Two body of same mass 'm' each are placed at distance d from each other, then gravitational force between them is F. If 50% mass is transferred from one body to another and distance between them increased by 50% then new gravitational force between them will be :

Options

A. `(3F)/(4)`
B. `F/9`
C. `F/3`
D. `(4F)/(9)`

Explanation

To solve the problem, we need to analyze the changes in mass and distance and their effect on gravitational force. Let's break it down step by step.

### Step 1: Initial Setup
We start with two bodies, each having mass \( m \), separated by a distance \( d \). The gravitational force \( F \) between them can be expressed using Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant, \( m_1 = m \), \( m_2 = m \), and \( r = d \).

### Step 2: Calculate Initial Gravitational Force
Substituting the initial values into the formula:
\[ F = \frac{G m \cdot m}{d^2} = \frac{G m^2}{d^2} \]

### Step 3: Mass Transfer
Now, we transfer 50% of the mass from one body to the other. After the transfer:
- The first body will have \( m - \frac{m}{2} = \frac{m}{2} \).
- The second body will have \( m + \frac{m}{2} = \frac{3m}{2} \).

### Step 4: Increase in Distance
The distance between the two bodies is increased by 50%. The new distance \( r' \) becomes:
\[ r' = d + 0.5d = 1.5d = \frac{3d}{2} \]

### Step 5: Calculate New Gravitational Force
Now, we can calculate the new gravitational force \( F' \) using the new masses and the new distance:
\[ F' = \frac{G m_1' m_2'}{(r')^2} \]
Substituting the new values:
\[ F' = \frac{G \left(\frac{3m}{2}\right) \left(\frac{m}{2}\right)}{\left(\frac{3d}{2}\right)^2} \]

### Step 6: Simplifying the New Force Equation
Calculating the denominator:
\[ \left(\frac{3d}{2}\right)^2 = \frac{9d^2}{4} \]
Now substituting this back into the equation for \( F' \):
\[ F' = \frac{G \cdot \frac{3m}{2} \cdot \frac{m}{2}}{\frac{9d^2}{4}} = \frac{G \cdot \frac{3m^2}{4}}{\frac{9d^2}{4}} = \frac{3G m^2}{9d^2} = \frac{G m^2}{3d^2} \]

### Step 7: Relating New Force to Initial Force
Recall that the initial gravitational force \( F \) was:
\[ F = \frac{G m^2}{d^2} \]
Thus, we can express \( F' \) in terms of \( F \):
\[ F' = \frac{1}{3} F \]

### Conclusion
The new gravitational force \( F' \) after transferring 50% of the mass from one body to another and increasing the distance by 50% is:
\[ F' = \frac{F}{3} \]

### Final Answer
The new gravitational force between the two bodies is \( \frac{F}{3} \).

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