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, we need to determine the acceleration of the center of mass (COM) of two particles with masses M and 2M when they are at a distance D/2 apart. We'll follow these steps:

### Step-by-Step Solution:

1. **Identify the System**: We have two particles with masses M and 2M, initially separated by a distance D. They are attracting each other due to their mutual gravitational force.

2. **Understand the Concept of Center of Mass**: The center of mass (COM) of a system of particles is defined as the point where the total mass of the system can be considered to be concentrated for the purpose of analyzing translational motion.

3. **Calculate the Position of 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, let’s assume particle M is at position 0 (x1 = 0) and particle 2M is at position D (x2 = D). Thus,
\[ x_{cm} = \frac{M \cdot 0 + 2M \cdot D}{M + 2M} = \frac{2MD}{3M} = \frac{2D}{3} \]

4. **Consider the Forces Acting on the Particles**: The only forces acting on the particles are their mutual gravitational attractions. However, these forces are internal to the system.

5. **Apply the Principle of Conservation of Momentum**: Since there are no external forces acting on the system, the momentum of the center of mass remains constant. This implies that the acceleration of the center of mass is zero.

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

### Final Answer:
The acceleration of the center of mass when the two particles are D/2 apart is **0**.