Question

Which of the following is not correct about the centre of mass?
A) It depends on the choice of frame of reference
B) In the centre of mass frame, the momentum of a system is always zero
C) Internal forces may affect the motion of the centre of mass
D) Centre of mass and centre of gravity coincide in a uniform gravitational field.

Answer 1

Hint : The centre of mass of any object is defined relative to an object or a system of objects. It can also be thought of as the weighted average position of all the different objects in the system.

Complete step by step answer
In option (A), the centre of mass does depend on the choice of frame of reference. If we were to calculate the centre of mass of two objects in which one is moving and the other is stationary, we can define a frame that is we can move with such a velocity that is equal to the centre of mass in which case the centre of mass would seem stationary to us. However, if we were to stay with the first block, the centre of mass would seem to be moving.
In option (B), the centre of mass frame is an inertial frame that is moving along with the centre of mass as we discussed above. In this case, the centre of mass is stationary and the momentum of the system which can be defined using the momentum of the centre of mass, is also zero since the centre of mass is stationary in this frame, that is it has zero velocity.
In option (C), internal forces cannot affect the motion of the centre of mass. This can be thought of using an example where a bomb breaks an object into two parts. The two parts will be ejected in different directions in such a way that the centre of mass stays in its initial position. Here the bomb acts as an internal force however it cannot affect the motion of the centre of mass. So option (C) is not true about the centre of mass.
Hence the option (C) is the correct answer.

Note
In option (A) and (B), the reasoning we provided is only applicable to inertial frames of reference. These frames of reference are such frames that move with constant velocity with respect to a rest frame.