Question

Statement-1: KE of rotating rigid body in CM frame is ${\displaystyle \frac{1}{2}}{I}_{cm}{\omega }^2$ where symbols have usual meaning.
Statement-2: In a CM frame rigid body can have only pure rotational motion.
(A) Statement-1 is true, Statement-2 is true and Statement-2 is the correct explanation for statement-1.
(B) Statement-1 is true, Statement-2 is true and Statement-2 is not the correct explanation for statement-1.
(C) Statement-1 is true, Statement-2 is false.
(D) Statement-1 is false, Statement-2 is true.

Answer 1

Hint :When a rigid body is in pure rotational motion, all particles in the body rotate through the same angle during the same time interval. Thus, all particles have the same angular velocity and the same angular acceleration.
The centre-of-mass (CM) frame is a special case of the centre-of-momentum (COM) frame. It is an inertial frame in which the centre of mass (Physical point) remains at the origin. In all COM frames, the centre of mass is at rest, but it is not necessarily at the origin of the coordinate system.

Complete Step By Step Answer:
The two types of motion a rigid body (sphere) can undergo are Translational Motion (circle motion) and Rotational Motion. In a circular motion, the object just moves in a circle. In rotational motion, the object rotates about an axis. For example, Earth rotates on its own axis.
The rigid body (sphere) is stationary at one point when the CM frame of reference is selected. Only the rotation of the rigid body is the relevant motion. Thus, the KE in this case is ${\displaystyle \frac{1}{2}}{I}_{cm}{\omega }^2$ .
Hence the correct Option is A. Statement-1 is true, Statement-2 is true and statement-2 is the correct explanation for statement-1.

Note :
In physics, a rigid body is also known as a rigid object. It is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it.