Pendulum Calculator

In short The period of a one-metre pendulum is 2.0064 seconds by the familiar formula T = 2π√(L/g) — but that formula is not the period of a pendulum. It is the period in the limit of zero amplitude, and every calculator that prints it alone is quietly answering a different question. The real period is T = 4√(L/g)·K(sin(θ₀/2)), where K is the complete elliptic integral of the first kind, and it grows with how far the pendulum swings. This page computes both and shows the gap. At 10° the approximation is 0.19% low, which is why it survives at all; the error passes 1% at 22.8°, reaches 4.00% at 45° and 18.03% at 90°. So a clock pendulum swinging a few degrees is fine, and a playground swing is not. Two consequences follow. Mass does not appear anywhere, because gravity supplies both the restoring force and the inertia and they cancel — a heavy bob and a light one keep the same time. But amplitude DOES, which means a pendulum is not truly isochronous: it slows as it swings wider. That is exactly why a pendulum clock must hold its amplitude constant to keep time, and why a real one, losing amplitude to friction, very slightly speeds up as it dies away. The length that gives a two-second period is 0.9936 m, which is what a grandfather clock is built around.