A pendulum is a weight hanging from a pivot, free to swing. For small swings its rhythm is remarkably steady and depends on only two things: the length of the pendulum and the strength of gravity. That steadiness made pendulum clocks the most accurate timekeepers in the world for nearly three centuries. This calculator finds the period from the length, the length needed for a given period, or the local gravity from a measured period. It also corrects for large release angles, where the familiar textbook formula starts to drift.
How to use the pendulum calculator
- Choose to Solve for the period, the length or gravity (g).
- Pick Where is the pendulum? — Earth’s standard gravity, the Moon, a planet, or a custom g.
- Enter the Length from the pivot to the center of the bob, in m, cm, mm, in or ft, and/or the Period, the time for one full swing out and back.
- Set the Release angle measured from vertical. Use a small angle such as 5° to reproduce the classic formula.
- Read the result with the frequency, the small-angle period for comparison, the bob’s speed at the bottom, a swing diagram and a table of periods at other release angles.
Pendulum period formula
For small swings:
For a release angle θ0, the exact period is
where AGM is the arithmetic–geometric mean. Rearranging gives the length, L = g(T × AGM ÷ 2π)2, and gravity, g = L(2π ÷ (T × AGM))2. The bob’s top speed follows from energy conservation: v = √(2gL(1 − cos θ0)).
Worked example
A 1-meter pendulum released at 10°
Small-angle period: T0 = 2π × √(1 ÷ 9.80665) = 2.00641 s.
Correction: AGM(1, cos 5°) = 0.998096, so T = 2.00641 ÷ 0.998096 = 2.01024 s, 0.19% longer.
The bob rises 1.52 cm at each end of the swing and passes the bottom at 0.546 m/s. Take the same pendulum to the Moon (g = 1.62 m/s²) and the period stretches to 4.95 s.
Working backward. To build a pendulum with a 2-second period in standard gravity, solve for length with a release angle of 0°: L = 9.80665 × (2 ÷ 2π)² = 0.9936 m. Release it at 10° and you would need it 0.38% shorter to keep the same period.
How the period grows with the swing
| Release angle | Period of a 1 m pendulum | Longer than 2π√(L/g) by |
|---|---|---|
| 5° | 2.0074 s | 0.05% |
| 15° | 2.0150 s | 0.43% |
| 30° | 2.0413 s | 1.74% |
| 60° | 2.1532 s | 7.32% |
| 90° | 2.3683 s | 18.0% |
Large swings slow the pendulum because the restoring force grows more slowly than the angle (it follows sin θ, not θ). Clockmakers keep swings small for this reason; a clock whose swing shrinks as its spring winds down would otherwise run fast. Christiaan Huygens found that guiding the string between curved cheeks makes the bob follow a cycloid, for which the period does not depend on amplitude at all.
Real pendulums versus the model
The calculator treats the pendulum as simple: a point mass on a weightless, rigid string. A real bob has size, and a real rod has mass, which make it a physical pendulum with an effective length slightly different from the measured one. Air drag gradually shrinks the swing, and a stretchy string lengthens under load. For precise gravity measurements, time at least 20 swings, keep the release angle under 5° and measure to the bob’s center.
Related tools: the free fall calculator shows what g does to a dropped object, the weight calculator compares gravity across the Solar System, and the potential energy calculator gives the energy the bob trades back and forth.
Frequently asked questions
Does the mass of the bob change the period?
No. A heavier bob feels more gravity but also has more inertia, and the two effects cancel. A 1 m pendulum swings with the same period whether the bob is a steel ball or a cork, as long as air drag is small.
How long is a pendulum that beats seconds?
A seconds pendulum takes one second per swing in each direction, so its full period is 2 s. With standard gravity and a small swing its length is 0.9936 m, which is why tall clock pendulums are close to a meter long.
When is the small-angle formula good enough?
Below about 15°. At a 10° release the exact period is only 0.19% longer than 2π√(L/g), and at 15° it is 0.43% longer. At 60° the error grows to 7.3%, and near 180° the period becomes very long.
Can I measure gravity with a pendulum?
Yes. Time many swings to get an accurate period, measure the length from the pivot to the center of the bob, and solve for g. Timing 20 swings instead of one shrinks the effect of your reaction time twentyfold.
How is the large-angle correction calculated?
The exact period involves a complete elliptic integral. The calculator evaluates it with the arithmetic–geometric mean: T = 2π√(L/g) ÷ AGM(1, cos(θ₀/2)). This converges in a handful of steps and is exact to machine precision.