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 @. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity (a) $\frac{\omega(M-2 m)}{M+2 m}$ (b) $\frac{\omega M}{M+2 m}$ (c) $\frac{\omega M}{M+m}$ (d) $\frac{\omega(M+2 M)}{M}$

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity @. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity
(a) $\frac{\omega(M-2 m)}{M+2 m}$
(b) $\frac{\omega M}{M+2 m}$
(c) $\frac{\omega M}{M+m}$
(d) $\frac{\omega(M+2 M)}{M}$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Conservation of Angular Momentum

In an isolated system where no external torques act, the total angular momentum remains constant. When the configuration of a rotating body changes, such as by attaching additional masses, the distribution of mass and hence the moment of inertia changes. However, to conserve angular momentum, the angular velocity adjusts so that the product of moment of inertia and angular velocity remains the same as before the change.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, dependent on the mass distribution relative to the axis of rotation. Different shapes (like a ring or point masses) have specific formulas for calculating their moment of inertia, and these calculations are crucial when additional masses are added to a rotating system.

Application of Conservation Laws in Rotational Dynamics

Using the principle of conservation of angular momentum, one can equate the initial and final angular momentum of a system. When extra masses are attached to a rotating object, their contribution to the moment of inertia must be accounted for, and the resulting decrease in angular velocity ensures that angular momentum is conserved, highlighting the interplay between mass distribution and rotational speed.

Please give Ace some feedback