Simple Harmonic Oscillator
A simple harmonic oscillator is a system — such as a mass on a spring — that experiences a restoring force directly proportional to its displacement from equilibrium, producing sinusoidal motion in time.
Interactive simulation
The physics
| Symbol | Quantity | SI Unit |
|---|---|---|
| x(t) | Displacement from equilibrium at time t | metre (m) |
| x₀ | Amplitude (initial displacement, released from rest) | metre (m) |
| ω | Angular frequency | rad/s |
| m | Mass | kilogram (kg) |
| k | Spring constant | N/m |
Velocity and acceleration follow by differentiation: v(t) = −x₀ω sin(ωt), and a(t) = −x₀ω² cos(ωt) = −ω²x(t). This last relationship — acceleration proportional to and opposite in sign to displacement — is the defining mathematical signature of simple harmonic motion, and follows directly from Newton's second law applied to Hooke's law, F = −kx: ma = −kx, so a = −(k/m)x = −ω²x.
Total mechanical energy, E = ½kx₀², remains constant throughout the motion (in this idealised, frictionless model), continuously exchanging between kinetic energy (maximum at x = 0) and spring potential energy (maximum at the extremes of motion, x = ±x₀) — visible directly in the simulation's live energy readouts.
Sources
- Hooke, R. (1678). De Potentia Restitutiva — original statement of Hooke's law.
- Taylor, J.R. (2005). Classical Mechanics. University Science Books, Ch. 5.