Question

A pendulum bob is made from a sphere filled with water. What would happen to the frequency of vibration of this pendulum if there were a hole in the sphere that allowed the water to leak out slowly?

Short answer

The center of mass of the bob moves back to the center of sphere when the last drops of water dribble out. It results to increase the frequency of the pendulum to its original value quickly.

Step-by-step solution

Concept of a simple harmonic motion

A simple pendulum of length Lcan be modeled to move in simple harmonic motion for small angular displacements from the vertical. Its period is

$T=2\pi \sqrt{\frac{L}{g}}$

A physical pendulum is an extended object that, for small angular displacements, can be modeled to move in simple harmonic motion about a pivot that does not go through the center of mass. The period of this motion is

$T=2\pi \sqrt{\frac{I}{mgd}}$, and$f=\frac{1}{T}=\frac{1}{2\pi }\sqrt{\frac{mgd}{I}}$

Where I is the moment of inertia of the object about an axis through the pivot and d is the distance from the pivot to the center of mass of the object.

Find what would happen to the frequency of vibration of this pendulum

Let’s assume that the bob’s diameter is not very small compared with the cord length. The effective length of the pendulum is increasing and is lowering the frequency slightly when the water leaks out and the bob’s center of mass moves down. The center of mass of the bob moves back to the center of sphere when the last drops of water dribble out. It results to increase the frequency of the pendulum to its original value quickly.