AP Physics Chapter 13 Vibrations and Waves

AP Physics Chapter 13Vibrations and Waves

Chapter 13: Vibrations and Waves 13.1 Simple Harmonic Motion 13.2 Equations of Motion 13.3 Wave Motion 13.4 Wave Properties 13.5 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.

13.1: Simple Harmonic Motion

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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. • The block is at the equilibrium position, x = 0. • 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 from Chapter 5, the total potential energy stored in a spring is: U = ½ kx2 On Gold Sheet The total kinetic and potential energies of a spring-mass system is equal to its total mechanical energy. E = K + U = ½ mv2 + ½ kx2 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 kA2 **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. • What is the total energy of the system? • What is the amplitude? • At what location are the values for the potential and kinetic energies the same?

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. • What is the amplitude? • What is the period? • What is the frequency?

13.1: Simple Harmonic Motion: Check for Understanding • A particle in SHM: • has variable amplitude • has a restoring force in the form of Hooke’s Law • has a frequency directly proportional to its period • has its position represented graphically by x(t) = at + b Answer: b

13.1: Simple Harmonic Motion: Check for Understanding • The maximum kinetic energy of a spring-mass system in SHM is equal to: • A • A2 • kA • kA2/2 Answer: d

13.1: Simple Harmonic Motion: Check for Understanding 3. If the amplitude of an object in SHM is doubled: • how is the energy affected? • how is the maximum speed affected? • Answer: • Since E = ½kA2, the energy is four times as large. • b. Since vmax = k A , the maximum speed is twice as large. • m

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 kx2

13.2: Equations of Motion

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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.” View the animation: http://137.229.52.100/physics/p103/applets/ref_circle.html 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 • The shadow of an object in uniform circular motion has the same horizontal motion as the object on a spring in SHM. • The motion equation can be written x = A cos Ɵ or x = A cost.

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

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. •The equations of motion for an object in SHM is a combination of simple harmonic and uniform circular motion. They are: y = A sin (t + ) where y is the vertical displacement (in meters) A is the amplitude (in meters)  is the angular frequency of motion (in rad/sec)  is the phase constant (in rad) Recall,  = 2f = 2 Remember to set your calculator to radians! T •  is the phase constant. It is determined by the initial displacement and velocity direction. It will help you decide whether to use the sine or cosine function to describe a particular case of SHM.

The phase difference between sine and cosine is 90° or /2.

13.2: Equations of Motion • If y=0 at t=0, and the motion is initially upward, the curve corresponds with a sine wave. • If the initial condition has positive amplitude, the wave, the curve corresponds with a cosine wave. • Here, the motion is initially downward and y = 0 at t = 0. A is negative; it is a sine wave. • Finally, the initial amplitude negative; it is a cosine wave.

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 cos (t + ) acceleration:a = -2 A sin (t + ) = - 2 y

13.2: Equations of Motion a) The mass is held, then released. b) The weight of the mass makes it drop. c) The restoring force of the spring pulls back. d)The mass is in SHM. • Velocity is /2 out of phase with displacement. • Acceleration is  out of phase with displacement.

13.2: Equations of Motion damped harmonic motion - without a driving force, the amplitude or energy of an oscillating body will decrease with time. View simulation: http://physics.bu.edu/~duffy/semester1/c19_damped_sim.html

13.2: Equations of Motion T = 2  m Period of an object oscillating on a spring k The frequency f of the oscillation is equal to 1/T. Therefore, f = 1 k Frequency of an object oscillating on a spring 2  m or  = 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 consists of a small, heavy object on a string. For small angles of oscillation (Ɵ < 10°), a good approximation for period is: T = 2  L Period of a simple pendulum On Gold Sheet 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. • What is the amplitude A? • What is the period T? • What is the frequency f?

13.2: Equations of Motion • Example 13.4: The pendulum of a grandfather clock is 1.0 m long. • What is its period on the Earth? • What would its period be on the Moon where the acceleration due to gravity is 1.7 m/s2?

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 • amplitude A • period T • maximum speed

13.2: Equations of Motion: Check for Understanding • 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

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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 - the number of wavelengths that passes by a given point in a second. 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 The rope “particles” oscillate vertically in simple harmonic motion. The distance between two successive points that are in phase (at identical points on the wave form) is the wavelength. Question: What is the phase difference between the first (red) and last (blue) waves? Answer: /2

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 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/waves/wavemotion.html Rarefaction

13.3: Wave Motion: Check for Understanding

AP Physics Chapter 13 Vibrations and Waves