How To Calculate The Period Of A Pendulum

The period of a simple pendulum is the time for one complete swing, out and back. The formula is two pi times the square root of the length divided by the acceleration of gravity.

The result is famously independent of mass. Hang a heavy bob or a light one at the same length and they swing together, because gravity pulls harder on the heavy one by exactly the factor that makes it harder to accelerate.

It is also independent of amplitude, at least for small swings. This is the isochronism Galileo noticed watching a lamp swing in a cathedral, and it is what made accurate clocks possible.

The small angle approximation is the fine print. The formula assumes the restoring force is proportional to the displacement, which holds to about a quarter of a percent at 10 degrees but drifts to roughly 1.7 percent slow at 30 degrees.

Only two inputs matter: length and gravity. Everything else, the bob's mass, the string's thickness, the material, drops out of the mathematics entirely.

Longer pendulums swing slower, but not proportionally. Doubling the length multiplies the period by the square root of two, which is 1.414214, so a 2 metre pendulum takes 2.8375 seconds against the 2.0064 of a 1 metre one.

A seconds pendulum is the classic special case. It takes exactly one second to swing each way, giving a two second period, and that needs a length of 0.993621 metres at standard gravity.

That number is why the metre was nearly defined by a pendulum. A one second swing is so close to a metre that the two were proposed as the same standard, before the Earth proved too irregular.

Gravity genuinely varies across the Earth's surface. It is about 9.780 at the equator and 9.832 at the poles, so a pendulum clock calibrated in one place runs slow or fast in another by a fraction of a percent.

Pendulums were the best way to measure gravity for two centuries. Kater's reversible pendulum turned the formula around and read g off a measured period to five significant figures.

Clock makers had to deal with thermal expansion. A steel rod grows with temperature, the period lengthens, and the clock loses time, which is why compensation pendulums using two metals were invented.

The formula describes a simple pendulum: a point mass on a weightless string. A real swinging rod is a compound pendulum, and its period depends on how its mass is distributed as well as its length.

Angular frequency is the square root of gravity over length. For a 1 metre pendulum that is 3.1316 radians per second, and the ordinary frequency is that divided by two pi, which is 0.4984 hertz.

On the Moon, where gravity is about a sixth of Earth's, the same pendulum swings roughly two and a half times slower. Nothing about the pendulum changed, only g.

For a 1 metre pendulum at standard gravity the period is 2.0064 seconds, close enough to two seconds that the approximation is worth memorising.