + ½ kx 2 At a point of maximum displacement, (-A or +A), the instantaneous velocity is zero. Therefore all the energy at this point is potential. E = ½ m(0)2 + ½ k( A) 2 Simplifying, the total energy in SHM of a spring: E = ½ k. A 2 **Energy is proportional to the square of the amplitude**

13. 1: Simple Harmonic Motion Example 13. 1: A 0. 50 kg object is attached to a spring of spring constant 20 N/m along a horizontal frictionless surface. The object oscillates in simple harmonic motion and has a speed of 1. 5 m/s at the equilibrium position. a) What is the total energy of the system? b) What is the amplitude?

13. 1: Simple Harmonic Motion Example 13. 2: An object is attached to a spring of spring constant 60 N/m along a horizontal, frictionless surface. The spring is initially stretched by a force of 5. 0 N on the object and let go. It takes the object 0. 50 s to get back to its equilibrium position after its release. a) What is the amplitude? b) What is the period? c) What is the frequency?

1. A particle in SHM: a. has variable amplitude b. has a restoring force in the form of Hooke’s Law c. has a frequency directly proportional to its period d. has its position represented graphically by x(t) = at + b Answer: b

13. 1: Simple Harmonic Motion: Check for Understanding 2. The maximum kinetic energy of a spring-mass system in SHM is equal to: a. A b. A 2 c. k. A d. k. A 2/2 Answer: d

Simple Harmonic Motion: Check for Understanding 3. If the amplitude of an object in SHM is doubled: a. how is the energy affected? b. how is the maximum speed affected? Answer: a. Since E = ½k. A 2, the energy is four times as large. b. Since vmax = k m A , the maximum speed is twice as large.

13. 1: Simple Harmonic Motion: Check for Understanding 4. If the period of a system in SHM is doubled, its frequency is: a. doubled b. halved c. four times as large d. one-quarter as large Answer: b, because f = 1/T

13. 1: Simple Harmonic Motion: Check for Understanding 5. When a particle in SHM is at the equilibrium position, the potential energy of the system is: a. zero b. maximum c. negative d. none of the above Answer: a, because U = ½ kx 2

13. 2: Equations of Motion

13. 2: Equations of Motion Simple Harmonic Motion can be defined using a reference circle as follows: “If a particle is undergoing uniform circular motion then its projection on any diameter of its circular path performs Simple Harmonic Motion. ” http: //www. cabrillo. edu/~jmccullough/Applets/Flash/Fluids, %20 Oscillations%20 and%20 Waves/SHM 1. swf For this chapter, we will be using radians. Make sure to adjust your calculator! is measured in radians (rad) is measured in radians/second (rad/sec) A is the radius of the reference circle.

13. 2: Equations of Motion The reference circle for horizontal motion a. The shadow of an object in uniform circular motion has the same horizontal motion as the object on a spring in SHM. b. The motion equation can be written x = A cos Ɵ or x = A cos t or x = A cos (2Πft)

13. 2: Equations of Motion The reference circle for vertical motion a. The shadow of an object in uniform circular motion has the same vertical motion as the object oscillating on a spring in SHM. a. The motion equation can be written as y = A sin Ɵ or y = A sin t or y = A sin (2Πft)

13. 2: Equations of Motion equation of motion - gives the object’s position as a function of time. ex: for constant acceleration, we use kinematics formulas, such as x = vo + at. Simple harmonic motion does NOT have constant acceleration, so we can’t use kinematics equations. Recall, = 2 f = 2 T Remember to set your calculator to radians!

13. 2: Equations of Motion Observe how sinusoidal curve is traced out on the moving paper. Since the object’s initial displacement is +A, the equation can be written as y = A cos t

13. 2: Equations of Motion Two other equations of motion for an object in SHM are: velocity: v= A acceleration: a = - 2 A

13. 2: Equations of Motion damped harmonic motion - without a driving force, the amplitude or energy of an oscillating body will decrease with time.

13. 2: Equations of Motion T=2 m k Period of an object oscillating on a spring The frequency f of the oscillation is equal to 1/T. Therefore, f= 1 k 2 m or = Frequency of an object oscillating on a spring k m • Note that the time period depends on the mass of the object and the spring constant, but does not depend on the acceleration due to gravity. • The greater the mass, the longer the period. The greater the spring constant (the stiffer the spring), the shorter the period.

13. 2: Equations of Motion A simple pendulum heavy object on a string. consists of a small, For small angles of oscillation (Ɵ < 10°), a good approximation for period is: T=2 L Period of a simple pendulum g where L is the length of the string g is the acceleration due to gravity • Note that the period of the simple pendulum is independent of the mass of the bob and the amplitude of the oscillations.

13. 2: Equations of Motion Example 13. 3: An object with a mass of 1. 0 kg is attached to a spring with a spring constant of 10 N/m. The object is displaces by 3. 0 cm from the equilibrium position and let go. a) What is the amplitude A? b) What is the period T? c) What is the frequency f?

13. 2: Equations of Motion Example 13. 4: The pendulum of a grandfather clock is 1. 0 m long. a) What is its period on the Earth? b) What would its period be on the Moon where the acceleration due to gravity is 1. 7 m/s 2?

13. 2: Equations of Motion Example 13. 5: The position of an object in simple harmonic motion is described by y = (0. 25 m) sin ( /2 t). Find the a) amplitude A b) period T c) maximum speed

13. 2: Equations of Motion: Check for Understanding 1. The equation of motion for a particle in SHM a. is always a cosine function b. reflects damping action c. is independent of the initial conditions d. gives the position of the particle as a function of time Answer: d

13. 2: Equations of Motion: Check for Understanding 2. If the length of a pendulum is doubled, what is the ratio of the new period to the old one? a. 2 b. 4 c. 1/2 d. 2 Answer: d

13. 2: Equations of Motion: Check for Understanding 3. Which of the following does not affect the period of a vibrating mass on a spring? a. mass b. spring constant c. acceleration due to gravity d. frequency Answer: c

Homework for Chapter 13. 1 & 13. 2 • HW 13. A: pp. 447 -448: 8, 9, 11, 12, 15, 16, 17, 20, 31 -38, 41, 42.

13. 3: Wave Motion

Warmup: Type Casting Waves that cause a medium to be disturbed in a direction perpendicular to the direction in which the wave is traveling are called transverse waves. When the medium is disturbed in a direction parallel to the direction in which the wave is traveling, the wave is called longitudinal. *********************************************** Complete the table below by identifying each wave as being either transverse or longitudinal. Wave Source A sound wave longitudinal A water wave caused by a boat moving transverse A wave in a rope caused by one end being moved up and down transverse A wave in a coiled spring caused by pushing one end in and out repeatedly longitudinal A light wave transverse

13. 3: Wave Motion

13. 3: Wave Motion wave motion - the propagation of a disturbance (energy and momentum) through a material. • Only energy is transferred, not matter. periodic wave - requires a disturbance from an oscillation source. • If the driving source maintains constant amplitude of the wave, the result is SHM. A periodic wave can be characterized by the following: amplitude - the magnitude of displacement of the particles of the material from their equilibrium position. wavelength - the distance between two successive crests or troughs. frequency in a second. - the number of wavelengths that passes by a given point wave speed - the speed of wave motion (speed of a crest or trough) given by: v = f = /T where is wavelength, f is frequency, and T is period

13. 3: Wave Motion Example 13. 6: A student reading her physics book on a lake dock notices that the distance between two incoming wave crests is about 2. 4 m, and she then measures the time of arrival between wave crests to be 1. 6 s. What is the approximate speed of the waves? Answer: = 2. 4 m T = 1. 6 s v = /T = 2. 4 m / 1. 6 s v = 1. 5 m/s

13. 3: Wave Motion

13. 3: Wave Motion transverse wave - the particle motion is perpendicular to the direction of the wave velocity. ex: guitar string; electromagnetic wave longitudinal wave - the particle oscillation is parallel to the direction of the wave velocity. • also called a compressional wave • can propagate in solids, liquids, or gases ex: sound waves Combination of transverse and longitudinal waves: ex: seismic, water View Wave Motion: http: //paws. kettering. edu/~drussell/Demos/w aves/wavemotion. html Rarefaction

13. 3: Wave Motion: Check for Understanding

13. 4: Wave Properties

13. 4: Wave Phenomena interference - when waves meet or overlap principle of superposition - at any time, the waveform of two or more interfering waves is given by the sum of the displacements of the individual waves at each point in the medium. constructive interference - if the amplitude of the combined wave is greater than that of any of the individual waves. destructive interference - if the amplitude of the combined wave is smaller than that of any of the individual waves. Wave Interference 2: http: //zonalandeducation. com

13. 4: Wave Phenomena

total constructitve interference - when two waves of the same frequency and amplitude are exactly in phase. ex: the crest of one wave is aligned with the crest of another. total destructive interference - when two waves of the same frequency and amplitude are completely 180° out of phase. ex: the crest of one wave meets the trough of another.

13. 4: Wave Phenomena reflection - when a wave strikes and object or comes to a boundary of another medium and is at least partly bounced back. ex: an echo is a reflected sound wave ex: a mirror reflects light waves a) When a wave is reflected from a fixed boundary, the reflected wave is inverted, or undergoes a 180° phase shift. b) If the string is free to move at the boundary, there is no phase shift of the reflected wave.

$13. 4: Wave Phenomena refraction - when a wave crosses a boundary into another$

13. 4: Wave Phenomena refraction - when a wave crosses a boundary into another medium and the transmitted wave moves in a different direction. • When a wave crosses a boundary into another medium, its speed changes. • When the incident wave enters at an angle, the transmitted wave moves in a different direction. • Generally, when a wave strikes the boundary, both reflection and refraction occur. a) Refraction of water waves. As the crests approach the beach, their left edge slows as it enters the shallow water first. The whole crest rotates, approaching the beach more or less head on.

13. 4: Wave Phenomena dispersion - waves of different frequencies spread apart form one another. • Nondispersive waves travel at the same speed which is determined solely by the properties of the medium. They do not depend on the wavelength (or frequency) or the wave. ex: a wave on a string, sound • Dispersion examples: light in water, rainbows b) Surface wave traveling in deep water are dispersive. The wave speed depends on frequency or wavelength.

$13. 4: Wave Phenomena diffraction - the bending of waves around and object. ex:$

13. 4: Wave Phenomena diffraction - the bending of waves around and object. ex: A person in a room with an open door can hear sound from outside the room, partially due to diffraction. lectureonline. cl. msw. edu/~mmp/kap 13/cd 372. htm

13. 4: Wave Phenomena: Check for Understanding 1. When waves meet each other and interfere, the resultant waveform is determined by a) reflection b) refraction c) diffraction d) superposition Answer: d

$13. 4: Wave Phenomena: Check for Understanding 2. Refraction a) involves constructive interference b)$

13. 4: Wave Phenomena: Check for Understanding 2. Refraction a) involves constructive interference b) refers to a change in direction at the media interfaces c) is synonymous with diffraction d) occurs only for mechanical waves, not light Answer: b

13. 4: Wave Phenomena: Check for Understanding

13. 5: Standing Waves and Resonance

Warmup: Foot Stompin’ Physics Warmup #117 Resonance occurs when the frequency of a forced vibration on an object matches the object’s natural frequency. This causes a great increase in amplitude, which increases the power transmitted by the object. In 1940, the Tacoma Narrows suspension bridge collapsed when wind-driven oscillations produced resonance in the bridge. Films of its collapse have become favorites among physics teachers an their students. Subsequent designs have incorporated such innovations as separate parallel roadways as a way to keep this type of disaster from happening again. *********************************************** In the 1800’s, English soldiers marching across a small suspension bridge caused it to collapse when their marching set it into resonance. Their marching was in rhythm with the bridge’s natural frequency. Since that time, soldiers and marching bands have been told to not march in step when crossing any type of suspension bridge. Give another example of the disastrous effects of resonance and describe how it happens.

13. 5: Standing Waves and Resonance standing wave - occur when interfering waves of the same frequency and amplitude traveling in opposite directions, such as in a rope • Standing waves can be generated in a rope by more than one driving frequency. • The higher the frequency, the more “loops” in the rope. nodes - the points on the rope that are always stationary due to destructive interference • adjacent nodes are separated by a half wavelength, or antinodes - the points of maximum amplitude, where constructive interference is greatest • adjacent antinodes are separated by a half wavelength, or

13. 5: Standing Waves and Resonance natural frequencies or resonant frequencies which large-amplitude standing waves are produced. - the frequencies at • The resulting standing wave patterns are called normal, characteristic, or resonant, modes of vibration. • A stretched string or rope fixed at both ends can be analyzed to determine its natural frequencies. • The number of loops of a standing wave that will fit between the nodes at the ends is equal to an integral number of half-wavelengths. L = n n 2 or n = 2 L (for n = 1, 2, 3, …) n where n is the number of loops or half-wavelengths, L is the length of string, and is the wavelength • A stretched string can have standing waves only at certain frequencies. These frequencies correspond to the number of half-wavelength loops that will fit on the string between the ends.

13. 5: Standing Waves and Resonance

13. 5: Standing Waves and Resonance • The natural frequencies of oscillation for waves on a string are: fn = v = n v n 2 L = n f 1 (for n = 1, 2, 3, …) where v is the speed of waves on a string fundamental frequency (f 1) - the lowest natural frequency • All other natural frequencies are integral multiples of the fundamental frequency. fn = n f 1 ( for n = 1, 2, 3, …) harmonic series - the set of frequencies f 1, f 2=2 f 1, f 3=3 f 1, …, • f 1, the fundamental frequency, is called the first harmonic • f 2, is called the second harmonic, etc.

13. 5: Standing Waves and Resonance Natural frequencies also depend on factors such as the mass of the string and the tension.

13. 5: Standing Waves and Resonance • The natural frequencies of a stretched string can also be written as: fn = n v 2 L = n 2 L FT = n f 1 (for n = 1, 2, 3, …) • Note, the greater the linear mass density of a string, the lower its natural frequencies. ex: the low note strings on a guitar are thicker, or more massive, than the higher note strings ex: tightening a string increases all the frequencies of that string

13. 5: Standing Waves and Resonance Example 13. 7: A 50 m long string has a mass of 0. 010 kg. A 2. 0 m segment of the string is fixed at both ends and when a tension of 20 N is applied to the string, three loops are produced. What is the frequency of the standing wave?

13. 5: Standing Waves and Resonance resonance - when the driving frequency of an external source matches a natural frequency of the system. • the vibrational amplitude is greatly enhanced Pushing a person in a swing is a common example of resonance. The loaded swing, a pendulum, has a natural frequency of oscillation, its resonant frequency, and resists being pushed at a faster or slower rate. Resonance can be desirable and undesirable. Read Insight, p. 445.

13. 5: Standing Waves and Resonance: Check for Understanding 1. For two traveling waves to form standing waves, the waves must have the same a) frequency b) amplitude c) speed d) all of the preceding Answer: d

13. 5: Standing Waves and Resonance: Check for Understanding 2. When a stretched violin string oscillates in its third harmonic mode, then the standing wave in the string will exhibit a) 3 wavelengths b) 1/3 wavelength c) 3/2 wavelengths d) 2 wavelengths Answer: c

13. 5: Standing Waves and Resonance: Check for Understanding

13. 5: Standing Waves and Resonance

Homework for 13. 3, 13. 4, & 13. 5 • HW 13. B: pp 449 -450: 51, 52, 54, 55, 58, 60, 62, 64, 71, 76 – 80.