What this calculator does
A simple pendulum's period depends on its length and on gravity, and on nothing else. Mass does not appear in the formula and neither does the size of the swing, provided the swing stays small, which is the result that made pendulums the basis of accurate clocks for centuries.
The relationship involves a square root, so a pendulum four times as long does not swing four times as slowly. It swings twice as slowly, which is why grandfather clocks are tall but not absurdly so.
The formula
The period is 2π times the square root of length divided by gravity. Frequency is the reciprocal of the period. Rearranging the same formula gives the length needed for any chosen period, which is how clock pendulums are set.
| Term | Meaning |
|---|---|
| Period (T) | The time for one complete swing, out and back to the starting point. |
| Small-angle approximation | The assumption that the swing is small enough for the period to be independent of amplitude. It holds well below roughly 15 degrees. |
| Frequency | Swings per second, the reciprocal of the period, in hertz. |
The inputs explained
| Field | What to enter |
|---|---|
| Pendulum length (m) | The length from the pivot to the centre of mass of the bob, in metres. |
| Gravity (m/s²) | Local gravitational acceleration. 9.80665 is the standard value; it varies slightly with latitude and altitude. |
When to use it
Setting a clock pendulum
A clock runs fast or slow depending on pendulum length, and the formula gives the length needed for the required period.
Measuring local gravity
Because the formula contains gravity, timing a pendulum of known length is a classic way to measure g experimentally.
Designing a swing or a metronome
Anything that oscillates on a pivot follows the same relationship, so the period can be predicted before building it.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the period change with pendulum length?
A range of pendulum lengths at Earth gravity.
| Length | Period | Swings in 60 s |
|---|---|---|
| 0.25 m | 1.003 s | 59.8 |
| 0.5 m | 1.419 s | 42.3 |
| 1 m | 2.006 s | 29.9 |
| 2 m | 2.837 s | 21.1 |
| 4 m | 4.013 s | 15.0 |
Questions
Does a heavier bob swing more slowly?
No. Mass cancels out of the equation entirely. A heavy bob and a light one on identical strings swing at the same rate, which surprises most people the first time they see it demonstrated.
Does the size of the swing matter?
Only slightly, and only once the swing gets large. Below about 15 degrees the period is essentially independent of amplitude, which is the small-angle approximation this calculator uses. At large angles the real period is slightly longer.
Why is a one-second pendulum about 0.25 m?
It falls out of the formula at Earth gravity. A pendulum of about 0.248 metres has a period of one second, and one of about 0.994 metres has a period of two seconds, which is the classic seconds pendulum that ticks once per swing.
Would a pendulum clock work on the Moon?
It would swing, but far more slowly, since lunar gravity is about a sixth of Earth's. The period would be roughly two and a half times longer, so the clock would run badly slow unless the pendulum were shortened to match.
For gravity as an acceleration in other contexts, see the force, mass and acceleration calculator. For circular motion rather than swinging, see the angular momentum calculator.