If the distance between two bodies is increased by `25%`, then the `%` change in the gravitational force is

Options

A. Decreases by `36%`
B. Incraeses by `36%`
C. Increases by `64%`
D. Decreases by `64%`

Explanation

To solve the problem of how the gravitational force changes when the distance between two bodies is increased by 25%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Gravitational Force Formula**:
The gravitational force \( F \) between two masses \( M_1 \) and \( M_2 \) separated by a distance \( D \) is given by:
\[ F = \frac{G M_1 M_2}{D^2} \]
where \( G \) is the gravitational constant.

2. **Calculate the New Distance**:
If the distance \( D \) is increased by 25%, the new distance \( D' \) can be calculated as:
\[ D' = D + 0.25D = 1.25D \]

3. **Calculate the New Gravitational Force**:
The new gravitational force \( F' \) when the distance is \( D' \) is:
\[ F' = \frac{G M_1 M_2}{(D')^2} = \frac{G M_1 M_2}{(1.25D)^2} = \frac{G M_1 M_2}{1.5625 D^2} \]

4. **Express the New Force in Terms of the Original Force**:
We can relate \( F' \) to \( F \):
\[ F' = \frac{1}{1.5625} \cdot F = \frac{16}{25} F \]

5. **Calculate the Percentage Change in Force**:
The percentage change in the gravitational force can be calculated using the formula:
\[ \text{Percentage Change} = \frac{F' - F}{F} \times 100 \]
Substituting \( F' \):
\[ \text{Percentage Change} = \frac{\frac{16}{25} F - F}{F} \times 100 = \left(\frac{16}{25} - 1\right) \times 100 \]
Simplifying:
\[ = \left(\frac{16 - 25}{25}\right) \times 100 = \left(-\frac{9}{25}\right) \times 100 = -36\% \]

6. **Interpret the Result**:
The negative sign indicates that the gravitational force has decreased. Therefore, the gravitational force decreases by 36%.

### Final Answer:
The gravitational force decreases by **36%**.