EQUATION

SHM Position

Gives the displacement of a simple harmonic oscillator at any time t: x(t) = A·cos(ωt + φ)

§ 01

The equation

EQ.SHM-POSITION

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

§ 02

What it solves

Gives the displacement of a simple harmonic oscillator at any time t: x(t) = A·cos(ωt + φ). The amplitude A and phase φ are determined from initial conditions x(0) and v(0).

§ 03

When to use it

Any undamped oscillator once ω, A, and φ are known. Use x(0) = A·cos φ and v(0) = −Aω·sin φ to extract A and φ from given initial values.

§ 04

When NOT to use it

Does not apply to damped oscillators — replace with x(t) = A·e^(−γt/2)·cos(ω_d·t + φ). Also invalid outside the linear (small-amplitude) regime.

§ 05

Common mistakes

Assuming φ = 0 when the oscillator starts at maximum displacement but with a non-zero velocity. Mixing radians and degrees in ωt + φ. Using f (Hz) instead of ω = 2πf in the argument of cosine.

§ 06

Topics that use this equation

oscillators everywhere