Waves+-+Simple+Harmonic+Motion+(SHM)

Simple harmonic motion (SHM) is a common type of oscillation which is defined in terms of the acceleration of any point on the oscillator, as follows:

Definition of SHM

 * //An oscillation where the acceleration of a point is always directed towards the equilibrium position, and the acceleration is proportional to the distance the point is from the equilibrium position.// ||

As a result of this property, all instances of SHM will have the following characteristics:
 * The period of oscillation is independent of the amplitude (isochronous)
 * The restoring force is proportional to the displacement

The next sections deal with (i) where might you find SHM, (ii) The correlation between circular motion and SHM, and (iii) the mathematics of SHM.

i) Where might you find SHM:

 * Mass on a spring
 * Bendy ruler (cantilever)
 * Simple Pendulum (note: by "simple" we mean an idealised pendulum with a point mass on a massless string. This is approximated by a heavy ball on a light string)
 * Vertical Oscillations in a liquid
 * In the oscillations of a hanging lamp (This final point is included as Galileo is understood to have been inspired to investigate SHM by his observations of the oscillations of a swinging candelabra during a religious service. By timing the oscillations using his pulse he was able to determine the isochronous property of their oscillations).

ii) Circular Motion and SHM
It is possible to set up a simple pendulum and, say, a ball spinning round on a record player, in such a way that the shadows of both move in sync with each other. This is shown in the video clip below.

media type="youtube" key="0IaKcqRw_Ts" height="315" width="420"

Why does this happen?

If take a conker on a string and spin it round your hand, it is apparent that it is the piece of string which is stopping the conker spinning round, rather than shooting off. In other words, the only net force acting on the string to keep it moving in a circle is the tension in the string. Using newtons second law, we know that F=ma, so the fact there is a force means that there is an acceleration, or change in the direction of motion, and that this acceleration is acting in the direction of the force, i.e. the acceleration is always acting towards the centre of the circle.

And this is very similar to the requirement for SHM (see the definition above) that the acceleration is towards the equilibrium position. The motion described by SHM can therefore be thought of as a two dimensional version of circular motion, as demonstrated by the above video, where the 2 two dimensional projection of the circular motion (the shadow) can be made to move in time with the oscillations of the pendulum.

The next section uses the maths of circular motion to enable us to analyse SHM in more detail.


 * iii) The mathematics of SHM**