A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?

The solid sphere is rotating in free space, which implies there is no external torque acting on the system.

$$ \tau_{ext} = 0 $$

According to the law of conservation of angular momentum, if the net external torque on a system is zero, its total angular momentum ($$L$$) remains conserved (constant).

$$ L = \text{constant} $$

Therefore, the angular momentum will not be affected.

Let's briefly analyze why the other quantities are affected:

Moment of Inertia ($$I$$):The moment of inertia of a solid sphere is given by $$ I = \frac{2}{5}MR^2 $$. If the mass ($$M$$) is kept constant and the radius ($$R$$) is increased, the moment of inertia $$I$$ will increase.

Angular Velocity ($$\omega$$): The relationship between angular momentum, moment of inertia, and angular velocity is $$ L = I\omega $$. Since $$L$$ is constant and $$I$$ increases, the angular velocity $$\omega$$ must decrease.

Rotational Kinetic Energy ($$K$$): The rotational kinetic energy can be expressed in terms of angular momentum as $$ K = \frac{L^2}{2I} $$. Since $$L$$ is constant and $$I$$ increases, the rotational kinetic energy $$K$$ will decrease.

Answer: Option (B) angular momentum