SHM Acceleration
Gives the acceleration of a simple harmonic oscillator: a(t) = −Aω²·cos(ωt + φ) = −ω²·x(t)
The equation
What it solves
Gives the acceleration of a simple harmonic oscillator: a(t) = −Aω²·cos(ωt + φ) = −ω²·x(t). Acceleration is always directed toward equilibrium (opposite to displacement) and is proportional to the displacement.
When to use it
When the problem asks for instantaneous acceleration or the maximum acceleration Aω² at the turning points. The relation a = −ω²x is useful for identifying SHM in any physical system: if acceleration is proportional to displacement, it is SHM.
When NOT to use it
This form assumes undamped SHM. In a damped oscillator, the acceleration has an additional term proportional to velocity. Do not confuse this sinusoidal time-domain expression with the amplitude envelope.
Common mistakes
Forgetting the ω² factor and writing a = −A·cos(…) instead of a = −Aω²·cos(…). Confusing maximum acceleration (at amplitude, x = A) with maximum velocity (at equilibrium, x = 0). Using ω instead of ω² when computing the maximum acceleration.