What will happen to the gravitational force between two bodies if they are brought closer by half of their initial separation ?

Explanation

To solve the problem of what happens to the gravitational force between two bodies when they are brought closer by half of their initial separation, we can follow these steps:

### 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 Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant.

### Step 2: Determine the New Distance
If the two bodies are brought closer by half of their initial separation, the new distance \( r' \) becomes:
\[ r' = \frac{r}{2} \]

### Step 3: Calculate the New Gravitational Force
Now we can calculate the new gravitational force \( F' \) using the new distance \( r' \):
\[ F' = \frac{G m_1 m_2}{(r')^2} = \frac{G m_1 m_2}{\left(\frac{r}{2}\right)^2} \]
This simplifies to:
\[ F' = \frac{G m_1 m_2}{\frac{r^2}{4}} = \frac{4 G m_1 m_2}{r^2} \]

### Step 4: Relate the New Force to the Initial Force
We can express the new force \( F' \) in terms of the initial force \( F \):
\[ F' = 4 \cdot \frac{G m_1 m_2}{r^2} = 4F \]

### Conclusion
Thus, when the two bodies are brought closer by half of their initial separation, the gravitational force between them increases to four times the initial gravitational force. Therefore, the answer is:
\[ \text{The new gravitational force is } 4F. \]

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