Question

A pendulum consisting of a small bob of mass m at the end of an effectively massless rigid rod of length L is set into motion by releasing its bob from a small angle theta 0 from the vertical. a) What maximum angular and linear speed does the pendulum achieve? Where in its path is this speed reached? b) What are the angular and linear oscillation frequencies of the pendulum [or what is its period].

Answer 1

a)
Initially, before the bob is released, it had only potential energy. As the bob falls, it gains kinetic energy and at the bottom, kinetic energy is maximum and potential energy is zero.
mgL (1 - coso) = 1/2 mv²
Where v is the maximum linear speed.
v² = 2gL (1 - coso)
v = SQRT[2gL (1 - coso)]

Maximum angular speed = = v/L
= SQRT[2gL (1 - coso)] / L
= SQRT[2g (1 - coso) / L]

As explained before, speed will be maximum at the bottom of the swing.

b)
Linear oscillation frequency, T = 2 SQRT[L/g]
Where g is the acceleration due to gravity.

Angular oscillation frequency, = 2/T
= 2/{2 SQRT[L/g]}
= SQRT[g/L]