SHM Is the Shadow of a Circle: Phase, Velocity and Acceleration

The reference circle

Shine a light from the side on a ball going round and round a circle at constant speed — uniform circular motion. Its shadow on the wall slides back and forth in exactly simple harmonic motion. The radius of the circle is the amplitude A, and the ball's steady angular speed is our \omega. This is the reference circle, and it is the fastest way to see everything about SHM at once.

x(t) = A\cos(\omega t + \varphi)

The horizontal projection of the circling point.

The angle \omega t + \varphi that the ball has swept is the phase of the motion, and \varphi is the phase constant — the ball's starting angle at t = 0. Phase tells you exactly where in the cycle you are: phase $0$ is the far right, \pi/2 is passing through the centre, \pi is the far left, and so on all the way round.

Angular frequency, properly

The reference circle finally makes \omega intuitive: it is simply how fast the phase angle advances, in radians per second. One full trip round the circle is 2\pi radians and takes exactly one period T, so the three quantities are locked together.

\omega = \frac{2\pi}{T} = 2\pi f

Angular frequency links period and frequency (measured in rad/s).

Velocity and acceleration come free

Because x is the shadow of circular motion, we get velocity and acceleration just by differentiating (or by reading off the circling ball's own velocity and acceleration components).

v(t) = \frac{dx}{dt} = -A\omega\sin(\omega t + \varphi)

Velocity in SHM.

a(t) = \frac{dv}{dt} = -A\omega^2\cos(\omega t + \varphi) = -\omega^2 x

Acceleration in SHM — note a = -ω²x reappears, confirming the defining equation.

Read off the peaks: the speed maxes out at v_{\max} = A\omega and the acceleration at a_{\max} = A\omega^2. And crucially, these peaks happen at different moments — they are a quarter-cycle out of step.

v_{\max} = A\omega, \qquad a_{\max} = A\omega^2

Peak speed and peak acceleration.

Reading phase differences

Displacement is a cosine, velocity a negative sine, acceleration a negative cosine. In phase language, the velocity leads the displacement by 90° (a quarter period), while the acceleration is 180° out of phase with the displacement — always pointing the opposite way. This same phase vocabulary is exactly what you will reuse for AC circuits and for travelling waves.

A quick worked case: a 2.0 Hz oscillator with amplitude 5.0 cm. First \omega = 2\pi f = 2\pi(2.0) \approx 12.6 rad/s; then v_{\max} = A\omega \approx 0.63 m/s and a_{\max} = A\omega^2 \approx 7.9 m/s². Notice how the single number \omega turns amplitude into speed and acceleration.

\omega = 2\pi f = 4\pi \approx 12.6\ \text{rad/s}, \quad v_{\max} = A\omega \approx 0.63\ \text{m/s}, \quad a_{\max} = A\omega^2 \approx 7.9\ \text{m/s}^2

Worked numbers for a 2.0 Hz, 5.0 cm oscillator.