Simple Harmonic Motion (Edexcel A-Level Further Mathematics): Revision Notes

8.3.1 Simple Harmonic Motion

Simple Harmonic Motion (SHM)

Consider a particle moving in a circle of radius r.

If we rotate the situation 90 degrees and view the particle from the side, the particle undergoes motion similar to a pendulum.

In this situation, acceleration is always directed towards the centre of the circle.

Definition (SHM)

Simple harmonic motion is motion in which the acceleration is proportional to the distance from the centre of motion.

Worked Example

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Example: Find the general solution to $\ddot{x} = -\omega^2 x$. (Note: A dot above a variable indicates differentiation with respect to time, $\frac{d}{dt}$.)

$$\frac{d^2x}{dt^2} + 0\frac{dx}{dt} + \omega^2 x = 0$$

Characteristic equation (C.F.):

$$m^2 + \omega^2 = 0 \quad \Rightarrow \quad m = \pm i\omega$$

Thus the general solution is:

$$x = A\sin(\omega t) + B\cos(\omega t)$$

Note that this can easily be transformed to the form:

$$x = R\sin(\omega t + \alpha) \quad \text{or} \quad R\cos(\omega t + \alpha)$$