Pendulum Period Calculator
From a single length, get the time one swing takes and how many swings happen each second — the two numbers behind every clock, metronome, and physics demo.
One input, two answers
Enter the length and the calculator returns the period (2π√(L ÷ g)) and the frequency (1 ÷ period) at once.
Use SI units
Enter the length in metres (m) — then the period comes back in seconds (s) and the frequency in hertz (Hz).
What is a pendulum's period?
Length in, period and frequency out
The period is the time one full swing — over and back — takes. For a simple pendulum it is fixed by a single quantity, the length, because the only other thing it depends on is gravity, which is essentially constant near Earth's surface, g = 9.80665 m/s². The relationship is period = 2π√(L ÷ g). Once you know the length, the period is set — and so is the frequency, the number of swings per second, frequency = 1 ÷ period. Remarkably, neither the mass of the bob nor (for small swings) the size of the swing changes the answer. That makes length the one input you need for clocks, metronomes, and physics demonstrations.
Enter the length in metres to get the period and frequency instantly.
Two short formulas, both built from the length (L) and the fixed standard gravity (g).
period = 2π√(L ÷ g)The period is two pi times the square root of the length divided by gravity. The frequency — the number of swings per second — is just the reciprocal of the period (1 ÷ period). Because g is fixed, the period depends on the length alone, and it grows with the square root of the length: longer pendulums swing more slowly.
Suppose a pendulum is exactly one metre long, that is L = 1 m.
Period
2 × π × √(1 ÷ 9.80665) = 2.006409 s — about 2 seconds for one full swing.
Frequency
1 ÷ 2.006409 = 0.498403 Hz — just under half a swing each second.
Cross-check
Quadruple the length to 4 m and the period roughly doubles to 4.012819 s — period scales with the square root of length.
The two outputs answer two different practical questions. The period (about 2 s for a 1 m pendulum) is how long one full swing lasts in time — it sets the tick of a clock or the beat of a metronome. The frequency (about 0.498 Hz here) is how many swings happen each second, the reciprocal view of the same motion. The key insight is the square-root relationship: the period grows with the square root of the length, not in proportion to it, so to double the period you must quadruple the length, and to halve it you cut the length to a quarter. A pendulum about one metre long takes roughly two seconds per swing — the basis of the famous "seconds pendulum" used in early precision clocks. Longer pendulums swing slower and shorter ones faster, which is exactly why a grandfather clock is tall while a wristwatch escapement is tiny. Remember that the formula assumes small swings, so it is most accurate when the amplitude stays modest.
The formula is exact in the small-angle limit, but a couple of practical points are worth keeping in mind.
Small angles and ideal conditions
This tool uses the small-angle approximation, accurate for swings up to about 15°–20°; for larger amplitudes the true period is slightly longer than the formula returns. It also assumes an ideal simple pendulum — all the mass at a single point on a weightless, inextensible string, with no air resistance or friction at the pivot. The value of gravity used (9.80665 m/s²) is the standard value; local gravity varies slightly with latitude and altitude. Keep the length in metres so the period comes back in seconds and the frequency in hertz.