Pendulum Period T = 2π√(L/g): Galileo and Foucault

Galileo and the lamp

A weight on a string, swinging freely, is a pendulum. In 1602 Galileo wrote a letter that describes his experiments with it, noting that swings of different size take almost the same time. A popular story says he first noticed this as a young student, watching a lamp sway in the cathedral of Pisa and timing it by his pulse. Historians think that is a legend, told only after his death. What is certain is the idea: a pendulum is a clock. Count its swings and you count time.

The period ignores the mass

The time of one full swing, out and back, is the period T. For small swings it is T = 2π√(L/g), where L is the length of the string in metres and g = 9.81 m/s². Note what is missing: the mass. A pendulum 2 m long has T = 2π√(2 / 9.81) ≈ 2.84 s, whether its bob weighs 1 kg or 3 kg. The animation shows two pendulums of 2 m, one three times heavier, swinging in perfect step. The heavier bob is pulled harder by gravity, but it is also harder to move, and the two effects cancel exactly.

A shorter string swings faster

Length matters, but only through a square root. Shorten the string from 2 m to 0.5 m, a quarter, and the period halves: T = 2π√(0.5 / 9.81) ≈ 1.42 s. The short pendulum makes twice as many swings in the same time. Gravity matters too: the 2 m pendulum swings in 6.98 s on the Moon (g = 1.62 m/s²) and in 1.78 s on Jupiter (g = 24.79 m/s²). In the sandbox, drag the length between 0.3 m and 2.6 m and choose Earth, Moon or Jupiter, then compare the period with the formula.

Height becomes speed

At the top of its swing the bob stops for an instant and is highest: it stores energy in height, E = m·g·h. At the bottom it is lowest and fastest, and the energy is now motion, ½·m·v². With no losses the two are equal, so v = √(2·g·h). A 2 m pendulum pulled out by 30° starts h = 2 × (1 − cos 30°) ≈ 0.27 m above its lowest point, and passes the bottom at v = √(2 × 9.81 × 0.268) ≈ 2.29 m/s. Mass cancels again. Bigger swings also take slightly longer: at 60° the period is about 7 % more, 3.04 s instead of 2.84 s.

Where the swing goes

Real pendulums slow down and stop. The string rubs at the pivot and the bob pushes air aside, and every bit of friction takes energy from the swing. The energy is not destroyed; it becomes heat, warming the air and the pivot by a tiny amount. This is the law of conservation of energy: the total stays the same, but it spreads out. Clockmakers found a cure. In 1656 Christiaan Huygens built the first pendulum clock, in which a falling weight or a spring tops up the swing with small pushes. In the animation, the heat bar grows as the swing shrinks.

Foucault's pendulum

In 1851 the French physicist Léon Foucault hung a 28 kg ball on a 67 m wire inside the Panthéon in Paris. Its period was T = 2π√(67 / 9.81) ≈ 16.4 s. Once set swinging, the pendulum kept its plane of swing while the floor slowly turned beneath it, so visitors saw the line of swing rotate: proof, in a single room, that the Earth spins. The rate depends on latitude φ: it is 15° per hour × sin φ. In Paris (φ ≈ 48.9°) that is about 11.3° per hour, a full turn in roughly 32 hours. At the North Pole it would take 24 hours, and at the equator the plane would not turn at all.

The pendulum