Pendulum Calculator — Period, Length, and Frequency
Calculate pendulum period and frequency using T = 2π√(L/g). Covers simple vs compound pendulums, gravity variations, and clock-making examples.
A pendulum's period depends only on its length and the local gravity — not on the mass of the bob or the swing amplitude (for small swings). Galileo noticed this while watching a chandelier swing in a cathedral, timing it with his pulse. That insight eventually led to accurate mechanical clocks and a way to measure local gravity precisely.
The CalcHub pendulum calculator computes period, frequency, or required length for any pendulum.
The Formula
T = 2π × √(L/g)- T = period (time for one complete swing, seconds)
- L = pendulum length (meters)
- g = gravitational acceleration (m/s²)
Classic Pendulum Lengths
| Period | Required Length (at g = 9.81) |
|---|---|
| 1 second | 24.8 cm |
| 2 seconds | 99.4 cm (~1 meter) |
| 5 seconds | 6.21 m |
| 10 seconds | 24.8 m |
Amplitude and the Small-Angle Limitation
The formula T = 2π√(L/g) assumes small amplitudes (< ~15°). For larger swings, the actual period is longer. At 90°, the pendulum takes about 18% longer per swing than the formula predicts.
For a simple clock, keeping swings to about 5–10° ensures the formula remains accurate to within 0.2%.
Worked Example
A grandfather clock uses a pendulum with period 2 seconds. How long must it be?
L = g × T² / (4π²) = 9.81 × 4 / (4 × 9.87) = 39.24 / 39.48 ≈ 0.994 m
Just under a meter. That's why grandfather clocks are typically about 2 m tall — the pendulum needs room to swing.
Why doesn't mass affect pendulum period?
Because gravity accelerates all masses equally. A heavier bob has more gravitational force, but also more inertia (resistance to acceleration) in exactly the same proportion. The two effects cancel, and mass drops out of the equation entirely — just like in free fall.
Can pendulums be used to measure local gravity?
Yes, and historically they were. Measure the period T and length L of a pendulum precisely: g = 4π²L/T². This method was used to map Earth's gravitational variations, which correspond to underground density differences — useful for mineral and oil exploration.
What is a compound or physical pendulum?
A simple pendulum assumes all mass is at the end of a massless rod. A compound (physical) pendulum has mass distributed along its length. Its period uses the moment of inertia: T = 2π√(I/(mgh_cm)), where h_cm is the distance from pivot to center of mass. A physical pendulum swings slower than a simple pendulum of the same length.