The Simple Pendulum and the Small-Oscillation Approximation

Setting up the equation

A simple pendulum consists of a point mass m at the end of an inextensible string of length L, oscillating in a vertical plane under the effect of gravity g. Describing its position by the angle theta(t) relative to the vertical, the angular momentum theorem gives:

theta''(t) + (g/L).sin(theta(t)) = 0

This equation is not linear because of the sin(theta) term: this is the main difficulty of the physical pendulum.

Small-angle approximation

For small theta (typically theta <= 10-15 degrees), we use the Taylor expansion sin(theta) ~= theta. The equation becomes linear:

theta''(t) + (g/L).theta(t) = 0

This is the canonical form of the harmonic oscillator with omega0 = sqrt(g/L), and T0 = 2.pi.sqrt(L/g). Note that the period depends neither on the mass nor on the amplitude (isochronism of small oscillations).

Beyond small angles

For larger amplitudes, the actual period increases slightly with the amplitude (isochronism is only approximate). The exact calculation involves elliptic integrals, beyond the standard curriculum but important to know qualitatively.

Common pitfall

The most common mistake is forgetting that the approximation sin(theta) ~= theta is only valid in radians and for small angles. Using this approximation for theta = 60 degrees gives a completely wrong result.