Simple harmonic motion

Simple harmonic motion is the motion of a mass under a restoring force proportional to its displacement: $m x'' = -k x$, or $x'' = -\omega^2 x$ with $\omega = \sqrt{k/m}$. Every solution is $x = A\cos(\omega t - \varphi)$, and its period $T = 2\pi/\omega$ does not depend on the amplitude A. It is also often written $x = A\sin(\omega t + \varphi)$, with another phase. The page shows why, for a mass on a spring, and compares a pendulum.

The three layers share one x axis. On the phase plane the state is the point (x, v/ω). Its velocity there has two parts, v across and a/ω = −ωx up or down; together they are always perpendicular to the radius and ω times as long, so every point of the plane turns about the centre at the same rate ω, like a record. The mass on the rail is the point's shadow on the x axis, which is why x is a cosine; the distance from the centre is the amplitude, the direction is the phase, and the radius squared times ½k is the energy — the spring's share is ½kx², the motion's ½k(v/ω)². Dividing v by ω is what makes the orbit a circle: with v itself it would be an ellipse, and the point would not turn evenly. The paper tape under the plane records x against time, ticked every 0.1 s, with a mark at each zero crossing; crossings come every T/2 whatever the amplitude, and a second oscillator's marks line up with the first one's.

The pendulum obeys $s'' = -g\sin(s/L)$ with arc length $s = L\theta$: the force is not proportional to s, so the motion is not simple harmonic. Its energy curves on the plane (s, ṡ/ω₀) are ovals, not circles, and its period grows with the swing, $T = T_0/\mathrm{AGM}(1, \cos(\theta_{\max}/2))$, about $T_0(1 + \theta_{\max}^2/16)$ for a small swing $\theta_{\max}$ in radians, where $T_0 = 2\pi\sqrt{L/g}$. Here $\theta_{\max}$ is the swing set by the energy; it equals the start angle only when the bob starts at rest. The pendulum's tape is a numerical solution, checked against that exact period; the spring's is the exact solution. Past the dashed separatrix a pendulum goes over the top; this page keeps the swing within 170°.