The Pendulum That Beats Seconds Is Not One Metre Long

What the formula leaves out

Neither the mass of the bob nor the size of the swing appears anywhere in it, and the first of those is exactly true while the second is only nearly so. The independence from amplitude — a wide swing taking the same time as a narrow one — holds for small angles and drifts as the angle grows: at 30 degrees the real period is about 1.7 per cent longer than the formula says, enough to lose close to twenty-five minutes a day in a clock. Pendulum clocks are therefore built to swing through only a few degrees, and their escapements exist to feed in just enough energy to replace what friction takes without widening the arc.

A pendulum is a gravimeter

Turn the formula round and it measures gravity instead of time: time enough swings of a known length and g falls out. That is precisely how the local strength of gravity was mapped for two centuries, and how the variation between the poles and the equator — about half a per cent — was first pinned down. It also means a pendulum clock keeps different time in different places: carried from Paris to the equator it loses roughly two minutes a day, which is why marine chronometers use a balance spring rather than a pendulum. A clock corrected by moving the bob a millimetre is doing arithmetic on the square root, and a millimetre on a metre changes the period by about a part in two thousand — some forty seconds a day.

Frequently asked questions

Why doesn't the mass matter?

Because it appears on both sides of the problem and cancels. A heavier bob is pulled harder by gravity, but it is also correspondingly harder to accelerate, and the two effects are proportional to the same mass. It is the same cancellation that makes a heavy and a light object fall side by side in a vacuum, and a pendulum is one of the cleanest demonstrations of it available on a tabletop: hang a lead weight and a cork of the same size on identical strings and they keep time together, even though nothing about them feels alike.

Does the length mean the string or the whole thing?

In this formula the length runs from the pivot to the centre of the bob, not to the top of it and not to the bottom. Anything else is a different problem: a real pendulum with a rod that has mass of its own swings according to its moment of inertia, and its effective length is not simply the distance to any single point. For a light string and a compact heavy bob the two descriptions agree closely, which is why the simple version survives in textbooks — but a broom handle balanced on a nail is not a simple pendulum, and treating it as one gets the period noticeably wrong.