The gravitational force between two bodies is F. If the mass of each body is doubled and the distance between them is halved , them the force between them will be

Options

A. `(4F)/9`
B. `(9F)/4`
C. 16F
D. `F/16`

Answer

To solve the problem, we need to use the formula for gravitational force, which is given by:

\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two bodies,
- \( r \) is the distance between the centers of the two bodies.

### Step 1: Identify the initial conditions
Let the initial masses of the two bodies be \( m_1 \) and \( m_2 \), and the initial distance between them be \( r \). The initial gravitational force is given as:

\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

### Step 2: Modify the masses and distance
According to the problem, the masses of each body are doubled, so the new masses become:

\[ m_1' = 2m_1 \]
\[ m_2' = 2m_2 \]

The distance between the two bodies is halved, so the new distance becomes:

\[ r' = \frac{r}{2} \]

### Step 3: Substitute the new values into the gravitational force formula
Now, we can find the new gravitational force \( F' \) using the modified masses and distance:

\[ F' = \frac{G \cdot m_1' \cdot m_2'}{(r')^2} \]

Substituting the new values:

\[ F' = \frac{G \cdot (2m_1) \cdot (2m_2)}{\left(\frac{r}{2}\right)^2} \]

### Step 4: Simplify the expression
Now, simplify the expression:

\[ F' = \frac{G \cdot 4m_1 \cdot m_2}{\frac{r^2}{4}} \]

This can be rewritten as:

\[ F' = \frac{G \cdot 4m_1 \cdot m_2 \cdot 4}{r^2} \]

\[ F' = 16 \cdot \frac{G \cdot m_1 \cdot m_2}{r^2} \]

### Step 5: Relate it back to the initial force
Since \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \), we can substitute this into our equation:

\[ F' = 16F \]

### Conclusion
Thus, the new gravitational force \( F' \) between the two bodies when their masses are doubled and the distance is halved is:

\[ F' = 16F \]

### Final Answer
The force between them will be \( 16F \).

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