Gravitational force between a point mass m and M separated by a distance is F. Now if a point mass 2m is placed next to m is contact with it. The force on M due to `m` and the total force on M are

Options

A. 2F, F
B. F, 2F
C. F, 3F
D. F, F

Explanation

To solve the problem, we will analyze the gravitational forces acting on mass \( M \) due to the masses \( m \) and \( 2m \).

### Step-by-Step Solution:

1. **Understanding the Initial Force**:
The gravitational force \( F \) between the point mass \( M \) and mass \( m \) is given by the formula:
\[ F = \frac{G \cdot M \cdot m}{r^2} \]
where \( G \) is the gravitational constant and \( r \) is the distance between the two masses.

2. **Introducing the Second Mass**:
Now, we place a second mass \( 2m \) next to mass \( m \). The distance between mass \( M \) and mass \( m \) remains the same, while the distance between mass \( M \) and mass \( 2m \) is also the same.

3. **Calculating the Force on \( M \) Due to \( 2m \)**:
The gravitational force \( F' \) between mass \( M \) and mass \( 2m \) can be calculated as:
\[ F' = \frac{G \cdot M \cdot 2m}{r^2} \]
This can be simplified as:
\[ F' = 2 \cdot \frac{G \cdot M \cdot m}{r^2} = 2F \]

4. **Calculating the Total Force on \( M \)**:
The total gravitational force \( F_{total} \) on mass \( M \) due to both masses \( m \) and \( 2m \) is the sum of the individual forces:
\[ F_{total} = F + F' = F + 2F = 3F \]

5. **Conclusion**:
Therefore, the force on \( M \) due to mass \( m \) is \( F \), and the total force on \( M \) due to both masses \( m \) and \( 2m \) is \( 3F \).

### Final Answer:
- The force on \( M \) due to \( m \) is \( F \).
- The total force on \( M \) is \( 3F \).