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

Answer 1

The Correct Option is A

To solve this problem, we need to analyze the effects of increasing the radius of a rotating solid sphere while keeping its mass constant. Let's evaluate each given physical quantity:

1. Angular Momentum: Angular momentum L is given by the formula: L = I\omega, where I is the moment of inertia and \omega is the angular velocity. For a solid sphere, the moment of inertia I is equal to \frac{2}{5}MR^2, where M is the mass and R is the radius. Since the mass is constant, when the radius increases, the moment of inertia also increases. To conserve angular momentum in the absence of external torques, the angular velocity must decrease. Thus, the angular momentum remains constant.
2. Angular Velocity: As mentioned earlier, when the radius of the sphere increases, the moment of inertia increases and, to conserve angular momentum, the angular velocity \omega must decrease. Hence, the 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. When the radius increases, the moment of inertia increases and angular velocity decreases. Since the angular momentum is constant, the rotational kinetic energy would change. Therefore, the rotational kinetic energy does not remain constant.
4. Moment of Inertia: The moment of inertia I = \frac{2}{5}MR^2. As the radius increases, the moment of inertia increases proportionally to the square of the radius. Therefore, the moment of inertia does not remain constant.

Therefore, the physical quantity that would remain constant as the radius of the rotating solid sphere is increased, keeping its mass constant, is the Angular Momentum.