Last updated: 2026-06-14

Simple Pendulum Calculator

A simple pendulum is a mass swinging on a light string or rod under gravity. For swings through a small angle, the time it takes to complete one full back-and-forth cycle — the period — depends only on the length of the pendulum and the strength of gravity: $T = 2\pi\sqrt{L/g}$. Galileo first noticed that the period is nearly independent of how far the pendulum swings, an observation that made the pendulum the heart of accurate clocks for three centuries.

This calculator finds the period and frequency from the length and gravity, or, in the other mode, works backwards from a measured period to the length that would produce it.

Why length is what matters

The period grows with the square root of the length. Quadrupling the length only doubles the period, so a pendulum has to be surprisingly long to swing slowly. A pendulum about one metre long takes roughly two seconds per full swing on Earth — the basis of the old "seconds pendulum" used in grandfather clocks. Neither the mass of the bob nor the size of the swing (within the small-angle limit) changes the period, which is what makes a pendulum such a reliable timekeeper.

Formula

Quantity	Symbol	Meaning
Period	$T$	Time for one full swing, $T = 2\pi\sqrt{L/g}$
Length	$L$	Distance from pivot to centre of the bob
Gravity	$g$	Local gravitational acceleration
Frequency	$f$	Swings per second, $f = 1/T$

Because gravity appears in the formula, the same pendulum runs at a different rate elsewhere: on the Moon, where g is about 1.62 m/s², a one-metre pendulum swings far more slowly, with a period near 4.9 seconds.

Worked example

A pendulum is 1 metre long and sits on Earth, where $g = 9.80665$ m/s². Its period is:

$$\begin{aligned} T &= 2\pi\sqrt{L/g} \\ &= 2\pi\sqrt{1 / 9.80665} \\ &= 2\pi \times 0.3193 \\ &= 2.006\ \text{s} \end{aligned}$$

The frequency is $f = 1/T = 0.499$ Hz, just under one swing per second each way. Entering a length of 1 m reproduces this. To go the other way — say you timed a pendulum at exactly 2 seconds and want its length — switch to "From period" and the calculator returns $L = g \cdot (T/2\pi)^2 \approx 0.994$ m.

The small-angle assumption

The clean formula $T = 2\pi\sqrt{L/g}$ is an approximation that holds when the swing amplitude is small, below roughly 15°. In that range the restoring force is very nearly proportional to the displacement, which is the condition for simple harmonic motion. For wider swings the period lengthens slightly — about 1% at 20° — because the true restoring force grows more slowly than the displacement. Exact results for large amplitudes require an elliptic integral, but for clocks, metronomes and most laboratory pendulums the small-angle formula is more than accurate enough.

Limitations

This model treats the string as massless and the bob as a point, ignores air resistance and friction at the pivot, and assumes a constant gravitational field. A real pendulum with a heavy rod or an extended bob is a physical pendulum, whose period depends on its moment of inertia and the distance to its centre of mass rather than a single length.

Frequently Asked Questions (FAQ)

What is the formula for a pendulum period?

For a simple pendulum swinging through a small angle, the period — the time for one complete back-and-forth swing — is T = 2π√(L/g), where L is the length from the pivot to the centre of the bob and g is the local gravitational acceleration. The frequency, the number of swings per second, is the reciprocal: f = 1/T.

How does length affect the period?

The period grows with the square root of the length, so to double the period you must make the pendulum four times longer. A pendulum 1 metre long has a period of about 2.0 seconds on Earth, which is why a "seconds pendulum" — one that ticks once per second each way — is just under a metre long. Switch this calculator to "From period" to find the exact length for any target period.

Why does the angle not appear in the formula?

The formula T = 2π√(L/g) is the small-angle approximation: it holds when the swing amplitude is small (below about 15°), where the restoring force is very nearly proportional to the displacement. For larger swings the period increases slightly — by roughly 1% at 20° and more at wide angles — and the exact period requires an elliptic integral rather than this simple expression.

Does the mass of the bob matter?

No. The period of a simple pendulum depends only on its length and the local gravity, not on the mass of the bob. A heavy and a light pendulum of the same length swing in step, because gravity accelerates all masses equally — the same reason objects of different mass fall together. Mass would matter only if air resistance or friction were significant.