Why a pendulum keeps time

Pull the bob aside and let go. Height turns into speed and back again. Change the string, gravity or mass and see which of them the period actually cares about.

String and bob: the swinging massPotential energy: bob height above its lowest pointKinetic energy: speed, shown by the red arrowAt the top of each swing the bob stops and all its energy is height (blue). Through the bottom it is all speed (red). The black tick on the base is the total, which stays put unless damping is on.

Measured periodtiming…

Small-angle formula T₀–

True period at this swing–

Small swing: the formula is within 1%

Potential 0Kinetic 0Total 0 (scales with mass)Mass 1.0 kg: not in the formula

String length

Gravity

Bob mass

Air drag (damping)

Click the figure, then: ← → pull the bob (Shift for bigger steps) · Space release · ↑ ↓ length · PgUp PgDn gravity · M mass · D drag · Home reset

Longer string, slower swing

The ideal period is T₀ = 2π√(L/g). Four times the length doubles it; stronger gravity shortens it. Mass is not in the formula: gravity pulls a heavier bob harder, but it also takes more force to move it, and the two cancel. Try the mass slider and watch the period stay put while the total energy changes.

A caveat: small angles only

The formula replaces sin θ by θ, which holds for small swings. Bigger swings take longer: about 7% at 60°. The "true period" card uses the exact result, so you can see the formula drift as you pull the bob farther. Air drag slowly shrinks the swing without changing a small swing's period much.