Two pendulums have the same dimensions (length {L}) and total mass (m). Pendulum A is a very small ball swinging at the end of a uniform massless bar. In pendulum B, half the mass is in the ball and half is in the uniform bar. 1. Find the period of pendulum A for small oscillations.
2. Find the period of pendulum B for small oscillations.

1) , 2) , where .

Explanation:

1) Pendulum A is a simple pendulum, whose period () is determined by the following formula:

(1)

Where:

- Length of the massless bar.

- Gravitational acceleration.

2) Pendulum B is a physical pendulum, whose period () is determined by the following formula:

(2)

Where:

- Total mass of the pendulum.

- Gravitational acceleration.

- Length of the uniform bar.

- Moment of inertia of the pendulum with respect to its suspension axis.

The moment of inertia can be found by applying the formulae of the moment of inertia for a particle and the uniform bar and Steiner's Theorem:

(3)

By applying (3) in (2) we get the following expression: