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. decreased by `(G//d^2)`
C. becomes `d^2` times
D. increases by `(2G//d^2)`

Answer

To solve the problem, we need to analyze the gravitational force between two masses before and after transferring one unit of mass from one mass to the other.

### Step 1: Define the initial scenario
Let the two equal masses be \( m \) and \( m \), separated by a distance \( d \). The gravitational force \( F \) between them can be expressed using Newton's law of gravitation:
\[ F_1 = \frac{G \cdot m \cdot m}{d^2} = \frac{G m^2}{d^2} \]
where \( G \) is the gravitational constant.

### Step 2: Transfer one unit of mass
Now, we transfer one unit of mass from one of the masses to the other. This means:
- One mass becomes \( m - 1 \)
- The other mass becomes \( m + 1 \)

### Step 3: Calculate the new gravitational force
The new gravitational force \( F_2 \) between the two masses after the transfer is given by:
\[ F_2 = \frac{G \cdot (m - 1) \cdot (m + 1)}{d^2} \]
Using the identity \( (a - b)(a + b) = a^2 - b^2 \), we can simplify this:
\[ F_2 = \frac{G \cdot (m^2 - 1)}{d^2} \]

### Step 4: Relate the new force to the original force
Now, we can express \( F_2 \) in terms of \( F_1 \):
\[ F_2 = \frac{G m^2}{d^2} - \frac{G}{d^2} \]
This can be rewritten as:
\[ F_2 = F_1 - \frac{G}{d^2} \]

### Step 5: Conclusion
From the equation above, we see that the new force \( F_2 \) is less than the original force \( F_1 \) by \( \frac{G}{d^2} \). Therefore, the force decreases by \( \frac{G}{d^2} \).

### Final Answer
The correct option is that the force decreases by \( \frac{G}{d^2} \).

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