Question

A body having its center of mass at the origin. Then,
(This question having a multiple correct options)
A) $X$ coordinates of the particles may be all positive.
B) Total $K.E$ must be conserved.
C) Total $K.E$ must vary.
D) Total momentum shall vary.

Answer 1

Hint:-The point where the whole mass of the body is concentrated is termed as the center of mass. That means it balances the body at that point. We have to infer how the body acts at that point, and how the $K.E$ and momentum behaves.

Complete Step by step solution:
If the center of mass of the body is at origin, then the whole mass is concentrated in the origin. Thus the particles can have negative and positive $X$coordinates hence the left and right side of the origin represents the negative and positive coordinates respectively.
If all the particles have positive $X$coordinates then their center of mass will not be at the origin.
Thus option A is incorrect.
The body is trying to balance in that center of mass. The internal forces are present, hence their total kinetic energy will vary. Therefore the kinetic energy will not be conserved. Thus option B will be incorrect. And option C is correct.
If no external force is applied on the system, then the momentum will also change. Since the kinetic energy varied the momentum will also vary. Thus option D is correct.

The answer is option C and D.

Note: We have to note that the center of gravity and center of mass are different concepts. Center of mass is not an actual position and it may vary. It is the average of the position of the masses of all particles in the body.