Amplitude and the circular error

The pendulum law is a small-angle approximation. It is exact only if the swing is tiny. In a real pendulum, the push that brings the bob back to the middle gets weaker, relative to the distance, as the swing gets wider, so a wide swing takes slightly longer than a narrow one.

If the swing reaches an angle θ₀ on each side of vertical (the amplitude, in radians), the true period is longer than the small-angle value T₀ by about:

T ≈ T₀ × (1 + θ₀² / 16)

This means a clock whose pendulum swings wider will run slower. The effect is called the circular error, because the bob moves along a circle (an arc) and not along a straight line. A perfectly isochronous pendulum, one whose period is the same at every amplitude, would have no circular error. Real ones are only nearly so, and only for small swings.

How big is the effect? Convert the angle to radians first (10° = 0.1745 rad).

A clock's "loses 165 s per day" simply means that the pendulum takes 0.19 % too long, and a day has 86 400 s: 86 400 × 0.0019 ≈ 165 s. Wide swings are clearly bad.

So clock pendulums are made to swing through only a few degrees on each side (about 2 to 4 degrees is typical). But here is the subtle point. A constant error can be removed, because you can adjust the length until the clock shows the right time at its usual amplitude. The real enemy is a changing amplitude. If the amplitude drifts from 3° to 2° as the clock's driving force varies, the pace changes by roughly 8 s per day, and no setting of the length can fix that. The goal of a good design is to keep the amplitude constant, and to make the error small and flat around the working point.

Clockmakers also looked for a cleverer cure. Huygens proved that a bob that swings along a cycloid (the curve traced by a point on a rolling wheel) is perfectly isochronous. He fitted curved metal "cheeks" beside the suspension so that the pendulum's string wrapped against them and the bob followed a cycloid-like path. It worked in principle, but the cheeks introduced friction and errors of their own, and clockmakers found it simpler to use a small amplitude and a good escapement. In the next world you will see the escapement that did this best, the deadbeat.

The fact to remember is that amplitude is a pace-changing quantity, and that a good clock is designed to keep it small and steady.