A simple pendulum consists of a particle of mass m supported by a piece of string of len...

A simple pendulum consists of a particle of mass m supported by a piece of string of length L. Assuming that the pendulum is displaced through an angle θ₀ radians from the vertical and then released from rest, the resulting motion is described by the initial-value problem

(a) For small oscillations, θ << 1, we can use the approximation sin θ ≈ θ in Equation (1.11.28) to

obtain the linear equation Solve this initial-value problem for θ as a function of t . Is the predicted motion reasonable?

(b) Obtain the following first integral of (1.11.28):

(c) Show from Equation (1.11.29) that the time T (equal to one-fourth of the period of motion) required for θ to go from 0 to θ₀ is given by the elliptic integral of the first kind

(d) Show that (1.11.30) can be written as

where k = sin(θ₀/2). [Hint: First express cos θ and cos θ₀ in terms of sin₂(θ/2) and sin2(θ₀/2).]