Physics · Simulator 002
Swing a simple pendulum and time it. Change the length, gravity, starting angle and damping, then compare the measured period with the small-angle formula T = 2π√(L/g).
A point mass on a light rod of length L obeys
θ'' = −(g/L)·sin θ − c·θ'
where c is a damping rate in 1/s. For small angles, sin θ ≈ θ. The motion is then simple harmonic with period T₀ = 2π√(L/g), which does not depend on amplitude or mass.
The simulator integrates the full non-linear equation (RK4) and plots it next to the small-angle prediction. The exact undamped period for amplitude θ₀ is T = T₀ / AGM(1, cos(θ₀/2)), using the arithmetic–geometric mean. It grows as the swing gets wider.
The measured period is the time between successive crossings of θ = 0 in the same direction.
Assumes a massless, rigid rod, a point mass, no friction at the pivot except the damping term, and uniform gravity.
A grandfather clock pendulum ticks once per swing (a half period of 1 s). How long must it be on Earth? Will it run fast or slow if the swing is wider than intended?
T₀ = 2 s gives L = g·T₀² / 4π² ≈ 0.994 m. A wider swing lengthens the period, so the clock runs slow. That is why precision clocks keep the amplitude small and constant. Try L = 0.994 m at 3° and then at 20°.