A pendulum consists of a small bob of mass m suspended from a light, inextensible string of length L. The bob is displaced by a small angle θ (measured in radians) and released from rest. Neglect air resistance.
(a) Derive an expression for the net torque on the bob about the pivot in terms of m, g, L, and θ. Clearly state any physical assumptions you make.
(b) Show that, under an appropriate approximation, the angular motion of the pendulum satisfies the defining condition for SHM. Hence obtain an expression for the angular frequency of small oscillations.
(c) Explain physically why the approximation used in part (b) becomes invalid for larger angular displacements.
(d) For a pendulum of length 0.75m, estimate the maximum angular displacement (in degrees) for which the period differs from the small-angle value by no more than 2%. Support your estimate with quantitative reasoning.