Which of the following statements is correct ?

Options

A. If the centre of mass is at rest , then the net work done dy all the forces acting on the system is zero
B. If the velocity of the centre of mass remains zero, then the net external force acting on the system must zero
C. If the speed of centre of mass is changing , then there must be some work done by the internal forces on the system
D. If centre of mass of three - particle system is at rest and it is known that two of the particles are moving in different non - collinear lines , then third particle may be moving

Answer

To determine which of the statements is correct, we will analyze each statement one by one based on the principles of physics, specifically focusing on the center of mass and the forces acting on a system.

### Step-by-Step Solution:

1. **Statement Analysis**:
- **Statement 1**: "If the center of mass is at rest, then the net work done by all the forces acting on the system is zero."
- This statement is incorrect. The center of mass being at rest does not imply that the net work done is zero. The center of mass can remain at rest even if there are forces acting on the system, as long as these forces balance each other out (net force is zero).

2. **Statement 2**: "If the velocity of the center of mass remains 0, then the external force acting on the system must be 0."
- This statement is also incorrect. The velocity of the center of mass being zero does not necessarily mean that the external force is zero. The external forces can be present but balanced, resulting in no net force and hence no change in the velocity of the center of mass.

3. **Statement 3**: "If the speed of the center of mass is changing, then there must be some work done by the internal forces on the system."
- This statement is incorrect. The change in speed of the center of mass is due to external forces acting on the system, not internal forces. Internal forces do not affect the motion of the center of mass.

4. **Statement 4**: "If the center of mass of a three-particle system is at rest, and it is known that two of the particles are moving in different collinear lines, then the third particle may be moving."
- This statement is misleading. If the center of mass is at rest, the motion of the two particles must be such that their momentum balances out. Therefore, the third particle must be moving in such a way that it compensates for the motion of the other two particles to keep the center of mass at rest. Thus, it is not just "may be" but "must be" moving in a specific manner.

### Conclusion:
After analyzing all the statements, we can conclude that:
- All statements presented are incorrect based on the principles of mechanics regarding the center of mass and forces.