Two spheres of masses 2 M and M are intially at rest at a distance R apart. Due to mutual force of attraction, they approach each other. When they are at separation R/2, the acceleration of the centre of mass of sphere would be

Options

A. `0m//s^(2)`
B. `gm//s^(2)`
C. `3g m//s^(2)`
D. `12 gm//s^(2)`

Explanation

To find the acceleration of the center of mass of two spheres with masses 2M and M when they are at a separation of R/2, we can follow these steps:

### Step 1: Understand the System
We have two spheres:
- Sphere 1 with mass \( m_1 = 2M \)
- Sphere 2 with mass \( m_2 = M \)

Initially, they are at rest and separated by a distance \( R \).

### Step 2: Calculate the Force Between the Spheres
The gravitational force \( F \) between the two spheres when they are at a distance \( r \) apart is given by Newton's law of gravitation:

\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

Substituting the masses:

\[ F = \frac{G \cdot (2M) \cdot M}{R^2} = \frac{2GM^2}{R^2} \]

### Step 3: Determine the Accelerations of Each Sphere
Using Newton's second law, the acceleration \( a_1 \) of sphere 1 (mass \( 2M \)) due to the force \( F \) is:

\[ a_1 = \frac{F}{m_1} = \frac{\frac{2GM^2}{R^2}}{2M} = \frac{GM}{R^2} \]

The acceleration \( a_2 \) of sphere 2 (mass \( M \)) is:

\[ a_2 = \frac{F}{m_2} = \frac{\frac{2GM^2}{R^2}}{M} = \frac{2GM}{R^2} \]

### Step 4: Find the Center of Mass
The position of the center of mass \( x_{cm} \) of the system is given by:

\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]

Since both spheres are initially at rest, the center of mass does not change its position due to internal forces.

### Step 5: Calculate the Acceleration of the Center of Mass
The total external force acting on the system is zero because the forces between the two spheres are internal. Therefore, the acceleration of the center of mass \( a_{cm} \) is given by:

\[ a_{cm} = \frac{F_{net}}{m_{total}} = \frac{0}{m_1 + m_2} = 0 \]

### Conclusion
Thus, the acceleration of the center of mass of the system is:

\[ \boxed{0} \]