Pendulum Calculator
The swing period of a simple pendulum from its length — and the length that gives exactly one second, which is how the metre was nearly defined.
For small swings the period is 2π√(L ÷ g), which has a famous and counter-intuitive property: it does not depend on the mass of the bob at all, nor on the amplitude, provided the swing stays small.
How the pendulum calculator works
For small swings the period is 2π√(L ÷ g), which has a famous and counter-intuitive property: it does not depend on the mass of the bob at all, nor on the amplitude, provided the swing stays small. A heavy bob and a light one on the same string keep the same time.
The small-angle assumption is a real limit. Past about 20° the true period grows, by roughly 1% at 23° and 4% at 45°, because the restoring force stops being proportional to displacement. This page shows the corrected figure alongside.
Formula: T = 2π √(L ÷ g)
Worked examples
| Inputs | Period | Note |
|---|---|---|
| A one-metre pendulum | 2.00641 s | 2.0064 s — a full swing there and back |
| The seconds pendulum | 1.99998 s | a 2-second period — one second each way |
| On the Moon | 4.93654 s | 4.94 s — much slower |
FAQFrequently asked questions
Does a pendulum's period depend on the weight?
No. A heavy bob and a light one on the same string keep the same time — Galileo's observation, and it still surprises people.
What is the formula?
T = 2π √(L ÷ g), for small swings.
How long is a one-second pendulum?
About 0.994 m gives a two-second period, so each swing across takes one second. It was once proposed as the definition of the metre.
Does amplitude matter?
Only slightly at small angles — about 1% longer at 23° and 4% at 45°. The corrected figure is shown.
Why does gravity change it?
Gravity provides the restoring force. On the Moon, at a sixth of Earth's gravity, the same pendulum swings about 2.5 times slower.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Measurement Institute — Australia's measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.