Chapter 7.7, Problem 52E

Period of a pendulum A standard pendulum of length L swinging under only the influence of gravity (no resistance) has a period of

$T=\frac{4}{\omega }{\displaystyle {\int }_0^{\pi /2}\frac{d\phi }{\sqrt{1-k^2{\sin }^2\phi }}},$

where ω² = g/L, k² = sin² (θ₀/2), g ≈ 9.8 m/s² is the acceleration due to gravity, and θ₀ is the initial angle from which the pendulum is released (in radians). Use numerical integration to approximate the period of a pendulum with L = 1 m that is released from an angle of θ₀ = π/4 rad.