Pendulum Calculator

A simple pendulum consists of a mass (the “bob”) suspended from a fixed pivot by a string or rod of negligible mass. When displaced from its resting position and released, gravity causes it to swing back and forth in a regular, predictable rhythm. This regularity made pendulums the basis of clocks for over 300 years.

Period of a simple pendulum formula: T = 2π × √(L / g)

Frequency formula: f = 1 / T

What each variable means:

• T: period: the time for one complete oscillation (one full swing back and forth), in seconds
• L: length of the pendulum from pivot to center of mass of the bob, in meters
• g: gravitational acceleration = 9.81 m/s² on Earth’s surface
• f: frequency: oscillations per second (Hertz, Hz)
• π ≈ 3.14159

Worked example: A grandfather clock pendulum has a length of 0.9940 meters.

T = 2π × √(0.9940 / 9.81) = 2π × √(0.10133) = 2π × 0.31833 = 2.0000 seconds

That is why the classic grandfather clock pendulum is almost exactly 1 meter long. A 2-second period means 1 second each way, so the escapement ticks on the second.

Frequency = 1 / 2.0 = 0.5 Hz (half an oscillation per second)

Worked example 2: A child’s playground swing has a rope length of 3 meters. T = 2π × √(3 / 9.81) = 2π × 0.5530 = 3.47 seconds per full cycle

Key physics insight: The period depends only on length and gravity, not on the mass of the bob, and not on the amplitude (for small angles up to about 20°). Doubling the mass has zero effect on the period. Doubling the length increases the period by √2 ≈ 1.414×.

That amplitude caveat is where this page stops. It takes the length in metres, centimetres, feet or inches, works out how far to move the bob to regulate a clock, and assumes the swing is small.
If the swing is not small, the pendulum period calculator takes the amplitude as a third input and evaluates the exact correction instead of ignoring it. The two pages return the same small-angle period, so you can move between them freely.

Gravity variation: On the Moon (g = 1.62 m/s²), the same 1-meter pendulum would have a period of T = 2π × √(1/1.62) = 4.94 seconds, nearly 2.5× slower. Pick a world in the gravity dropdown and the calculator works in that gravity throughout.

Regulating a real clock

This is the part that matters if you own a pendulum clock rather than a physics problem. Since T is proportional to √L, a fractional change in length produces half that fractional change in period, so:

ΔL / L = 2 × Δt / 86400

for a gain of Δt seconds per day. Put numbers in it and the result is startling: a seconds pendulum near 0.994 m needs to move about 0.023 mm to gain one second a day. That is well under the thickness of a sheet of paper, and it is why the rating nut under the bob is threaded so finely. Raising the bob shortens the pendulum and speeds the clock up; lowering it slows the clock down.

Clockmakers rarely reach for the arithmetic and count turns of the nut instead. The arithmetic is still worth having, because it tells you when a clock is too far out to be regulated at all: if the length needed is a centimetre off what is on the rod, the pendulum is the wrong one for that movement.