## Simple Harmonic Motion

Simple harmonic motion of a body is motion in a straight line about a point such that the acceleration of the body is always directed towards the equilibrium point, labelled O below, and the magnitude of the acceleration is proportional to the distance from the point. We may illustrate this in the diagram below, in which the double headed arrows indicate the acceleration. In the diagram above the body is moving backwards and forwards about the point O.

At A the distance of the body from O is and the acceleration is towards O.

At B the distance of the body from O is and the acceleration is towards O.

At the point A it does not matter whether the particle is moving away from O or towards O: the acceleration is still towards O. At the point B it does not matter whether the particle is moving away from O or towards O: the acceleration is still towards O – the acceleration is always directed towards O. Mathematically we may write This equation states that the acceleration is proportional to the displacement from the equilibrium point and directed opposite to the displacement.

Simple harmonic motion is naturally related to motion in a circle. We can use simple trigonometry to project motion in a circle onto the axis in the diagram above using simple trigonometry: where is the rate of change of the angle with the axis, and is called the angular velocity.

If we differentiate this twice to get the acceleration, we obtain Compare this with to obtain  