By what percent will the gravitational force between the two bodies be increased if their masses are increased by 50% ?

Explanation

To determine the percentage increase in the gravitational force between two bodies when their masses are increased by 50%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Gravitational Force Formula**:
The gravitational force \( F \) between two bodies is given by the formula:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two bodies, and \( r \) is the distance between their centers.

2. **Increase the Masses by 50%**:
If the masses \( m_1 \) and \( m_2 \) are increased by 50%, the new masses become:
\[ m_1' = 1.5 m_1 \]
\[ m_2' = 1.5 m_2 \]

3. **Calculate the New Gravitational Force**:
The new gravitational force \( F' \) with the increased masses is:
\[ F' = \frac{G m_1' m_2'}{r^2} = \frac{G (1.5 m_1)(1.5 m_2)}{r^2} \]
Simplifying this gives:
\[ F' = \frac{G (1.5 \cdot 1.5) m_1 m_2}{r^2} = \frac{G (2.25) m_1 m_2}{r^2} = 2.25 \frac{G m_1 m_2}{r^2} = 2.25 F \]

4. **Calculate the Increase in Force**:
The increase in force is given by:
\[ \Delta F = F' - F = 2.25 F - F = 1.25 F \]

5. **Calculate the Percentage Increase**:
The percentage increase in the gravitational force is calculated as:
\[ \text{Percentage Increase} = \left( \frac{\Delta F}{F} \right) \times 100 = \left( \frac{1.25 F}{F} \right) \times 100 = 125\% \]

### Final Answer:
The gravitational force between the two bodies will be increased by **125%** if their masses are increased by 50%.