Simple Harmonic Motion Calculator

Because the sinusoidal solution for position, velocity, and acceleration only ever needs the angular frequency ω, and ω is derived directly from the period as ω = 2π/T. Mass and spring stiffness (or pendulum length and gravity) are two of the many possible ways to produce a given period, but once you know T, none of those underlying details change the shape of the motion — so the calculator skips straight to the period you'd measure with a stopwatch.

The engine never asks for mass, so it computes energy per unit mass instead — kinetic energy as ½v², potential energy as ½ω²x², and total energy as ½ω²A², all in units of velocity-squared rather than joules. Multiply any of these by your system's actual mass in kilograms to convert to real energy; the ratio between them stays valid regardless of mass.

The phase constant φ shifts where in the oscillation cycle the object sits at t = 0, since displacement is computed as A·cos(ωt + φ). Leaving it at zero starts the object at maximum positive displacement (x = A) with zero velocity; a nonzero phase moves that starting point earlier or later in the cycle without changing the amplitude, period, or energy of the motion.

Maximum velocity, A·ω, occurs when the object passes through equilibrium (x = 0), because that's where the restoring force — and therefore deceleration — is momentarily zero. Maximum acceleration, A·ω², occurs at the two endpoints of travel where displacement is greatest, since acceleration is always proportional to displacement and directed back toward equilibrium. The two maxima never happen at the same instant.