A swing keeps its own time.
Pull a pendulum back further and it has more distance to cover — but it also picks up more speed, and for small swings the two cancel out almost exactly. Push it far enough, though, and that stops being true.
Length L
L ∝ T²
0.1 m4× longer · 2× the period5 m
Amplitude θ₀
how far you pull it back
5°barely matters, until it does120°
Try
The equation shown is the small-angle approximation to the simple pendulum — it has no amplitude term at all, which is the point: for small swings the period genuinely doesn't depend on how far back you pull it. The period shown and animated above is the exact one instead, found from the pendulum's real equation of motion (θ″ = −(g/L)sinθ, integrated numerically) rather than the small-angle formula, so the gap between the two becomes visible once you push the amplitude past roughly 23°. "On the Moon" and "Jupiter" use real surface gravity for comparison; Jupiter has no solid surface to actually stand a pendulum on, so that one is hypothetical. Real pendulums also lose a little energy each swing to air resistance and friction at the pivot, which very slowly shrinks the amplitude — not modelled here, since it would only complicate the one relationship this page is about.