Chapter 13: Vibrations and Waves Suggested homework problems:12,33,47,54,58 Hooke’s Law Hooke’s law and oscillation A simple example of vibration motion: an object attached to a spring. No friction The negative sign means that the force exerted by the spring is always directed opposite the displacement of the object (restoring force). A restoring force always pushes or pulls the object toward the equilibrium position.

Hooke’s Law No friction A simple example of vibration motion: Hooke’s law and oscillation (cont’d) No friction A simple example of vibration motion: an object attached to a spring. Suppose the object is initially pulled a distance A to the right and released from rest. Then the object does simple harmonic motion. Simple harmonic motion occurs when the net force along the direction of motion obeys Hooke’s law – when the net force is proportional to the displacement from equilibrium point and is always directed toward the equilibrium point. periodic motion

Hooke’s Law No friction A simple example of vibration motion: Hooke’s law and oscillation (cont’d) No friction A simple example of vibration motion: an object attached to a spring. Terminology: - The amplitude A is the maximum distance of the object from its equi- librium position. –A<= x <= A. - The period T is the time it takes the object to move through one complete cycle of motion from x=A to x=-A and then back to x=A. - The frequency f is the number of complete cycles or vibrations per unit time. f=1/T. periodic motion

Hooke’s Law Example 13.1 : Measuring the spring constant Hooke’s law and oscillation (cont’d) Example 13.1 : Measuring the spring constant Harmonic oscillator equation

Elastic Potential Energy The energy stored in a stretched or compressed spring The conservation of energy: If there are only conservative forces If there are also non-conservative forces

Elastic Potential Energy Conservation of energy : - The velocity is zero at x=+A,-A. - The velocity is at its maximum at x=0

Comparing Simple Harmonic Motion with Uniform Circular Motion A circular motion and its projection As the turntable rotates with constant angular speed, the shadow of the ball moves back and forth with simple harmonic motion. c.f. simple harmonic motion

Comparing Simple Harmonic Motion with Uniform Circular Motion Period and Frequency Period of oscillation (T) One period is completed when the ball rotates 360o and moves a distance v0T. From conservation of energy, for simple harmonic oscillation of a spring system with the spring constant k at x=0,

Comparing Simple Harmonic Motion with Uniform Circular Motion Period and Frequency Frequency (f) and angular frequency (w) Frequency is how many complete rotations/cycles a simple harmonic oscillation or uniform circular motion makes per unit time. Units : hertz (Hz) = cycles per second Angular frequency is a frequency measured in terms of angle.

Position, Velocity, and Acceleration as a Function of Time x vs. time We can obtain an expression for the position of an object with simple harmonic motion as a function of time. if constant angular speed

Position, Velocity, and Acceleration as a Function of Time v vs. t We can obtain an expression for the velocity of an object with simple harmonic motion as a function of time. if constant angular speed

Position, Velocity, and Acceleration as a Function of Time Period and Frequency We can obtain an expression for the position of an object with simple harmonic motion as a function of time. sinusoidal if constant angular speed

Motion of a Pendulum Pendulum If a force is a restoring one, from an analogy of a Hooke’s law we can prove that the system under influence the force makes simple harmonic oscillation. In an analogy to Hooke’s law Ft=-kx, Ft= The motion of a pendulum is not simple harmonic in general but it is if the angle q is small.

Motion of a Pendulum Physical pendulum In general case, the argument for a pendulum system of a mass attached to a string can be used to an object of any shape. I: moment of inertia For a simple pendulum,

Damped Oscillation Oscillation with friction In any real systems, forces of frictions retard the motion induced by restoring forces and the system do not oscillate indefinitely. The friction reduces the mechanical energy of the system as time passes, and the motion is said to be damped.

Waves The world is full of waves: sound waves, waves on a string, Examples and sources of waves The world is full of waves: sound waves, waves on a string, seismic waves, and electromagnetic waves such as light, radio waves, TV signals, x-rays, and g-rays. Waves are produced by some sort of vibration: Vibration of vocal cords, guitar strings, etc sound Vibration of electrons in an antenna, etc radio waves water waves Vibration of water Types of waves Transverse waves The bump (pulse) travels to the right with a definite speed: traveling wave Each segment of the rope that is disturbed moves in a direction perpendicular to the wave motion: transverse wave

Waves Longitudinal waves Types of waves (cont’d) Longitudinal waves The elements of the medium undergo displacements parallel to the direction of wave motion: longitudial wave Their disturbance corresponds to a series of high- and low- pressure regions that may travel through air or through any material medium with a certain speed. sound wave = longitudinal wave C = compression R = rarefaction

Waves Types of waves (cont’d) Longitudinal-transverse waves

Frequency, Amplitude, and Wavelength Consider a string with one end connected to a blade vibrating according to simple harmonic oscillation. Amplitude A: The maximum distance the string moves. Wavelength l: The distance between two successive crests Wave speed v: v=Dx/Dt=l/T (wavelength/period) Frequency f: v=l/T=fl (wavelength/period)

Frequency, Amplitude, and Wavelength Examples Example 13.8: A traveling wave A wave traveling in the positive x-direction. Find the amplitude, wavelength, speed, and period of the wave if it has a frequency of 8.00 Hz. Dx=40.0 cm and Dy=15.0 cm.

Frequency, Amplitude, and Wavelength Examples (cont’d) Example 13.9: Sound and light A wave has a wavelength of 3.00 m. Calculate the frequency of the wave if it is (a) a sound wave, and (b) a light wave. Take the speed of sound as 343 m/s and that of light as 3.00x108 m/s. (a) (b)

Speed of Waves on Strings Two types of speed: The speed of the physical string that vibrates up and down transverse to the string in the y-direction The rate at which the disturbance propagates along the length of the string in the x-direction: wave speed For a fixed wavelength, a string under greater tension F has a greater wave speed because the period of vibration is shorter, and the wave advances one wavelength during one period. A string with greater mass per unit length m (linear density) vibrates more slowly, leading to a longer period and a slower wave speed. Dimension analysis: [F]=ML/T2, [m]=M/L, F/m=L2/T2, [qF/m]=L/T=[v]

Speed of Waves on Strings Example 13.10 A uniform string has a mass M of 0.0300 kg and a length L of 6.00 m. Tension is maintained in the string by suspending a block of mass m = 2.00 kg from one end. (a) Find the speed of the wave. (b) Find the time it takes the pulse to travel from the wall to the pulley.

Interference of Waves Superposition principle Tow traveling waves can meet and pass through each other without being destroyed or even altered. When two or more raveling waves encounter each other while moving through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point. Interference constructive interference (in phase) destructive interference (out of phase)

Interference of Waves Example 13.10 A uniform string has a mass M of 0.0300 kg and a length L of 6.00 m. Tension is maintained in the string by suspending a block of mass m = 2.00 kg from one end. (a) Find the speed of the wave. (b) Find the time it takes the pulse to travel from the wall to the pulley.

Reflection of Waves Reflected wave is inverted Reflection of waves at a fixed end Reflected wave is inverted

Reflection of Waves Reflected wave is not inverted Reflection of waves at a free end Reflected wave is not inverted