The linearized pendulum equation breaks down at large angles; the full nonlinear version has a longer period calculable via elliptic integrals.
Key Takeaways
The standard small-angle substitution (sin θ ≈ θ) works only for small displacements; at 60°, sin θ = 0.866 but θ = 1.047 radians – a 21% gap.
The nonlinear pendulum’s period is longer by a factor dependent on initial displacement θ0; the closed-form exact answer requires the complete elliptic integral of the first kind.
A practical approximation estimates the period increase as roughly 1 + θ0²/16; at 60° this gives a 7.46% stretch vs. the exact 7.32%.
Stretching the linear solution’s period by the exact correction factor matches the nonlinear solution to sub-pixel accuracy – useful for approximate numerical work.
A followup post refines the error analysis using Jacobi elliptic functions and finds the original numerical solver introduced spurious error roughly doubling the apparent deviation.
