Pendulum Lab

Full nonlinear pendulum against the small-angle formula, with a phase plot.

0/3challenges

t = 0.0 s

Small-angle approximationFull equation

Release angle°

Lengthm

Gravitym/s²

Damping c1/s

Small-angle period: 2.006s
Exact period (undamped): 2.041s
Measured period: ...s
Difference: 1.74%

The model

A point mass on a rigid, massless rod: θ″ = -(g/L) sin θ - c θ′. It is integrated with fourth-order Runge-Kutta, eight steps per frame.

Textbooks replace sin θ with θ, which gives the small-angle period T = 2π√(L/g), independent of amplitude. The exact undamped period grows with amplitude; it is computed here with the arithmetic-geometric mean.

Try this

At 10° the two periods agree to 0.2%. At 90° the real pendulum is 18% slower. Near 180° the period heads to infinity, because the bob balances on top.

Add damping and the phase plot spirals into the centre.

Assumes a rigid massless rod, a frictionless pivot and linear (viscous) damping. Educational model, not an engineering design or validation tool.

Achievements0/3

Seconds pendulum +30Tune a pendulum to a small-angle period of 2.00 s.
Over the top +20Release a pendulum from 170 degrees or more.
Settle down +20Let a damped pendulum fall under 1 degree.

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