What Determines a Pendulum Period

Quick Answer

A simple pendulum's period depends on just two things: its length and the local gravity. The small-angle formula is T = 2 pi times the square root of L / g. For example, a one-metre pendulum on Earth swings with a period near 2 seconds. What determines the period of a pendulum is length and gravity alone: the mass of the bob and, for small swings, the release angle do not change it. Open the simulator, drag the bob, and watch the pendulum period and the energy readouts update live.

What Determines the Period

Length sets the period most strongly. Because the period grows with the square root of length, making a simple pendulum four times longer only doubles its period. Gravity works the other way: stronger gravity pulls the bob back faster, so a shorter period. For example, on the Moon gravity is about one sixth of Earth's, so the same pendulum swings roughly 2.4 times slower. To calculate a pendulum period by hand, divide the length by g, take the square root, and multiply by 2 pi.

Why Mass Does Not Affect the Period

Mass does not affect the pendulum period, and the reason is the same one Galileo found for falling objects. A heavier bob is pulled harder by gravity, but it also resists changes in motion more, and the two effects cancel exactly. So a heavier bob swings at the same rate as a light one of the same length. Drag the mass control in the simulator and the period reading does not move, even though the energy values scale up.

Potential vs Kinetic Energy

A swing is a constant trade between potential energy and kinetic energy. At the top of the arc the bob is momentarily still, so it holds all potential energy and no kinetic energy. At the bottom it moves fastest, so potential energy has become kinetic energy. For example, release the bob from the left: watch potential energy fall to zero as it reaches the bottom, then rebuild on the right. The total stays constant, which is energy conservation in action, the same idea behind simple harmonic motion.

Small-Angle Period vs True Period

The tidy formula is an approximation. The small-angle formula is only an approximation that holds when the swing stays under about 15 degrees, where the motion is close to true simple harmonic motion. Push the release angle wider and the real period grows longer than the formula predicts, which is why a wide swing is slower than the formula says. For example, at 90 degrees the true period runs roughly 18 percent longer than T = 2 pi times the square root of L / g. The simulator uses the full physics, so you can compare the small-angle period against the true period by widening the angle.

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