EQUATION

Small-Angle Pendulum Period

Gives the oscillation period of a simple pendulum when the swing amplitude is small (θ₀ ≲ 15°)

§ 01

The equation

EQ.SMALL-ANGLE-PENDULUM-PERIOD

T = 2\pi\sqrt{\frac{L}{g}}

§ 02

What it solves

Gives the oscillation period of a simple pendulum when the swing amplitude is small (θ₀ ≲ 15°). The remarkable result is that T depends only on length and g, not on mass or amplitude.

§ 03

When to use it

Laboratory pendulums, grandfather clocks, small-angle limit problems. Accurate to within 0.5% for amplitudes below about 15°.

§ 04

When NOT to use it

Breaks down for large amplitudes — at θ₀ = 30° the error is about 1.7%; at 90° it exceeds 18%. The formula also assumes a massless, inextensible string and a point mass; physical pendulums require the moment of inertia.

§ 05

Common mistakes

Confusing period T with frequency f or angular frequency ω. Taking L as the string length without adding the radius of the bob. Forgetting the square root — writing T = 2π·L/g instead of T = 2π·√(L/g).

§ 06

Topics that use this equation

the simple pendulum