Animated Solution for Physics - Rotational Motion: A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with an angular speed $\omega$. Two particles having mass $m$ each are now attached at diametrically opposite points. The angular speed of the ring will become

The Spinning Ring

Imagine a thin circular ring of mass $M$ and radius $r$, spinning smoothly around its central axis. It possesses a certain amount of rotational inertia, which acts as its resistance to any change in its spinning motion. For a thin ring where all the mass is concentrated at the rim, this moment of inertia is simply given by:

Because it is spinning with an initial angular speed $\omega$, it carries an initial angular momentum. Think of angular momentum as the "amount of spin" the ring has. We can write this as:

Enter the Masses

Now, the problem states that two particles, each of mass $m$, are gently attached to diametrically opposite points on the ring. The word gently is the most critical clue here. It implies that no external twisting force, or torque, is applied to the system during this process.

According to the fundamental laws of physics, if the net external torque on a system is zero, its total angular momentum must remain perfectly conserved.

Conservation of Angular Momentum

When the two masses are attached, they become part of the rotating system. Since they are placed on the rim of the ring, they are at a distance $r$ from the axis of rotation.

Each mass adds its own moment of inertia, $m{r}^2$, to the system. Since there are two masses, the total final moment of inertia becomes:

Because the system's inertia has increased (it's now "heavier" to spin), the ring must slow down to keep the total angular momentum constant. Let's call this new, slower angular speed ${\omega }^{\prime }$. The final angular momentum is:

The New Angular Speed

By equating the initial and final angular momenta, we set up our master equation:

Notice how $r^2$ appears in every single term? It's a common factor that beautifully cancels out, showing that the final speed doesn't actually depend on the physical size of the ring, only on the mass ratio!

Finally, isolating ${\omega }^{\prime }$, we arrive at our elegant solution:

This result perfectly captures the physical intuition: adding mass to the rotating system increases its inertia, which in turn proportionally decreases its angular speed to conserve the sacred quantity of angular momentum.