🔴 The Pendulum: Small-Angle Lie vs the Real Nonlinear Swing

About this simulation

Every introductory physics course teaches that a pendulum's period is T₀ = 2π√(L/g), independent of amplitude. That formula is only the first term of a much longer story: it comes from replacing sin θ with θ in the pendulum's equation of motion, which is only accurate for small swings. Release the bob from a wide angle and the true restoring force is weaker near the top of the swing than the linear approximation assumes, so the real period grows longer than T₀.

How it works

• The red pendulum is integrated numerically from the exact nonlinear equation θ'' = -(g/L) sin θ.
• The pale ghost pendulum follows the small-angle solution θ(t) = θ₀ cos(ω₀t) with ω₀ = √(g/L), released from rest at the same instant and amplitude.
• The exact period is computed from the complete elliptic integral of the first kind: T = 4√(L/g) · K(sin(θ₀/2)), evaluated live via the arithmetic-geometric mean.
• At small amplitude the two pendulums swing locked together. As amplitude grows past roughly 30–40°, the red pendulum visibly falls behind — its period is longer — and the phase gap widens with every cycle.

Controls

• Amplitude θ₀ — the angle both pendulums are released from, up to 170°.
• Rod length — longer rods swing slower for both pendulums, but the relative error is unchanged.
• Gravity — try the Moon or Jupiter to see how g rescales the timescale without changing the amplitude-dependent error.
• Show small-angle ghost — toggle the SHM comparison pendulum on or off.
• ↺ Resync swings — releases both pendulums together again from rest. Drag to rotate, scroll to zoom.

Did you know?

At θ₀ = 90° the true period is already about 18% longer than the small-angle formula predicts, and it diverges to infinity as θ₀ approaches 180° — a pendulum released exactly upside-down would, in a frictionless world, take forever to complete its first swing.