Simple Harmonic Motion Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

Revision notes with simplified explanations to understand Simple Harmonic Motion quickly and effectively.

Learn about Simple Harmonic Motion for your A-Level Further Maths Core Pure Exam. This Revision Note includes a summary of Simple Harmonic Motion for easy recall in your Further Maths Core Pure exam

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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 toward the center of the circle.

Definition (SHM)

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

\frac{d^2x}{dt^2} = -\omega^2 x \quad \text{where}-\omega^2 \text{ is the constant of proportionality.}

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)

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