Result

Simple pendulum period

T = 2 π \frac{L}{g}

Statement

The period of a simple pendulum, for small oscillation angles.

Symbols and units

$T$	period	s
$L$	pendulum length	m
$g$	gravitational acceleration	m/s²

When it applies

Small oscillation angles (roughly under 15°); a rigid, massless string or rod and a point-like bob.

Don't use when

Not valid beyond small angles (roughly 15°) — the period grows and depends on amplitude once the small-angle approximation breaks down; also assumes a massless string/rod and a point-like bob.

Derivation

For small angles, sin θ ≈ θ, so the restoring torque is proportional to angular displacement:

τ = - m g L sin θ \approx - m g L θ

This matches the SHM condition with angular frequency:

ω = \frac{g}{L}

Giving the period:

T = \frac{2 π}{ω} = 2 π \frac{L}{g}

Watch out for

Valid only for small angles (roughly under 15°); at larger amplitudes the period grows and depends on the amplitude, unlike ideal SHM.

Connects to

Simple harmonic motion