Small-angle approximation

Intermediate

Pendulum is nearly SHM for angles below about 15°.

Key relationsinθ ≈ θ

For θ ≲ 15°, sinθ ≈ θ (in rad) and the pendulum equation becomes θ̈ = −(g/L)θ, which is SHM.

Worked example

At 10°, the error is 0.5%.

Common mistake

Use radians.

In more detail

For small angles, sin θ is approximately θ in radians, so the pendulum equation becomes linear and the motion is simple harmonic. At 10 degrees the error is only about half a percent.

Real-world uses

The approximation makes pendulum analysis simple in physics teaching.
Clocks are designed to swing with small amplitudes for accuracy.
Clock makers rely on the small-angle approximation to show that the pendulum period is almost independent of how far it swings.

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Frequently asked questions

How small must the angle be?

Typically less than about 15 degrees for good accuracy.

What happens at large angles?

The period increases with amplitude, and the motion is no longer SHM.

What angle is 'small' for a pendulum?

Roughly 15 degrees or less, where sine and the angle in radians differ by only a few percent.

Small-angle approximation

In more detail

Real-world uses

Frequently asked questions

Related topics in Oscillations & SHM