Pendulum

Does a pendulum's period follow the textbook small-angle law — and how does it change as the swing grows?

▶ Run the simulationSee the measured result

Units: dimensionless (T(90°)/T₀ = (2/π)·K(sin 45°); K(1/√2) = Γ(¼)²/(4√π))

How the lab tests it

Release from a set amplitude, time the period between turning points, and compare to both the naive √(L/g) law and the exact elliptic-integral period.

What it checks

T₀ = 2π√(L/g) and the exact T = T₀ / AGM(1, cos θ₀/2)

Pendulum period, length, gravity & large-amplitude correction calculator

A grandfather clock, and the reason it only works if the swing stays narrow. Galileo watched a cathedral lamp in 1638 and concluded that a pendulum keeps the same time whatever the arc — 'whether the arc is eighty degrees or two'. He was wrong, and this page prices exactly how wrong. Nothing about the period is stored here. T₀ is ASSEMBLED as 2π√(L/g) out of the two numbers you type, and the large-angle factor as 1/AGM(1, cos(θ₀/2)) — an arithmetic-geometric mean iterated on the spot, which IS the complete elliptic integral K(sin(θ₀/2)) by Gauss's identity, so no elliptic series and no stored 1.18034 appear anywhere in the code; zero the lab's own readings and not one computed number moves. The defaults are this world's bench, a 4 m arm at g = 9.81 released from 90°: T₀ = 4.012133 s, true period 4.7356839 s, a ratio of 1.1803405990161 and an 18.034060% excess. The simulation above reaches that same ratio down a completely different road — it integrates the bare equation of motion θ'' = −(g/L)·sinθ with a symplectic stepper and a stopwatch, with no elliptic integral, no AGM and no correction series anywhere in its recovery — and its 24-seed noisy read of 4.73568844 ± 4.76e-5 s lands 0.095480 bars from the closed form, which is the only reason the agreement is worth anything. Three things here are not restatements of the finding but consequences it never computes. Time a 90° swing, reach for g = 4π²L/T², and you get 7.041324 m/s² instead of 9.81 — gravity understated by 28.2230%, which is what the anharmonicity actually costs anyone using a pendulum as an instrument. 'Small angle' has an honest ceiling: 7.244087° if you want the small-angle law itself good to 0.1%. And the divergence is far lazier than it sounds — doubling the period needs 159.753876°, 88.7522% of the way to the vertical, where going merely half way buys 18.0341%. Four things this page will NOT do. It will not re-run the simulation above. It is the SIMPLE pendulum — a point bob on a massless rigid rod — with no moment of inertia, no finite-size bob and no rod mass, so a physical pendulum needs L replaced by I/(mℓ) before a line here applies. It has no air drag and no pivot friction, so the arc never decays — which is itself the thing that carried Galileo's lamp down into the narrow regime that fooled him. And it will not correct a single number the finding recovered: it prices closed forms, and the measured values belong to the oracle.