Pendulum Period: formula
Enter the pendulum length in meters, measured from the pivot to the center of the bob, and the gravitational acceleration. The period is 2 pi times the square root of the length divided by gravity, and the frequency is its inverse. The formula holds for small swings, under about 15 degrees.
The mass of the bob does not appear.
Pendulum Period: worked example
| Pendulum length (m) | 1 |
|---|---|
| Gravity (m/s²) | 9.8 |
| Period (s) | 2 |
| Frequency (Hz) | 0.498 |
The expression is the small-angle limit for an ideal simple pendulum. At a 30-degree amplitude the first correction θ²/16 is about 1.7%, not 4%. At larger amplitudes use the full amplitude-dependent period.
Pivot friction, air resistance and a distributed bob change a real pendulum.
Pendulum Period: changing Pendulum length (m)
| Pendulum length (m) | Period (s) | Frequency (Hz) |
|---|---|---|
| 0.8 | 1.8 | 0.557 |
| 0.9 | 1.9 | 0.525 |
| 1 (example) | 2 | 0.498 |
| 1.1 | 2.1 | 0.475 |
| 1.2 | 2.2 | 0.455 |
Pendulum Period: compare input ranges
| Input | Tested from | To | Period (s): low input | Period (s): high input | Swing |
|---|---|---|---|---|---|
| Pendulum length (m) | 0.9 | 1.1 | 1.9 | 2.1 | 0.201 (10%) |
| Gravity (m/s²) | 9 | 11 | 2.1 | 1.9 | 0.2 (10.0%) |
Pendulum Period: individual input comparisons
| Input | Value | Period (s) | Frequency (Hz) |
|---|---|---|---|
| Pendulum length (m) | 0.8 | 1.8 | 0.557 |
| 0.9 | 1.9 | 0.525 | |
| 1 | 2 | 0.498 | |
| 1.1 | 2.1 | 0.475 | |
| 1.2 | 2.2 | 0.455 | |
| Gravity (m/s²) | 8 | 2.2 | 0.45 |
| 9 | 2.1 | 0.477 | |
| 10 | 2 | 0.503 | |
| 11 | 1.9 | 0.528 | |
| 12 | 1.8 | 0.551 |
Pendulum Period: combined input changes
| Pendulum length (m) / Gravity (m/s²) | 8 | 10 | 12 |
|---|---|---|---|
| 0.8 | 2 | 1.8 | 1.6 |
| 0.9 | 2.1 | 1.9 | 1.7 |
| 1 | 2.2 | 2 | 1.8 |
| 1.1 | 2.3 | 2.1 | 1.9 |
| 1.2 | 2.4 | 2.2 | 2 |
Pendulum Period: which assumptions move the result?
Pendulum Period: values and definitions
| Quantity and definition | Value used |
|---|---|
| Pendulum length (m) | 1 |
| Gravity (m/s²) | 9.8 |
| Period (s) | 2 |
| Frequency (Hz) | 0.498 |
Pendulum Period: half and double Pendulum length (m)
| Pendulum length (m) | Period (s) |
|---|---|
| 0.5 | 1.4 |
| 1 | 2 |
| 2 | 2.8 |
Common error: Thinking a heavier bob swings slower. Mass cancels out of the formula, so only length and gravity set the period.
Physics & Mechanics calculators: Horsepower Calculator, Speed, Distance and Time Calculator, Kinetic Energy Calculator, Newton's Second Law Force Calculator, Uniform Acceleration Calculator, Hooke's Law Calculator.
A pendulum's swing time depends on its length and on gravity, and not on the weight of the bob. A pendulum a little under 1 meter long takes about 2 seconds for a full back and forth, which is why pendulums have long been used to keep time. Quadruple the length and the period doubles.
The rule holds for small swings; a wide swing takes a little longer than the formula says.
Measure the length from the pivot to the center of the bob, not to the bottom of it. A small error in length shifts the period.
See the spring case on the Hooke's law calculator, the energy on the potential and kinetic energy calculators and the wave relation on the wavelength and frequency calculator. The free fall and uniform acceleration pages cover related motion, and the length converter changes units.
What determines the period of a pendulum?
Its length and the local gravity. The mass and small swing size do not affect it.
How long is a pendulum with a 2 second period?
About 1 meter, or 0.994 meters more exactly, which is the classic seconds pendulum.
Does a heavier bob swing faster?
No, the period of an ideal pendulum is independent of the mass of the bob, and depends only on length and gravity.