Topic 4 Oscillations and Waves

4.1 Simple Harmonic Motion

Oscillations

There are many systems, both natural and man made, that vibrate back and forth around an equilibrium point.

These systems are said to regularly oscillate.

Common examples are:A mass on a spring

A pendulum

Electrons under alternating current

Key Terms

The equilibrium point is that point where the system will naturally rest.e.g. for a pendulum bottom centre

For a mass on a spring the point where the upwards pull of the spring equals the downward pull of the weight.

The displacement (x) of the system is the vector displacement of the system from its equilibrium point.Usually the displacement is considered in 1 dimension and is given the symbol x even if the displacement is vertical.

Key Terms

The amplitude (A) of an oscillation is the maximum displacement of the system.It is the height of a wave from its equilibrium point.

It is half the peak to trough height.

The wavelength () of a moving wave is the distance from peak to peak in the space dimension.

Key Terms

The time period (T) is the time taken in seconds to complete 1 complete cycle.This is the time from peak to peak in the time dimension.

A cycle is complete when the system is back in its initial state.e.g. for a pendulum, when the bob is at its lowest point and travelling in the same direction as at the start.

The frequency (f) of the system is the number of oscillations per second.It is the inverse of the time period.

Frequency is measured in Hz or s-1

Key Terms

A sine wave has a period of 2 radians and a time period of T seconds.

Therefore its angular displacement (on an x- graph) at any time is:

The angular frequency () in rad s-1 is therefore:

Questions

Calculate the frequency and angular frequency of:A pendulum of period 4s

A water wave of period 12s

Mains electricity of period 0.02s

Laser light with period 1.5 fs

Key Terms

A sinusoidal wave has is an oscillation with the following properties.It has an amplitude of 1.

It has a period of 2 radians

It has an initial displacement of +0.

x

0 /2 3/2 2

Key Terms

A cosine wave is identical to a sine wave excepting that it has an initial displacement of +1

It can be said that a cosine wave is a sine wave with a phase difference () of -/2

x

0 /2 3/2 2

Oscillating Systems

An oscillating system is defined as one that obeys the general equation:

Here the amplitude is x0

The angular frequency ensures that the real time period coincides with the angular period of 2 radians

The phase allows for an oscillation that starts at any point.

Oscillating Systems

If the oscillations begin at the equilibrium point where displacement is zero at the start then:

If the oscillations begin at the end point where displacement is a maximum at the start then:

This second form is more useful in more situations

Questions

A simple harmonic motion is initiated by releasing a mass from its maximum displacement. It has period 2.00s and amplitude 16.0cm. Calculate the displacement at the following times:t=0s

t=0.25s

t=0.50s

t=1.00s

Timexv

00+v0

T/4x00

T/20-v0

3T/4-x00

T0+v0

Oscillating Systems

The rate of change of displacement (the speed) is given by the gradient of the displacement curve.

Assuming that:

Then:

The speed is therefore:

Questions

A bored student holds one end of a flexible ruler and flicks it into an oscillation. The end of the ruler moves a total distance of 8.0cm and makes 28 full oscillations in 10s.What are the amplitude and frequency of the motion of the end of the ruler?

Use the displacement equation to produce a table of x and t for t=0,0.04,0.08,0.012,...,0.036

Draw a graph of x versus t

Find the maximum speed of the end of the ruler.

Oscillating Systems

The rate of change of speed (the acceleration) is given by the gradient of the speed curve.

Using similar logic:

This has a very similar form to the displacement equation therefore:

Note that the acceleration is:In the opposite direction to the displacement,

Directly proportional to the displacement.

The SHM Equation

Any system undergoing simple harmonic motion obeys the relationship:

It can be shown using calculus or centripetal motion that

Therefore:

The SHM Equations

For a system starting at equilibrium

The general SHM equation applies to all simple oscillating systems.

For a system starting at maximum displacement

The SHM Equations

One final equation can be formed by squaring the speed equation.

Because sin2 + cos2 =1

Questions

A body oscillates with shm decribed by:x=1.6cos3t

What are the amplitude and period of the motion

At t=1.5s, calculate the displacement, velocity and acceleration.

Questions

The needle of a sewing machine moves up and down with shm. If the total vertical motion of the needle is 12mm and it makes 30 stitches in 7.0s calculate:The period,

The amplitude,

The maximum speed of the needle tip

The maximum acceleration of the needle tip.

Energy Changes

An oscillating system is constantly experiencing energy changes.

At the extremes of displacement, the potential energy is a maximum.Gravitational potential for a pendulum, elastic potential for a spring

At the equilibrium position, the kinetic energy is a maximum

Kinetic Energy

Remember that kinetic energy is given by:

Substituting

Gives

Total Energy

Remember that at the equilibrium point ALL the energy of the system is kinetic.

The total energy of the system ET is therefore:

Note that the total energy of the system is proportional to the amplitude squared.

Potential Energy

The law of conservation of energy requires that the total energy of an oscillating system be the sum of the potential and kinetic energies.

Therefore

Questions

A pendulum of mass 250g is released from its maximum displacement and swings with shm. If the period is 4s and the amplitude of the swing is 30cm, calculate:The frequency of the pendulum

The maximum speed of the pendulum

The total energy of the pendulum

The maximum height of the pendulum bob.

The energies of the pendulum at t=0.2s.