Physics
Simple pendulum
Nonlinear oscillations: the period grows with amplitude.
What happens
A pendulum swinging beyond the textbook: at large angles the restoring force is no longer proportional, so the period grows with amplitude, and the exact value is shown live. Set the initial angle and watch the phase portrait bend; add damping or a periodic drive to explore resonance.
Try this
At Earth's g and small swings L = g/π² = 0.994 m. At 60° the exact period is 7 % longer - the card counts that too.
Equations used in the module
Equations θ'' + (g/L) sin θ = 0 T = 2π √(L/g) (small angles) T(θ₀) = 4 √(L/g) K(sin(θ₀/2)) (exact, AGM)
