Simple harmonic motion and circular motion

Simple harmonic motion (SHM) is the simplest oscillation. Its displacement from the mean position is x(t) = A cos(ωt + φ):

• A, the amplitude, is the greatest displacement on either side.
• ω is the angular frequency: ω = 2π/T = 2πν, in rad s⁻¹.
• (ωt + φ) is the phase. It tells you two things at once: where the body is, and which way it is moving. φ, its value at t = 0, is the phase constant (initial phase).
• Writing A sin(ωt + φ) is just as good: it only changes φ by π/2.

Example: x = 0.04 cos(5πt + π/3), in metres. Then A = 4 cm, ω = 5π rad s⁻¹, so ν = 5π/2π = 2.5 Hz and T = 0.4 s, and the phase constant is π/3.

SHM is a shadow of circular motion. Let a point go round a circle of radius A at a steady angular speed ω. Its angle at time t is ωt + φ, so its foot on a diameter is at A cos(ωt + φ): exactly SHM. This circle is the reference circle.

Differentiate x to get the velocity and the acceleration:

• v = –ωA sin(ωt + φ), which can also be written v = ±ω√(A² – x²). It is largest, ωA, at the mean position and zero at the extremes, where the body stops to turn.
• a = –ω²A cos(ωt + φ) = –ω²x. It is zero at the mean position and largest, ω²A, at the extremes, and it always points towards the mean.
• All three repeat with the same period T. The velocity is π/2 ahead of x, and the acceleration is π out of step with x.

Continuing the example at t = 0: x = 0.04 cos(π/3) = 2 cm, and v = –0.04 × 5π × sin(π/3) ≈ –0.54 m/s, so the body is moving towards the mean. Its greatest speed is ωA ≈ 0.63 m/s, and its greatest acceleration ω²A ≈ 9.9 m/s².

Two facts at t = 0, the starting position and the starting velocity, are enough to fix A and φ.