Pendulum Period Explained: How T = 2π√(L/g) Works

A pendulum is one of the few physics objects you can hold in your hand and time with a phone. Tie a weight to a string, let it swing, and the time it takes to go out and come back is shockingly steady. That steadiness is the whole point. It is what made pendulums the heart of accurate clocks for nearly 300 years, and it is what makes them a favorite first lab in any physics class.

The behavior comes down to one short equation. Once you see what each part of it does, a lot of confusing pendulum questions answer themselves.

The formula behind every swing

For a simple pendulum, the period is:

T = 2π√(L/g)

Here T is the period in seconds, L is the length of the pendulum in metres, and g is the local gravitational acceleration, about 9.81 m/s² on Earth. The period is the time for one full round trip: out and back to where it started.

The √ sign is the part worth staring at. Length sits under a square root, which means it has a square-law grip on time. To double the period you do not double the length, you have to make the pendulum four times longer. Cut the length to a quarter and the period only halves. This is why short clock pendulums tick fast and tall grandfather clocks tick slow, but never quite as slow as their height alone would suggest.

You can run the numbers yourself in the pendulum period calculator: enter a length, pick a gravity, and it returns the period, the frequency f = 1/T, and the half-period, which is the time for a single one-way swing.

A worked example: the 1 metre pendulum

Take a pendulum exactly 1 metre long on Earth. Drop the numbers in:

T = 2π√(1 / 9.81) = 2π × √(0.1019) = 2π × 0.3193 ≈ 2.006 seconds

So a 1 metre pendulum swings out and back in just over two seconds. Its frequency is f = 1/T ≈ 0.499 Hz, just under half a swing per second, and its half-period is about 1.003 seconds, meaning it crosses from one side to the other in almost exactly one second.

That near-one-second one-way crossing is not a coincidence of taste. Clockmakers chose lengths close to a metre on purpose, because a roughly metre-long rod gives a clean one-tick-per-second beat. The classic "seconds pendulum," which ticks once each way, is about 0.994 metres, a hair shorter than a full metre. When you size a clock or a metronome, this is the math doing the deciding.

Why a heavier bob does not swing faster

The single most surprising thing about pendulums is hiding in plain sight in the formula: there is no mass anywhere in T = 2π√(L/g). The period depends on length and gravity, and nothing else.

That means a heavy brass weight and a light cork, hung on strings of equal length, swing in perfect step. The mass cancels out of the equation of motion because the same gravity that pulls harder on a heavy bob also has to accelerate that heavier bob, and the two effects exactly offset. Galileo reportedly noticed this watching a lamp sway in a cathedral around 1602, timing it against his own pulse, and it remains the cleanest demonstration in introductory physics. A grandfather clock keeps time whether its pendulum weighs a kilogram or three.

This is also a great classroom moment. Show students the formula, point out that mass is absent, and let them predict that two very different weights will match. Then run the real swings. I tried this with two improvised pendulums on my desk, a heavy hex nut and a folded paper clip on threads cut to the same length, and watching them stay locked together swing after swing was more convincing than any line in a textbook. The eye refuses to believe it until it sees it.

The small-angle assumption, and where it breaks

There is a quiet condition attached to T = 2π√(L/g): it is the small-angle approximation. The formula assumes the swing stays narrow, under about 15 degrees from vertical. Inside that range it is accurate to better than half a percent, which is plenty for homework, lab predictions, and sizing a metronome.

Push the amplitude wider and the formula starts to drift. At 45 degrees the true period is roughly 4 percent longer than the formula predicts, so a wide swing runs slow. The real period depends weakly on amplitude, but the math for that involves an elliptic integral, which is exactly why the textbook version drops it and pins itself to small swings.

In practice this matters most in the lab. If you release a pendulum from far out of vertical and your stopwatch average comes out a few percent above the prediction, the amplitude is usually the culprit, not a measurement error. Keep the release angle small and the prediction tightens right up. The same logic applies when you tune a real clock: pendulum clocks are built to swing through only a few degrees so the period stays as constant as possible.

Solving it backwards: from beat to length

The formula runs in reverse just as cleanly. If you know the period you want and need the length to build it, rearrange:

L = g(T / 2π)²

Say you want a pendulum that completes one full swing per second, T = 1. On Earth:

L = 9.81 × (1 / 6.283)² = 9.81 × 0.02533 ≈ 0.248 metres

So a one-second pendulum is about 24.8 cm long. Want a slow, hypnotic three-second swing for a kinetic sculpture? Plug in T = 3 and you get about 2.24 metres. This is the practical side of the equation: you size the rod before you cut anything. The pendulum period calculator has a period-to-length mode that does this for you, and it lets you swap Earth gravity for the Moon (1.62) or Mars (3.72) to see how the same length behaves on another world. A 1 metre pendulum that takes 2.006 seconds on Earth needs about 4.93 seconds on the Moon, because T scales as 1/√g.

Common traps when calculating periods

A few mistakes show up again and again, and they are worth flagging before you trust a number:

Metres, not centimetres. The formula expects L in metres. A 50 cm pendulum is 0.5, not 50. Type 50 and you get an absurd 14-second period.
Period is a round trip. T is out and back. The time for a single one-way swing is T/2. A seconds pendulum ticks every 1 second, so its full period is 2 seconds, not 1.
Keep the swing small. The formula is only the small-angle case. Past about 15 degrees, expect the real period to run measurably longer.
Set the right gravity. The default is Earth's 9.81. A Moon or Mars problem solved with Earth gravity is wrong by the whole gravity ratio.

If your problem set mixes pendulum work with wave or frequency questions, the related frequency and wavelength calculator covers the f = 1/T side of oscillations and the c = fλ relationship for waves, which pairs naturally with the pendulum's frequency output.

A pendulum is a small machine for turning length into time, and the formula T = 2π√(L/g) is the whole instruction manual. Get the length in metres, keep the swing narrow, and forget the mass entirely. The rest is arithmetic you can check in seconds.

Made by Toolora · Updated 2026-06-13