A thin circular ring of mass m and radius R is rotating...

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity of $[2006]$ (A) $\frac{\omega m}{m+M}$ (B) $\frac{\omega m}{m+2 M}$ (C) $\frac{\omega(m+2 M)}{m}$ (D) $\frac{\omega(m-2 M)}{m+2 M}$

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity of $[2006]$
(A) $\frac{\omega m}{m+M}$
(B) $\frac{\omega m}{m+2 M}$
(C) $\frac{\omega(m+2 M)}{m}$
(D) $\frac{\omega(m-2 M)}{m+2 M}$

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Key Concepts

Effects of Mass Redistribution

Changes in the distribution of mass in a rotating system directly affect the moment of inertia, and, by extension, the angular speed of the system due to the conservation of angular momentum. An increase in moment of inertia, such as from attaching additional masses a certain distance from the rotation axis, results in a decrease in angular velocity to maintain the original angular momentum.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on both the mass of the object and how that mass is distributed with respect to the axis of rotation. When additional masses are added at a distance from the axis, the overall moment of inertia increases, leading to a decrease in angular velocity if angular momentum is conserved.

Angular Momentum Conservation

This principle states that if no external torques act on a system, the total angular momentum remains constant. In rotational dynamics, any change in the distribution of mass, and hence the moment of inertia, must be accompanied by a compensatory change in angular velocity so that the product (angular momentum) stays constant.

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