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If a number of oscillations are involved we can work out the time period by dividing the total time taken by the number of oscillations completed:

If a number of oscillations are involved we can work out the time period by dividing the total time taken by the number of oscillations completed:

The frequency, f, of oscillations is the number of oscillations undergone in one second, and is measured in hertz (Hz).

The frequency and the period can therefore be related as:

The displacement of an oscillating particle is the distance the particle has been moved from its equilibrium position.

The amplitude of an oscillation is the maximum displacement of the vibrating object from the equilibrium position (its usual position).

Simple Harmonic Motion (SHM) is a particular type of oscillation. It is useful because its time period stays the same even when its amplitude changes.

If we set this system oscillating by lifting the mass and letting it go, then the system starts with:

Maximum positive displacement (because it's above the middle).

Zero velocity (it's not moving at the first instant).

Maximum negative acceleration (because it is about to start moving down).

As it passes through the equilibrium position on the way down it's at maximum speed down (negative), its displacement is zero and because the spring is at its equilibrium position, there is no resultant force on the mass so it is not accelerating.

At the bottom, the mass stops momentarily as it changes direction, so velocity is zero. The displacement is a maximum in the negative direction, so the acceleration is a maximum in a positive direction as the spring tries to shorten again.

The important point to note is the phase difference between these three variables...

1. The velocity, v, is zero where there are stationary points at the peaks and troughs of the displacement graph and the velocity is a maximum when the displacement is zero. (Don't forget the gradient of the displacement graph will equal velocity.)

2. The displacement and acceleration graphs are 180 degrees out of phase and therefore look like a mirror image of each other in the time axis. (Don't forget the gradient of the velocity graph will equal acceleration.)

A body is undergoing SHMwhen the acceleration on the body is proportional to its displacement, but acts in the opposite direction.

a α - s

There are two common examples of simple harmonic motion:

Where m = mass (kg)and k = spring constant (Nm-1)

Where l = length of pendulum (m)g = acceleration due to gravity (ms-2)

The definition for simple harmonic motion tells us that:

a α -s

We can get rid of the proportionality sign by putting in a constant. In this case, the constant is (2πf)2, so:

a = - (2 πf)2 s

As shm oscillations follow a sine or cosine wave, we can find the displacement at any point using:

Where:

A = amplitude - not acceleration!

Velocity can be found using:

The total energy during an oscillation (for example, a pendulum on a string) is constant as long as no energy is lost to the environment:

The energy in the system changes from potential to kinetic and back every half cycle, but the total energy in the system is constant at all times (the dotted line is the sum of the P.E. and K.E.)

Note: That energy changes from KE to PE and back again twice in every cycle.

Total energy = Kinetic energy + potential energy

This applies to a mass oscillating on a spring, so we can easily calculate the total energy using the equations for kinetic energy of a mass and the potential energy stored in a spring.

KE = 1/2 mv2 and PE = 1/2 ks2

So,

Total energy = 1/2 mv2 + 1/2 ks2

Where:

m = mass on the spring (kg)

v = velocity of the mass (ms-1)

k = spring constant

s = displacement of mass (m)

The amplitude of a wave gives an indication of the amount of energy the oscillator has. This makes sense if you think of the spring and mass. The greater the amplitude the larger the amount of energy stored in the spring when it is extended. However,

PE = 1/2 ks2 so energy must be proportional to the amplitude2.

If energy is being removed from the system, the amplitude of the oscillations must become smaller and smaller, we say that the oscillations are being damped.

The amplitude of oscillations decrease with time.

The higher the damping, the faster the oscillations will reduce in size.

Critical damping is the damping required to make the oscillations stop in the quickest possible time without going past the equilibrium position

Note: That the lines in the graph never touch or cross. Also, note that if the system becomes heavily damped, the peak of the red line will move slightly to the left - to a slightly lower value of natural frequency.

Hit anything and it will vibrate. The amazing thing is that every time you hit it, it will vibrate with exactly the same frequency, no matter how hard you hit it.

The frequency of un-damped oscillations in a system, which has been allowed to oscillate on its own, is called thenatural frequency, f0.

The frequency with which the periodic force is applied is called the forced frequency. If the forced frequency equals the natural frequency of a system (or a whole number multiple of it) then the amplitude of the oscillations will grow and grow. This effect is known as resonance.

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