The Pendulum: Why Only Length Sets the Swing
By Lucian — builder & engineer, LK Forge
Hang a heavy weight and a light one on strings of the same length and swing them together: they keep time with each other exactly. Pull one out a little further and it still matches — up to a point. A simple pendulum ignores almost everything you might change about it, answering to just the length of the string, until you push the swing wide and the famous formula starts to lag. Here is the exact relationship, and a measurement of exactly where it breaks.
A swing in the simple-pendulum simulator. Change the mass or the release angle and the period barely moves; change the length and it changes at once.
| Length L | Period T |
|---|---|
| 0.25 m | 1.00 s |
| 0.5 m | 1.42 s |
| 1 m | 2.01 s |
| 2 m | 2.84 s |
| 4 m | 4.01 s |
g = 9.8 m/s². Note the pattern: 0.25 m → 1 s, and each ×4 in length is exactly ×2 in period. A 1-metre pendulum ticks almost exactly once per second each way — the reason grandfather clocks are about a metre tall.
Why mass and amplitude drop out
Mass disappears for the same reason all objects fall at the same rate: a heavier bob feels a proportionally stronger restoring pull, and the two effects cancel, so mass never enters the period. Amplitude drops out because for small angles the restoring force is very nearly proportional to the displacement — the condition for simple harmonic motion, where the period does not depend on how big the swing is. Push to large angles and that approximation frays: sin θ falls below θ, the restoring pull weakens, and each swing takes a little longer. Exactly how much longer is a thing you can measure — so we did.
How much a wide swing really slows it, measured 1° to 170°
To pin the number down, we ran the simulator’s own engine with no browser and no human: the exact same fourth-order Runge–Kutta integrator it ships, stepping the true equation of motion θ″ = −(g/L)·sinθ at its fixed 1/240-second step, released from rest at each amplitude from 1° to 170° (L = 1 m, g = 9.8 m/s²). The period is timed from the turning points, exactly as the live readout does. Each measured period is set against two references: the small-angle formula T₀ = 2π√(L/g), and the exact large-angle period from the elliptic integral.
The small-angle formula never moves; the true period (exact solution, with the measured RK4 points sitting on top of it) climbs away from it as the swing widens.
How far the small-angle formula T = 2π√(L/g) sits below the true period. Modest for a schoolroom swing, unbounded near 180°, where the real period diverges to infinity.
| Amplitude | Measured T | Exact T | Ideal T₀ | Formula error | Sim vs exact |
|---|---|---|---|---|---|
| 1° | 2.007 s | 2.007 s | 2.007 s | 0.00% | 0.008% |
| 5° | 2.008 s | 2.008 s | 2.007 s | +0.05% | 0.004% |
| 10° | 2.011 s | 2.011 s | 2.007 s | +0.19% | -0.004% |
| 15° | 2.016 s | 2.016 s | 2.007 s | +0.43% | 0.006% |
| 20° | 2.022 s | 2.022 s | 2.007 s | +0.77% | 0.001% |
| 23° | 2.027 s | 2.027 s | 2.007 s | +1.02% | 0.000% |
| 30° | 2.042 s | 2.042 s | 2.007 s | +1.74% | -0.008% |
| 45° | 2.087 s | 2.087 s | 2.007 s | +4.00% | -0.001% |
| 60° | 2.154 s | 2.154 s | 2.007 s | +7.32% | -0.001% |
| 70° | 2.212 s | 2.212 s | 2.007 s | +10.21% | -0.001% |
| 90° | 2.369 s | 2.369 s | 2.007 s | +18.03% | -0.004% |
| 110° | 2.600 s | 2.600 s | 2.007 s | +29.53% | 0.005% |
| 130° | 2.950 s | 2.950 s | 2.007 s | +46.98% | -0.002% |
| 150° | 3.537 s | 3.537 s | 2.007 s | +76.22% | 0.002% |
| 170° | 4.896 s | 4.896 s | 2.007 s | +143.94% | 0.000% |
Measured 9 September 2026, L = 1 m, g = 9.8 m/s². Formula error is the exact period above the small-angle T₀; sim vs exact is the shipped RK4 integrator against the elliptic solution — never more than 0.008% off, which is the whole case that the simulator is physically faithful.
Reproduce it yourself
The physics is four lines. The exact large-angle period is the small-angle T₀ divided by the arithmetic–geometric mean of 1 and cos(θ₀⁄2) — the same function the simulator uses. To get the measured column, step θ″ = −(g/L)·sinθ with RK4 (the sim’s solver) and time the turning points.
const g = 9.8, L = 1; // metres, m/s^2
const T0 = 2 * Math.PI * Math.sqrt(L / g); // small-angle period
// exact period at release amplitude deg, via the AGM
function exactT(deg) {
const th = deg * Math.PI / 180;
let a = 1, b = Math.cos(th / 2);
for (let i = 0; i < 16; i++) { [a, b] = [(a + b) / 2, Math.sqrt(a * b)]; }
return T0 / a;
}
exactT(45); // 2.087 s -> +4.0% over T0
exactT(90); // 2.369 s -> +18%
exactT(170); // 4.896 s -> +144%The full measured sweep, including the RK4 turning-point timing, is the small script behind this post; the integrator it drives is the one the game ships.
Watch it happen
Open the simulator, wind the amplitude up, and watch the measured period pull ahead of the ideal on the live period-vs-amplitude curve.