Harmonic Motion and Waves

Chapter 14

Hooke’s Law

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 (T).

m

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 ).

x = 0

m

x = 0

Hooke’s Law

The minus sign on the force indicates that it is a restoring force – it is directed to restore the mass to its equilibrium position.

The restoring force exerted by the spring depends on the displacement:

m

x

F

kxFs

m

x

F kxF

(k) is the spring constant

Displacement (x) is measured from the equilibrium point

Amplitude (A) is the maximum displacement

Period (T) is the time required to complete one cycle

Frequency (f) is the number of cycles completed per second - Units are called Hertz (Hz)

A cycle is a full to-and-fro motion

Hooke’s Law

If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force.

m

xo okxF

mgEquilibrium

Position

Hooke’s Law

Any vibrating system where the restoring force is proportional to the negative of the displacement

moves with simple harmonic motion (SHM), and is often called a simple harmonic oscillator.

kxF

Hooke’s Law

Hooke’s Law

Example: If 3 people (m = 250 kg) get into a car and the springs compress 5.0 cm, what is the spring constant (k)?

F kx

k F x2k (250 kg)(9.8m s ) (.05 m)

k 49000 N/m

Potential energy of a spring is given by:

The total mechanical energy is then:

The total mechanical energy will be conserved

2kx

2mvE

22total

Elastic Potential Energy

2

2kxPEs

If the mass is at the limits of its motion, the energy is all potential.

m

A 2kAPE

2

m

x = 0

vmax

If the mass is at the equilibrium point, the energy is all kinetic.

2mv

KE2max

Elastic Potential Energy

Hippo physics mass spring 1,3

An equation of motion gives the position of an object as a function of time.

Simple harmonic motion can be represented as a component of uniform circular motion:

Comparing Simple Harmonic Motion with Circular Motion

Hippo physics 1,4,5

Vibrations and Waves

A mass on a spring undergoes SHM. When the mass passes through the equilibrium position, its instantaneous velocity

(A) is maximum.

(B) is less than maximum, but not zero.

(C) is zero.

(D) cannot be determined from the information given.

Vibrations and Waves

A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its kinetic energy is a minimum?

(A) at either A or B

(B) midway between A and B

(C) one-fourth of the way between A and B

(D) none of the above

Vibrations and Waves

A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its potential energy is a minimum?

(A) at either A or B

(B) midway between A and B

(C) one-fourth of the way between A and B

(D) none of the above

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.

The Simple Pendulum

2 LTg

Vibrations and Waves

A simple pendulum consists of a mass M attached to a weightless string of length L. For this system, when undergoing small oscillations

(A) the frequency is proportional to the amplitude.

(B) the period is proportional to the amplitude.

(C) the frequency is independent of the length L.

(D) the frequency is independent of the mass M.

Frank Oppenheimer demonstrates that a pendulum swinging back and forth traces out a straight line over a stationary surface, and a sine curve when the surface moves at constant speed.

Wave Description

Wave

• Rhythmic disturbance that carries energy through matter or space

Wave pulse

• Single bump or disturbance that travels through a medium

Continuous wave

• A regularly repeating sequence of wave pulses

Transverse wave - a wave travels along its medium, but the individual particles just move up and down, vibrates perpendicular to the direction of the wave

Velocity of wave

Velocity of particle

Wave Motion

Wave Motion

Wave characteristics:• Amplitude, A• Wavelength, λ• Frequency f and period T• Wave velocity

Tfv

Amplitude of a wave• Maximum displacement from its position of rest, or

equilibrium

– Waves with larger amplitudes transfer more energy– For waves that move at the same speed, the rate at

which energy is transferred is proportional to the square of the amplitude

• Doubling the amplitude of a wave increases the amount of energy it transfers each second by a factor of four

The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal).

Types of Waves: Transverse and Longitudinal

Longitudinal wave

• The disturbance is in the same direction as, or parallel to, the direction of wave motion– Sound waves– Liquids and gases usually transmit only longitudinal waves

• Transverse waves can occur only in solids, whereas longitudinal waves can travel in solids, fluids, and gases. 

• Transverse motion requires that each particle drag with it adjacent particles to which it is tightly bound.  • In a fluid this is impossible, because adjacent

particles can easily slide past each other. 

• Longitudinal motion only requires that each particle push on its neighbors, which can easily happen in a fluid or gas. 

• The fact that longitudinal waves originating in an earthquake pass through the center of the earth while transverse waves do not is one of the reasons the earth is believed to have a liquid core.

Incident wave

• Wave that strikes the boundary between two mediums

Law of reflection*

• Angle of incidence is equal to the angle of reflection

Reflected wave*• A returning wave that is either inverted or displaced

in the same direction as the incident wave

Whether or not a wave is inverted upon reflection depends on whether the end is free to move or not.

Reflection and Transmission of Waves

A wave traveling from one medium to another will be partly reflected and partly transmitted.

Reflection and Transmission of Waves

Principle of superposition• The displacement if a medium caused by two or more waves

is the algebraic sum of the displacements of the individual waves.

The superposition principle says that when two waves pass through the same point, the displacement is the arithmetic sum of the individual displacements.

The figure below exhibits constructive and destructive interference.

Interference; Principle of Superposition

Wave interferenceDestructive interference

Superposition of waves withequal but opposite amplitudes

Wave interferenceConstructive interference

Superposition of waves resulting in increased wave displacement

These figures show the sum of two waves.In (a) they add constructively. (In phase)In (b) they add destructively. (Out of phase)In (c) they add partially destructively.

(a) (b) (c)

+ + +

= = =

Interference; Principle of Superposition

Surface waves

• Have characteristics of both transverse and longitudinal waves

– Waves deep in the ocean are longitudinal, at the surface of the water, the particles move in a direction that is both parallel and perpendicular to the direction of the wave motion.

Standing wave

Wave with stationary nodes and antinodes

Standing Waves

Node – Stationary point where two equal wave pulses meet and displacement is zero

Antinodes – The point of largest amplitude when two wave pulses meet of two superimposed waves

Standing Waves

Refraction of waves

• A change in the direction of waves crossing a boundary between two different media

Diffraction*

The bending of waves around a barrier