The pendulum law

A pendulum is just a weight (the bob) hanging from a rod or string of length L. For small swings, its period depends on only two things:

T = 2π √(L / g)

Here L is the length from the pivot to the centre of the bob, in metres, and g is the strength of gravity, about 9.81 m/s² at Earth's surface. (This law comes straight from simple harmonic motion, which the Forces & Motion lab derives; here we just use it.)

Read the formula slowly, because three surprises are hidden in it:

• Mass is missing. A heavy bob and a light bob of the same length swing with the same period. Gravity pulls harder on the heavy bob, but it also has more mass to move, and the two effects cancel exactly.
• L is under a square root. Make the pendulum 4 times longer and T becomes only 2 times longer (√4 = 2). To double the period you need 4 times the length.
• g matters. Weaker gravity means a slower swing.

The seconds pendulum. What length gives T = 2 s? Rearrange the law: L = g T² / (4π²). With g = 9.81 m/s² and T = 2 s:

L = 9.81 × 4 / (4 × 9.87) ≈ 0.994 m

That is very close to a metre, which is not a coincidence in the history of units: in the 1600s people proposed defining the metre as the length of the seconds pendulum. Gravity varies from place to place, and that idea was dropped, but the long, slim pendulum case of a grandfather clock is still about this tall for this reason.

The same law tells you what happens on other worlds. Take a 1 m pendulum:

On the Moon the pendulum swings more than twice as slowly, simply because gravity there is about a sixth of Earth's. A pendulum clock carried to the Moon would lose well over half of its time.

A little history. Galileo studied swinging lamps and pendulums in the late 1500s and early 1600s and noticed that the period barely depends on how wide the swing is (a story about a swinging cathedral lamp is often told, but the evidence for that exact tale is thin, so treat it as a legend). He understood that a pendulum could count time, but he never finished a working clock. The Dutch scientist Christiaan Huygens did: his pendulum clock of 1656 to 1657 was the first accurate one. Where the best earlier clocks could be off by many minutes a day, Huygens's first clocks kept time to within about a minute a day, and improved versions soon reached roughly ten seconds a day. That jump, from minutes to seconds, changed astronomy, navigation and science.

There is also a practical use of the law that clockmakers love: it is easy to adjust. If a pendulum clock runs slow, the pendulum is a little too long. A clockmaker corrects this by raising the bob slightly, which shortens it and speeds the clock up. Because T depends on the square root of L, even tiny changes matter: a 1% change in length changes the pace by about half a percent, which is around 7 minutes a day.