If the mass of one particle is increased by `50 %` and the mass of another particle llis decreased by `50%`, the force between them

Options

A. decreases by `25%`
B. decreases by `75%`
C. increases by `25%`
D. does not change

Explanation

To solve the problem step by step, we will analyze the changes in the masses of the two particles and how it affects the gravitational force between them.

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

### Step 2: Determine the New Masses
- The mass of the first particle \( M_1 \) is increased by 50%. Therefore, the new mass \( M_1' \) is:
\[ M_1' = M_1 + 0.5 M_1 = \frac{3}{2} M_1 \]

- The mass of the second particle \( M_2 \) is decreased by 50%. Therefore, the new mass \( M_2' \) is:
\[ M_2' = M_2 - 0.5 M_2 = \frac{1}{2} M_2 \]

### Step 3: Calculate the New Gravitational Force
Now, substituting the new masses into the gravitational force formula, we get the new force \( F' \):
\[ F' = \frac{G M_1' M_2'}{R^2} = \frac{G \left(\frac{3}{2} M_1\right) \left(\frac{1}{2} M_2\right)}{R^2} \]
Simplifying this, we find:
\[ F' = \frac{G \cdot \frac{3}{4} M_1 M_2}{R^2} = \frac{3}{4} \frac{G M_1 M_2}{R^2} \]

### Step 4: Compare the New Force with the Original Force
The original force \( F \) is:
\[ F = \frac{G M_1 M_2}{R^2} \]
Now, we can express the new force \( F' \) in terms of the original force \( F \):
\[ F' = \frac{3}{4} F \]

### Step 5: Calculate the Percentage Change in Force
To find the percentage change in the force, we use the formula:
\[ \text{Percentage Change} = \frac{F' - F}{F} \times 100 \]
Substituting the values:
\[ \text{Percentage Change} = \frac{\frac{3}{4} F - F}{F} \times 100 = \frac{-\frac{1}{4} F}{F} \times 100 = -25\% \]

### Conclusion
The force between the two particles decreases by 25%.