EASA / CAR Part-66 · Module 2 · 2.2 Mechanics · Cat B1/B2
Objective: show that a pendulum's period depends on its length and on gravity but not on its mass or, for small angles, its amplitude; and show where the small-angle approximation begins to break down.
At small angles the restoring force is very nearly proportional to displacement, so the motion is simple harmonic and the period is independent of amplitude.
A pendulum swings because gravity provides a restoring force that always acts back toward the rest position. For small angles that restoring force is very nearly proportional to the displacement, which is the defining condition for simple harmonic motion, and the period is then T = 2π√(L/g).
Two things about that formula catch candidates out. First, there is no mass term: a heavy bob and a light bob on the same string swing at the same rate, because increasing the mass increases both the restoring force and the inertia in exactly the same proportion. Second, there is no amplitude term, so within the small-angle range a wide swing and a narrow swing take the same time. Only length and gravity change the period, and because the length is under a square root, quadrupling it only doubles the period.
The approximation relies on sin θ ≈ θ. Beyond roughly 15 to 20 degrees the real restoring force falls short of that, so the true period grows and the pendulum runs progressively slower than the formula predicts. Energy meanwhile shuttles between potential at the extremes, where the bob is momentarily stationary, and kinetic at the lowest point, where it is fastest.