Table of Contents
Simple Pendulum Frequency
A simple pendulum consists of a point mass suspended by a massless, inextensible string. When displaced from equilibrium by a small angle, it oscillates with a frequency that depends only on the string length and gravitational acceleration. Remarkably, the frequency is independent of the mass of the bob and the amplitude of oscillation (for small angles).
Pendulums have been used for timekeeping since Galileo's observations and Huygens' invention of the pendulum clock. They are also used to measure gravitational acceleration, study harmonic motion, and as scientific instruments. Foucault's pendulum demonstrated Earth's rotation in 1851.
Formula
Valid for small angle oscillations (typically less than 15 degrees).
Key Factors
| Factor | Effect on Period |
|---|---|
| Longer string | Increases period |
| Stronger gravity | Decreases period |
| Heavier bob | No effect (small angle) |
| Larger amplitude | Slight increase (non-linear) |
Frequently Asked Questions
Why doesn't mass affect the period?
The restoring force (gravity component) is proportional to mass (F = mg sin θ), but the inertia resisting acceleration is also proportional to mass (F = ma). Mass cancels out in Newton's second law, just as it does in free fall. This is a consequence of the equivalence of gravitational and inertial mass.
What happens at large amplitudes?
For amplitudes beyond about 15 degrees, the small-angle approximation (sin θ ≈ θ) breaks down. The exact period involves an elliptic integral and is always longer than the small-angle prediction. At 90 degrees, the period is about 18% longer than the small-angle value.