Animated Solution for Physics - Rotational Motion: Initial angular velocity of a circular disc of mass $M$ is ${\omega }_1$. Then, two small spheres of mass $m$ are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?

Imagine you are standing in a physics lab, watching a heavy circular disc spinning smoothly on a frictionless vertical axis. It has a mass $M$ and radius $R$, and it's rotating with an initial angular velocity ${\omega }_1$.

Now, we perform a delicate operation. We take two small spheres, each of mass $m$, and place them very gently on the exact opposite edges of the spinning disc. The word "gently" is the secret key to this problem. It tells us that we haven't pushed or twisted the disc in any way. In physics terms, the net external torque on the system is exactly zero.

When the external torque is zero, the universe enforces a strict rule: the total angular momentum of the system must remain perfectly conserved.

To use this law, we need to determine the moment of inertia before and after the masses are added. Initially, the system is just the bare disc. The moment of inertia of a uniform disc about its central axis is:

Finally, the system consists of the disc plus the two small masses sitting at the rim (at a distance $R$ from the axis). Each mass acts like a point particle, adding $m{R}^2$ to the total inertia.

Now, we substitute these moments of inertia into our conservation equation. The initial angular momentum equals the final angular momentum:

Look closely at this equation. The $R^2$ term appears in every single part! This means the actual radius of the disc doesn't affect the final answer. We can cancel $R^2$ out completely:

To make the algebra cleaner, let's multiply the entire equation by 2 to eliminate the fractions:

Finally, we isolate our target variable, ${\omega }_2$:

And there we have it! The final angular velocity is reduced by a factor of $\frac{M}{M+4m}$. This makes perfect intuitive sense: by adding mass to the edges, we increased the rotational inertia of the system. To keep the angular momentum constant, the spinning speed had to decrease.