Question

The solid disk and circular hoop in FIGURE $Q15.8$have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at the top edge. Which, if either, has the larger period of oscillation?

Short answer

The solid disc has less inertia than the ring for the same mass and size.

The solid disc's rotation around the pivot will take less time.

The oscillation period of the ring will be longer.

Step-by-step solution

Definition of Oscillation

The process of periodic oscillations of any quantity or measure about its equilibrium value throughout time is known as oscillation.

A periodic variation in a matter's value between two values around its centre value is known as oscillation.

Explanation of physical pendulum

A physical pendulum is a weighted object that swings back and forth on a pivot due to gravity.

A physical pendulum's time period is stated as:

$T=2\pi \sqrt{\frac{I}{Mgl}}$

Where, $I$is the moment of inertia, $M$is the mass of the pendulum, $g$is the gravitational acceleration, and is the pivot's distance from the centre of mass.

Explanation of  larger period oscillation

The centre of mass will be at the physical centre of the bodies because both pendulums are the same shape and size.

As a result, both pendulums swing in the same direction. The period of a physical pendulum is independent of its mass.

Although the moment of inertia is also dependent on mass, the statement contains it.

The distribution of mass, as reflected by the body's moment of inertia, determines the time period of a physical pendulum.

For the same mass and size, solid discs have less inertia than the ring, hence the solid discs' time period around the centre will be shorter.

As a result, the ring's oscillation period will be prolonged.