Question

A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere ?

• A. Angular velocity
• B. Rotational kinetic energy
• C. Moment of inertia
• D. Angular momentum

Answer 1

The Correct Option is D

To determine which physical quantity remains constant when the radius of a solid sphere is increased while keeping its mass the same, we need to analyze the relationship between angular momentum, angular velocity, rotational kinetic energy, and moment of inertia.

1. Moment of Inertia:

The moment of inertia I for a solid sphere about its symmetry axis is given by:

I = \frac{2}{5} m r^2

where m is the mass and r is the radius of the sphere. When the radius r is increased, the moment of inertia increases, as the mass m is constant. Thus, the moment of inertia does not remain constant when the radius changes.

2. Angular Velocity:

The relationship between angular momentum L and angular velocity \omega is:

L = I \omega

Since the moment of inertia I increases with the radius, and if the angular momentum L is constant (as we will see), the angular velocity \omega must decrease to keep L constant. Thus, angular velocity does not remain constant.

3. Rotational Kinetic Energy:

The rotational kinetic energy K is given by:

K = \frac{1}{2} I \omega^2

With increasing I and decreasing \omega (as discussed), the rotational kinetic energy K will change. Therefore, it does not remain constant.

4. Angular Momentum:

Angular momentum L is given by:

L = I \omega

In the absence of external torques, the angular momentum L is conserved. Thus, even as I increases and \omega decreases, the product I \omega remains constant.

Therefore, the correct answer is that the Angular Momentum of the sphere remains constant when the radius is increased while keeping the mass the same.