Oscillations about Equilibrium

Forces and elastic materials

Elastic material• Capable of recovering shape after deformation• Rubber ball versus lump of clay

Spring forces1. Applied force proportional to distance spring is

compressed or stretched2. Internal restoring force arises, returning spring to

original shape3. Restoring force also proportional to stretched or

compressed distance

Forces and vibrations

• Vibration - repetitive back and forth motion

• At the equilibrium position, spring is not compressed

• When disturbed from equilibrium position, restoring force acts toward equilibrium

• Carried by inertia past equilibrium to other extreme

• Example of “simple harmonic motion”

Simple Harmonic MotionA spring exerts a restoring force that is proportional to the displacement from equilibrium:

Periodic Motion

Period: time required for one cycle of periodic motion

Frequency: number of oscillations per unit time

This unit is called the Hertz:

Describing vibrations

• Amplitude - maximum extent of displacement from equilibrium

• Cycle - one complete vibration

• Period - time for one cycle• Frequency - number of

cycles per second (units = hertz, Hz)

• Period and frequency inversely related

Simple Harmonic Motion

If we call the period of the motion T – this is the time to complete one full cycle – we can write the position as a function of time:

It is then straightforward to show that the position at time t + T is the same as the position at time t, as we would expect.

Connections between Uniform Circular Motion and Simple Harmonic Motion

An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion:

The Period of a Mass on a Spring

The period is

Energy Conservation in Oscillatory Motion

In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:

Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:

Energy Conservation in Oscillatory Motion

This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.

The Pendulum

A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).

The angle it makes with the vertical varies with time as a sine or cosine.

The Pendulum

Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).

The Pendulum

However, for small angles, sin θ and θ are approximately equal.

The Pendulum

Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:

Driven Oscillations and ResonanceAn oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.

Driven Oscillations and Resonance

If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.

Waves

• Periodic (traveling) disturbances transporting energy• Causes

– Periodic motion disturbing surroundings

– Pulse disturbance of short duration

• Mechanical waves– Require medium for propagation

– Waves move through medium

– Medium remains in place

Kinds of waves

Longitudinal waves

• Vibration direction parallel to wave propagation direction

• Particles in medium move closer together/farther apart

• Example: sound waves

• Gases and liquids - support only longitudinal waves

Kinds of waves, cont.Transverse waves• Vibration direction perpendicular to

wave propagation direction• Example: plucked stringSolids - support both longitudinal and

transverse wavesSurface water waves• Combination of both• Particle motion = circular

Waves in air

• Longitudinal waves only• Large scale - swinging

door creates macroscopic currents

• Small scale - tuning fork creates sound waves

• Series of condensations (overpressures) and rarefactions (underpressures)

Types of Waves

Water waves are a combination of transverse and longitudinal waves.

Describing waves

Graphical representation• Pure harmonic waves = sines

or cosines

Wave terminology• Wavelength

• Amplitude

• Frequency

• Period

Wave propagation speed

Waves on a String

The speed of a wave is determined by the properties of the material through which it propagates.

For a string, the wave speed is determined by:

1. the tension in the string, and

2. the mass of the string.

As the tension in the string increases, the speed of waves on the string increases as well.

Waves on a String

The total mass of the string depends on how long it is; what makes a difference in the speed is the mass per unit length. We expect that a larger mass per unit length results in a slower wave speed.

14-2 Waves on a String

As we can see, the speed increases when the force increases, and decreases when the mass increases.

Waves on a String

When a wave reaches the end of a string, it will be reflected. If the end is fixed, the reflected wave will be inverted:

Waves on a String

If the end of the string is free to move transversely, the wave will be reflected without inversion.

Sound Waves

Sound waves are longitudinal waves, similar to the waves on a Slinky:

Here, the wave is a series of compressions and stretches.

Sound Waves

In a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.

Sound Waves

The speed of sound is different in different materials; in general, the denser the material, the faster sound travels through it.

Sound Waves

Sound waves can have any frequency; the human ear can hear sounds between about 20 Hz and 20,000 Hz.

Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are called infrasonic.

Ultrasonic waves are familiar from medical applications; elephants and whales communicate, in part, by infrasonic waves.

Sound IntensityThe intensity of a sound is the amount of energy that passes through a given area in a given time.

Sound Intensity

Expressed in terms of power,

14-5 Sound Intensity

Sound intensity from a point source will decrease as the square of the distance.

Sound Intensity

When you listen to a variety of sounds, a sound that seems twice as loud as another is ten times more intense. Therefore, we use a logarithmic scale to define intensity values.

Here, I0 is the faintest sound that can be heard:

Sound Intensity

The quantity β is called a bel; a more common unit is the decibel, dB, which is a tenth of a bel.

The intensity of a sound doubles with each increase in intensity level of 10 dB.

The Doppler Effect

The Doppler effect is the change in pitch of a sound when the source and observer are moving with respect to each other.

When an observer moves toward a source, the wave speed appears to be higher, and the frequency appears to be higher as well.

The Doppler Effect

The Doppler effect from a moving source can be analyzed similarly; now it is the wavelength that appears to change:

The Doppler Effect

Combining results gives us the case where both observer and source are moving:

The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements.

Superposition and Interference

Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition.

Superposition and Interference

If two pulses combine to give a larger pulse, this is constructive interference (left). If they combine to give a smaller pulse, this is destructive interference (right).

Superposition and Interference

Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.

Superposition and Interference

Here is another example of an interference pattern, this one from two sources. If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.

Standing Waves

A standing wave is fixed in location, but oscillates with time. These waves are found on strings with both ends fixed, such as in a musical instrument, and also in vibrating columns of air.

Standing Waves

The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string. Higher frequencies are called harmonics.

Standing Waves

There must be an integral number of half-wavelengths on the string; this means that only certain frequencies are possible.

Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.

In order for different strings to have different fundamental frequencies, they must differ in length and/or linear density.

A guitar has strings that are all the same length, but the density varies.

Standing Waves