The pendulum looks like the simplest possible experiment: a weight on a string, swinging back and forth. It is also where one of physics’s most-quoted formulas quietly stops being true.
Simple Pendulum
Test whether the period formula really holdsThe small-angle formula
T = 2π√(L / g)
Two things stand out. The period does not depend on the mass — a heavy bob and a light one on the same string swing in identical time. And it does not depend on the amplitude either, which is genuinely surprising: a wide swing and a narrow swing should take the same time.
Except they don’t, quite.
Where the formula breaks
Deriving that formula requires replacing sin θ with θ, which is only valid for small angles. Push the starting angle up to 70° and the approximation fails: the real pendulum is measurably slower than the formula predicts.
The simulation integrates the true equation of motion, θ″ = −(g/L) sin θ, and times the actual swings, so you can read the error off directly. At 5° it is a fraction of a percent. At 70° it is several percent — small enough to miss in a school lab, large enough to matter in a precision clock.
Try this
- Set the angle to 5°. Measured period and formula agree almost exactly.
- Set it to 70°. Watch the “Formula error” readout climb.
- Shorten the string to 0.25 m. The period halves, because T scales as √L — you need four times the length to double the period.
- Switch gravity to the Moon. The pendulum swings far more slowly, which is why a pendulum clock calibrated on Earth would run slow there.
Why it matters
Pendulums ran the world’s most accurate clocks for three centuries, and the discovery that the period is nearly independent of amplitude is exactly what made that possible. The same small-angle mathematics appears everywhere in physics — in oscillating circuits, in vibrating molecules, and in any system near a stable equilibrium.