The simple pendulum
A simple pendulum is a small weight hanging from a string or a light rod, free to swing. For small swings, the time it takes to go back and forth, its period, depends only on its length and on gravity. That regularity made pendulums the heart of accurate clocks for nearly three centuries.
The formulas
- Period T = 2π × √(L ÷ g)
- Length for a given period: L = g × (T ÷ 2π)²
- Frequency = 1 ÷ T, in hertz
- Full swings per minute = 60 ÷ T
L is the length from the pivot to the center of the weight, and g is the acceleration of gravity, 9.80665 m/s² on Earth. The period is a full back-and-forth swing; a single swing from one side to the other takes half of it.
Worked example
A pendulum 3 ft long swings on Earth.
- 3 ft = 0.914 m.
- T = 2π × √(0.914 ÷ 9.81) = 1.919 seconds for a full swing.
- Frequency: 0.521 Hz, or 31.3 full swings per minute.
- Each one-way swing takes 0.959 seconds.
Switch the mode to “the period” and enter 2 seconds: the calculator returns 0.994 m, about 39.1 inches, the classic seconds pendulum whose every swing lasts exactly one second.
Periods at a glance
| Length | Period on Earth |
|---|---|
| 4 in (0.1 m) | 0.63 s |
| 1 ft | 1.11 s |
| 3 ft | 1.92 s |
| 39.1 in (0.994 m) | 2.00 s |
| 10 ft | 3.50 s |
| 220 ft (67 m), Foucault’s pendulum in Paris | 16.4 s |
To double the period, you need four times the length.
What does not matter
- The mass: a heavy bob and a light one on the same string swing at the same rate, because a heavier bob is pulled harder but is also harder to accelerate.
- The amplitude, as long as it is small. The formula is accurate to 0.2% up to about 10°. At 30° the true period is 1.7% longer, and at 60° about 7% longer.
Gravity and pendulums
Because the period depends on g, pendulums were once used to measure gravity itself. On the Moon, a 3 ft pendulum would take 4.7 seconds per swing; on Jupiter, only 1.2 seconds. Try the gravity selector to compare.
Real pendulums
A child’s swing, a crane hook and a clock pendulum all behave approximately like a simple pendulum. When the mass of the rod is not negligible, or the weight is large compared with the length, it becomes a physical pendulum, whose period is somewhat shorter than this formula predicts for the same overall length.