AP® Physics 1: Algebra-Based · Unit 7 · Topic 7.2
Period of a Pendulum
A simple pendulum's period depends only on its length and the local gravitational acceleration, following T = 2 π √(L/g). Mass and swing size (for small angles) don't matter.
A pendulum is a mass swinging on a string, restored toward the bottom by gravity. For small swing angles, the restoring force is proportional to displacement, so the motion is simple harmonic, and that lets us write a clean formula for the period: T = 2 π √(L/g). L is the length from pivot to the bob's center, g is the local gravitational acceleration. Bigger L means a longer, slower swing; bigger g means a faster restoring pull and a shorter period.
Mass never appears in the formula because of how gravity works. The restoring force on the bob is proportional to its mass (heavier bob, more force pulling it back), but so is its inertia (heavier bob, more resistance to that force). Those two mass factors cancel in Newton's second law, the same cancellation that makes all objects fall at the same rate in free fall. So a bowling ball and a golf ball on identical strings swing in exact step.
This sits next to the formula for a mass on a spring, T = 2 π √(m/k), and the contrast matters: spring period depends on mass, pendulum period does not, because springs don't involve gravity pulling proportionally on mass. Both formulas share the feature this unit keeps returning to: period is independent of amplitude, as long as the amplitude is small enough that the small-angle approximation holds.
Key terms
- Simple pendulum
- A point mass swinging on a massless, inextensible string from a fixed pivot.
- Period (T)
- The time for one complete back-and-forth swing, measured in seconds.
- Small-angle approximation
- The assumption that swing angles are small enough that the restoring force is proportional to displacement.
- g
- The local gravitational acceleration, about 9.8 m/s2 on Earth, that sets how fast the pendulum restores.
Watch it explained
- How it worksHow does the pendulum period depend on length and g but not at all on the bob's mass?
- Worked exampleFind the period of a pendulum 0.65 m long swinging on Earth.
- Worked exampleFind the length a pendulum needs to tick out exactly one second per swing.
- Common mistakeWhy doesn't a heavier bob on the same string slow the pendulum down?
- Going furtherWhat happens to a pendulum clock carried to the Moon?
Check yourself
End of topic.