Pendulum Period Formula
Calculate the period of a simple pendulum with T = 2pi sqrt(L/g).
Learn how length and gravity affect swing time with examples.
The Formula
The period of a simple pendulum depends only on the length of the string and the acceleration due to gravity. It does not depend on the mass of the bob or the amplitude of the swing (for small angles).
Variables
| Symbol | Meaning |
|---|---|
| T | Period — time for one complete swing (measured in seconds, s) |
| π | Pi (approximately 3.14159) |
| L | Length of the pendulum (measured in meters, m) |
| g | Acceleration due to gravity (9.81 m/s² on Earth) |
Example 1
A grandfather clock has a pendulum 1 meter long. What is its period?
Identify: L = 1 m, g = 9.81 m/s²
T = 2π × √(1 / 9.81)
T = 2π × √(0.1019)
T = 2π × 0.3193
T ≈ 2.006 s (about 2 seconds — this is why grandfather clocks use 1 m pendulums)
Example 2
A playground swing has chains 3 meters long. How long does one full swing take?
Identify: L = 3 m, g = 9.81 m/s²
T = 2π × √(3 / 9.81)
T = 2π × √(0.3058)
T = 2π × 0.5530
T ≈ 3.47 s
When to Use It
Use the pendulum period formula for oscillation and timing problems:
- Designing clocks and timing devices
- Calculating the period of swings, chandeliers, or hanging objects
- Measuring gravitational acceleration by timing a pendulum
- Understanding simple harmonic motion in physics courses
Key Notes
- Formula: T = 2π√(L/g): T is the period (time for one complete swing), L is the pendulum length, and g ≈ 9.81 m/s². Notably absent: mass. A heavy bob and a light bob on the same length pendulum swing at exactly the same rate.
- Small-angle approximation: The formula is accurate only for amplitudes below ~15°. At 30°, the true period is ~1.7% longer; at 45°, ~4% longer. For large amplitudes, the exact period requires an elliptic integral.
- Amplitude independence (isochronism): For small swings, the period is independent of amplitude — Galileo reportedly observed this using his pulse to time a swinging chandelier. This property made pendulums valuable for timekeeping before electronic clocks.
- Gravitational variation: g varies from 9.780 m/s² at the equator to 9.832 m/s² at the poles. A pendulum clock that keeps perfect time in London would gain about 5 minutes per day near the equator (lower g → longer period → clock runs slow, not fast).
- Applications: Pendulums are used in grandfather clocks, seismometers (hanging mass detects ground motion), laboratory measurements of g, Foucault pendulums (demonstrate Earth's rotation), and as a reference in physics education for simple harmonic motion.