The pendulum formula quietly assumes a small swing
The pendulum tool opens showing 2.021 s. The formula in every textbook says 2.0061 s. Neither is a rounding error.
Open the Pendulum Simulator and change nothing. A one metre pendulum, released at 20 degrees, and the readout says the period is 2.021 s.
The formula everyone is taught, T = 2π√(L/g), gives 2.0061 s for the same pendulum. The tool and the textbook disagree in the third digit before you have touched a control.
The tool is right.
What the missing control tells you
Before the disagreement, notice what you are allowed to change: length, release angle, gravity, damping. Four sliders.
There is no mass slider, and its absence is not an oversight. Gravity pulls harder on a heavier bob, and a heavier bob resists acceleration in exactly the same proportion. The two effects are the same number, and they cancel before the period is computed. Mass cannot appear in the answer, so the interface does not offer to change it.
A missing control is sometimes the clearest statement a tool can make.
Where the approximation enters
The true equation of motion for a pendulum contains sin θ. That equation has no solution in ordinary functions, which is inconvenient, so the standard derivation replaces sin θ with θ.
For small angles that substitution is excellent: at 0.1 radians the two differ by about one part in 600. And it changes the problem into simple harmonic motion, whose period is 2π√(L/g), independent of amplitude.
That independence is entirely a property of the replacement. The real pendulum's period does depend on how far you pull it back, and the exact result involves an elliptic integral rather than a formula you would want on a slide.
Drag the angle slider and watch the readout, which is the fastest way to see the size of the lie:
- 5° → 2.0070 s, 0.05% above the formula
- 20° → 2.0215 s, 0.77% above (the default)
- 45° → 2.0863 s, 4.00% above
- 85° → 2.3228 s, 15.79% above
A 16% error is not a subtlety. A clock built on the textbook formula and swinging at 85 degrees would lose about three and a half hours a day.
Why clocks worked anyway
This was understood by the people who needed it. A pendulum clock keeps its swing small, a few degrees, where the error falls below a part in a thousand. The escapement exists to top up the energy lost to friction so the amplitude stays put, because an amplitude that drifts is a clock that drifts.
The seconds pendulum, one second per swing and therefore a 2 s period, comes out at L = g/π², which is 99.4 cm. That near miss against the metre is not a coincidence: a pendulum beating seconds was seriously proposed as the definition of the metre before the meridian survey won.
Pendulum Phase Space shows why the amplitude matters in a way the period readout cannot. Small swings trace near-perfect ellipses, the signature of simple harmonic motion. Push the amplitude up and the closed curves distort, and the distortion is the same sin θ that the derivation threw away.
And then there are two of them
One pendulum is the most predictable object in introductory physics. Attach a second to the end of the first and it becomes a standard example of chaos.
The Double Pendulum has no period to print, and the reason is worth stating carefully: it is not that the system is random or that the simulation is imprecise. The equations are as deterministic as the single pendulum's. It is that two starting angles a thousandth of a degree apart diverge into completely different motions within seconds, so a prediction requires knowing the start to a precision nobody has.
The distance between those two tools is one hinge.