Two particles of mass M and 2M are at a distance D apart. Under their mutual force of attraction they start moving towards each other. The acceleration of their centre of mass when they are D/2 apart is :

Options

A. `2 GM//D^(2)`
B. `4 GM//D^(2)`
C. `8 GM//D^(2)`
D. Zero

Answer

To solve the problem of finding the acceleration of the center of mass of two particles of masses \( M \) and \( 2M \) when they are \( \frac{D}{2} \) apart, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two particles: one with mass \( M \) and the other with mass \( 2M \).
- They are initially at a distance \( D \) apart and start moving towards each other due to their mutual gravitational attraction.

2. **Identifying Forces**:
- The force of attraction between the two masses is internal to the system. Therefore, there are no external forces acting on the system.
- According to Newton's laws, if the net external force on a system is zero, the acceleration of the center of mass (COM) of the system is also zero.

3. **Finding the Center of Mass**:
- The position of the center of mass \( x_{cm} \) for two particles can be calculated using the formula:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
- Here, \( m_1 = M \), \( m_2 = 2M \), and the positions \( x_1 \) and \( x_2 \) can be defined based on their initial positions.

4. **Acceleration of the Center of Mass**:
- Since there are no external forces acting on the system, the acceleration of the center of mass \( a_{cm} \) is given by:
\[ a_{cm} = \frac{F_{net}}{M_{total}} = 0 \]
- Here, \( F_{net} = 0 \) because the forces are internal, and \( M_{total} = M + 2M = 3M \).

5. **Conclusion**:
- Therefore, the acceleration of the center of mass when the two particles are \( \frac{D}{2} \) apart is:
\[ a_{cm} = 0 \]

### Final Answer:
The acceleration of the center of mass when the two particles are \( \frac{D}{2} \) apart is \( 0 \).