Consider a simple pendulum consisting of a small bob of mass m suspended by a massless string of length ℓ. The angular displacement θ is measured from the vertical, with θ>0 indicating the bob is to the right of the equilibrium position.
(a) Write an expression for the net torque τ about the pivot in terms of m, g, ℓ, and θ. Clearly indicate the sign convention you choose.
(b) Identify the angular position(s) at which the pendulum is in rotational equilibrium. State whether each equilibrium position is stable or unstable, and justify your answer based on your torque expression.
(c) For small-angle oscillations about the stable equilibrium, derive the differential equation governing θ(t) and show that the motion satisfies the mathematical form of simple harmonic motion.
(d) Obtain an expression for the period T of small-amplitude oscillations and explain why this period does not depend on the mass m of the bob.