Pendulum Calculator
Calculate the period and frequency of a simple pendulum from its length and gravity — or find the length needed for a given period.
How to use this calculator
- 1Choose whether to find the period from a length, or the length for a desired period.
- 2Enter the value you know.
- 3Adjust gravity to model the Moon, Mars, or another world — a pendulum runs slower in weaker gravity.
How it works
Simple pendulum period
T = 2π √(L ÷ g) T = period (seconds), L = length (m), g = gravity (m/s²) frequency f = 1 ÷ T length for a period: L = g (T ÷ 2π)² independent of the bob's mass and (for small swings) amplitude
A pendulum's period — the time for one complete back-and-forth swing — depends only on its length and the local gravity, provided the swing is small. Remarkably, it does not depend on the mass of the bob or on how far it swings: a heavy weight and a light one on strings of equal length keep the same time. Galileo noticed this constancy watching a swinging lamp, and it is what made the pendulum the heart of accurate clocks for nearly three centuries.
Worked example
A one-metre pendulum on Earth (g = 9.81) has a period of 2π √(1 ÷ 9.81) ≈ 2.006 seconds — almost exactly two seconds. A pendulum that ticks once per second (a 'seconds pendulum', period 2 s) is very close to one metre long, which is no coincidence: it was an early proposed definition of the metre.
Pendulum Calculator: the complete guide
Why mass doesn't matter
The most surprising thing about a pendulum is what is absent from its formula: the mass of the bob. A cannonball and a marble on strings of equal length swing in perfect step. The reason is the same one behind Galileo's falling bodies — gravity accelerates all masses equally, so a heavier bob feels a proportionally larger restoring force but also has proportionally more inertia resisting it, and the two cancel exactly.
This mass-independence is what made the pendulum such a precise timekeeper. Only length and gravity set the period, both of which can be held very stable, so a pendulum clock's rate depends on nothing that drifts with temperature of the bob or the force of the push. It is also why the same pendulum can be used to measure local gravity: time its swing, and the formula gives g.
The small-angle assumption
The clean formula T = 2π√(L/g) is an approximation that holds only for small swings — up to about 15 to 20 degrees. Within that range the period is genuinely independent of amplitude, a property called isochronism, and this is the regime a pendulum clock deliberately stays in. Push the swing wider and the period lengthens slightly, because the restoring force no longer grows in exact proportion to the displacement.
For a swing of 90 degrees the period is several percent longer than the simple formula predicts, and the exact solution requires elliptic integrals rather than a neat closed form. This is why grandfather clocks use a small, controlled swing: it keeps them in the isochronous zone where the period is constant, so the clock's rate doesn't change as the driving spring or weight gradually weakens and the amplitude drifts.
Pendulums, gravity, and the shape of the Earth
Because a pendulum's period responds directly to gravity, it became one of the first precision instruments for measuring g and, through it, the Earth itself. In the 17th and 18th centuries, expeditions carried pendulum clocks around the world and found they ran slightly slow near the equator — evidence that gravity is weaker there, which in turn showed the Earth bulges at the equator rather than being a perfect sphere.
The connection runs the other way too. A pendulum on the Moon, where gravity is a sixth of Earth's, swings about 2.5 times slower for the same length, because the period varies inversely with the square root of gravity. Setting the gravity field above lets you see this directly: the same pendulum keeps very different time on different worlds, which is why a pendulum clock brought to the Moon would run badly slow until its length was shortened.
Frequently asked questions
What is the period of a pendulum?
The time for one complete back-and-forth swing, given by T = 2π√(L/g) for small swings, where L is the length and g is gravity. A one-metre pendulum on Earth has a period of about 2 seconds.
Does a pendulum's period depend on the mass of the bob?
No. The period depends only on the length and local gravity — the mass cancels out, exactly as all objects fall at the same rate. A heavy and a light bob on strings of equal length keep identical time.
Does how far a pendulum swings change its period?
For small swings (under about 15–20°), no — the period is constant regardless of amplitude, a property called isochronism. For large swings the period lengthens slightly. Pendulum clocks stay in the small-angle range to keep accurate time.
How does gravity affect a pendulum?
The period varies inversely with the square root of gravity, so weaker gravity means a slower swing. A pendulum on the Moon (gravity 1.62 m/s²) swings about 2.5 times slower than the same pendulum on Earth. This is why pendulums can be used to measure local gravity.