Two objects of equal masses placed at certain distance from each other attracts each other with a force of F. If one-third mass of one objects is transferred to the other object, then new force will be :

Options

A. `(2)/(9)F`
B. `(16)/(9)F`
C. `(8)/(9)F`
D. F

Explanation

To solve the problem, we will use Newton's law of gravitation, which states that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( R \) is given by:

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

### Step 1: Identify the initial conditions
Let the masses of the two objects be \( m \) and \( m \). The initial gravitational force between them is given as \( F \).

\[ F = \frac{G \cdot m \cdot m}{R^2} \]

### Step 2: Determine the new masses after transferring mass
According to the problem, one-third of the mass from one object is transferred to the other.

- The mass of the first object after transferring one-third of its mass:
\[ m_1' = m - \frac{m}{3} = \frac{2m}{3} \]

- The mass of the second object after receiving one-third of the mass:
\[ m_2' = m + \frac{m}{3} = \frac{4m}{3} \]

### Step 3: Calculate the new gravitational force
Now, we can calculate the new gravitational force \( F' \) between the two new masses \( m_1' \) and \( m_2' \):

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

Substituting the new masses:

\[ F' = \frac{G \cdot \left(\frac{2m}{3}\right) \cdot \left(\frac{4m}{3}\right)}{R^2} \]

### Step 4: Simplify the expression for \( F' \)
Now, simplify the expression:

\[ F' = \frac{G \cdot \frac{8m^2}{9}}{R^2} \]

### Step 5: Relate \( F' \) to the original force \( F \)
We know that:

\[ F = \frac{G \cdot m^2}{R^2} \]

Now, we can express \( F' \) in terms of \( F \):

\[ F' = \frac{8}{9} \cdot \frac{G \cdot m^2}{R^2} = \frac{8}{9} F \]

### Conclusion
The new gravitational force after transferring one-third of the mass from one object to the other is:

\[ F' = \frac{8}{9} F \]