Question

A pendulum consists of a uniform rod of total mass m and length L that can pivot freely around one of its ends. The moment of inertia of such a rod around the pivot point is 1/3mL^2 The torque around the pivot point of the pendulum due to gravity is 1/2mgLsinθ, where θ is the angle the rod makes with the vertical and g is the acceleration due to gravity.

a) Write down the equation of motion for the angle θ including an appropriate derivative of θ

b) Assuming the angle θ to be sufficiently small that θ ≈ sinθ at all times, show explicitly and step by step that the expression θ (t) =Bsin(ωt−φ1),where B, ω and φ 1 are constants, is a solution to the equation of motion, as long as a particular relation holds between ω, L and g. State what this relation is as part of the proof.

Observing the motion of such a pendulum over a long time interval, it is observed that the amplitude

gradually decreases (due to internal friction and air resistance) according to B (t) = B0e−t/τ, where τ is a constant.

c) Find an expression for the time taken for the pendulum reduce its amplitude by half.

d) Find a mathematical expression for the rate at which energy is dissipated as a function of

time.

Answer 1