Two equal masses separated by a distance (d) attract each other with a force (F). If one unit of mass is transferred from one of them to the other, the force

Options

A. does not change
B. decreases by `(G//d^(2))`
C. becomes `d^2` times
D. increases by `(2G//d^2)`

Answer

To solve the problem, we start by analyzing the gravitational force between two equal masses and how it changes when one unit of mass is transferred from one to the other.

### Step-by-Step Solution:

1. **Initial Setup**:
Let the two equal masses be \( m \) each, separated by a distance \( d \). The gravitational force \( F \) between them is given by Newton's law of gravitation:
\[ F = \frac{G m^2}{d^2} \]
where \( G \) is the gravitational constant.

2. **Transfer of Mass**:
When one unit of mass is transferred from one mass to the other, the new masses become:
- Mass 1: \( m - 1 \)
- Mass 2: \( m + 1 \)

3. **New Force Calculation**:
The new gravitational force \( F' \) between the two masses is given by:
\[ F' = \frac{G (m - 1)(m + 1)}{d^2} \]
Expanding this, we get:
\[ F' = \frac{G (m^2 - 1)}{d^2} \]

4. **Finding the Ratio of Forces**:
To find how the new force \( F' \) compares to the original force \( F \), we take the ratio:
\[ \frac{F'}{F} = \frac{\frac{G (m^2 - 1)}{d^2}}{\frac{G m^2}{d^2}} = \frac{m^2 - 1}{m^2} \]
This simplifies to:
\[ \frac{F'}{F} = 1 - \frac{1}{m^2} \]

5. **Expressing the New Force**:
Now we can express \( F' \) in terms of \( F \):
\[ F' = F \left(1 - \frac{1}{m^2}\right) \]
Therefore, the change in force can be written as:
\[ F' = F - \frac{F}{m^2} \]

6. **Relating to Gravitational Constant**:
Since \( F = \frac{G m^2}{d^2} \), we can substitute this into our equation:
\[ F' = F - \frac{G m^2}{m^2 d^2} = F - \frac{G}{d^2} \]

7. **Conclusion**:
Thus, the new force \( F' \) is:
\[ F' = F - \frac{G}{d^2} \]
This shows that the force decreases by \( \frac{G}{d^2} \).

### Final Answer:
The force decreases by \( \frac{G}{d^2} \).