Simple Pendulum Period
Period of a simple pendulum.
Formula
Variables
Example
A 1 m pendulum swings with period T = 2π√(1/9.8) ≈ 2.0 s.
How it works
This formula gives the time for one complete back-and-forth swing of a simple pendulum when the angle is small and air resistance is negligible. It comes from the fact that gravity provides a restoring force that is approximately proportional to the displacement along the arc, making the motion very close to simple harmonic motion. The result shows that the period depends only on the pendulum length and the local gravitational field strength, not on the bob’s mass.
When to use it
Use this for idealized pendulums in physics problems, especially small-angle oscillations in labs or textbook calculations. It is also useful for estimating local gravity from a measured period and length, provided the pendulum is close to the simple-pendulum assumptions.
Worked example, step by step
A pendulum has length L = 2.50 m on Earth. Use g = 9.81 m/s². T = 2π√(L/g) T = 2π√(2.50/9.81) T = 2π√(0.2548) T = 2π(0.5048) T ≈ 3.17 s Final answer: the period is about 3.17 seconds.
Common mistakes to avoid
- Using the bob’s mass in the formula; mass does not affect the ideal period.
- Forgetting that L must be the pendulum length from pivot to center of mass, not just the string length.
- Applying the formula to large swing angles where the small-angle approximation is no longer accurate.
Did You Know?
A pendulum's period is independent of its swing size and mass — the discovery that made accurate clocks possible.
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