A level Physics · Edexcel 9PH0 · Topic 13A

Simple harmonic motion

Key points

SHM condition: $a = -\omega^2x$ — acceleration is proportional to displacement and in the opposite direction (restoring force).
Solutions: $x = A\cos\omega t$ (starting at maximum) or $x = A\sin\omega t$ ; velocity $v = -A\omega\sin\omega t$ and maximum speed $v_{max} = A\omega$ ; maximum acceleration $= A\omega^2$ .
$\omega = 2\pi f = \frac{2\pi}{T}$ . The period of SHM is independent of amplitude (isochronous).
Mass–spring: $T = 2\pi\sqrt{\frac{m}{k}}$ . Simple pendulum: $T = 2\pi\sqrt{\frac{l}{g}}$ for small angles (under about 10°).
Graphs: displacement, velocity and acceleration are all sinusoidal; velocity is a quarter cycle out of phase with displacement and acceleration is in antiphase with it.
Energy: kinetic energy is greatest at the centre and potential energy at the extremes; the total is constant.
Core practical: time oscillations of known masses on a spring, plot $T^2$ against $m$ and use the graph to find an unknown mass; time many oscillations to reduce uncertainty.

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Simple harmonic motion is the most important type of oscillation because so many systems behave this way for small displacements. One defining equation leads to all of its properties.

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