Circular motion. A point moves round a circle of radius A at a steady rate, in a period T. Its angle is θ = ωt + φ where ω = 2π/T is the angular velocity in radians per second and φ is the angle it started at.

The shadow. Shine a light from above and look at where the point's shadow falls on a line (the x axis). The shadow is the x component of the position of the point:

x = A cos(ωt + φ)

This shadow moves to and fro between +A and −A: it is in simple harmonic motion. Tilting the view until the circle is edge-on shows that you can think of SHM as circular motion seen from the side.

Velocity and acceleration. The point on the circle has a speed Aω (tangent to the circle) and a centripetal acceleration Aω² (towards the centre). The velocity and the acceleration of the shadow are the x components of these:

v = −Aω sin(ωt + φ), a = −Aω² cos(ωt + φ) = −ω² x

The last equation is the signature of SHM: the acceleration is proportional to the displacement and always points towards the centre (the minus sign). At the ends of the motion (x = ±A) the speed is zero and the acceleration is greatest. At the centre (x = 0) the speed is greatest (Aω) and the acceleration is zero. The speed at any position is v = ω√(A² − x²).

The graphs show the same motion against time. The position and acceleration graphs are mirror images (a = −ω²x), and the velocity graph is a quarter of a cycle ahead of the position graph. A mass on a spring, a pendulum swinging through small angles and the vibration of a guitar string are all SHM.

The size of the arrows is scaled so that the biggest possible value is the same length whatever the amplitude and period.