Force law for SHM: F = −kx

Newton's second law with a = −ω²x gives the force on a particle of mass m in SHM: F = ma = −mω²x, written F = −kx with k = mω².

So ω = √(k/m) and the period is T = 2π√(m/k). A stiffer spring makes it faster; a heavier block makes it slower.

The force always points towards the mean position, which is why it is called the restoring force.

SHM can be defined in either of two equivalent ways: by the displacement x(t) = A cos(ωt + φ), or by the force law F = −kx. Differentiating the first twice gives the second; integrating the second twice gives back the first.

A body acted on by a force exactly proportional to x is a linear harmonic oscillator. If the force also contains small terms in x², x³ and so on, it is a non-linear oscillator.

Test for SHM: the acceleration must be a negative constant times x. a = −(constant) × x is SHM; an acceleration proportional to +x, to x² or to x³ is not.

Worked example: a block of mass m held between two identical springs of constant k fixed on either side. Displaced by x, one spring stretches and the other compresses by x, and both push it back, so F = −2kx and T = 2π√(m/2k).

The period of SHM does not depend on the amplitude, the energy or the phase constant; it is set only by m and k.