Animated Solution for Physics - Rotational Motion: A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc

A Spinning World

Imagine you are looking down at a playground merry-go-round spinning smoothly. Now, picture a tiny insect sitting right at the edge of this spinning disc. The insect decides to go for a walk, straight across the diameter, passing right through the center to the other side. The question is: how does the insect's journey affect the spinning speed of the disc?

To solve this, we need to look at the forces—or rather, the lack of them. In this system, there is no external twisting force, which physicists call torque, acting on the disc-insect system about its vertical axis.

Conservation of Angular Momentum

Because the net external torque is zero, a beautiful law of nature takes over: the Conservation of Angular Momentum.

Mathematically, angular momentum ($L$) is the product of the moment of inertia ($I$) and the angular velocity ($\omega$):

This equation is the master key to our problem. It tells us that if the moment of inertia ($I$) changes, the angular velocity ($\omega$) must change in the exact opposite way to keep their product constant. If $I$ goes down, $\omega$ must go up, and vice versa. This is exactly why an ice skater spins faster when she pulls her arms in!

Let's break down the moment of inertia of our system. It's simply the sum of the disc's inertia and the insect's inertia:

Here, $M$ and $R$ are the mass and radius of the disc, $m$ is the mass of the insect, and $x$ is the distance of the insect from the center.

Notice that the disc's inertia ($\frac{1}{2}M{R}^2$) is a fixed number. The only thing changing is $m{x}^2$, which depends entirely on where the insect is!

Phase 1: Walking to the Center As the insect walks from the rim towards the center, its distance $x$ is decreasing. Because $x$ is getting smaller, the total moment of inertia $I$ is decreasing. To keep the angular momentum constant, the angular speed $\omega$ must increase. The disc spins faster and faster, reaching its maximum speed exactly when the insect steps on the center ($x=0$).

Phase 2: Walking to the Other Edge Once the insect crosses the center and continues towards the opposite edge, its distance $x$ starts increasing again. This means the total moment of inertia $I$ is increasing. Consequently, the angular speed $\omega$ must decrease to compensate.

By following the insect's path, we've uncovered a dynamic, changing system. The angular speed doesn't just do one thing; it responds perfectly to the insect's position.

Therefore, the angular speed of the disc first increases and then decreases. It's a perfect demonstration of how mass distribution directly controls rotational motion!