Two particles of masses `m_1` and `m_2` , `m_1 gt m_2` move in circular paths under the action of their gravitational attraction. While doing so, their separation remains constant and equals ‘r’. Radius of circular path of `m_2` is:

Options

A. `r/2`
B. `m_1/m_2r`
C. `m_2/(m_1+m_2)r`
D. `m_1/(m_1+m_2)r`

Explanation

To find the radius of the circular path of mass \( m_2 \) in a two-particle system where both masses \( m_1 \) and \( m_2 \) are in circular motion under their mutual gravitational attraction, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
We have two masses, \( m_1 \) and \( m_2 \), with \( m_1 > m_2 \). They are separated by a distance \( r \) and are in circular motion due to their gravitational attraction.

2. **Identifying Forces**:
The gravitational force between the two masses provides the necessary centripetal force for their circular motion. The gravitational force \( F \) can be expressed as:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant.

3. **Center of Mass**:
The center of mass (CM) of the system can be calculated using the formula:
\[ r_{CM} = \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2} \]
where \( r_1 \) is the distance from \( m_1 \) to the CM, and \( r_2 \) is the distance from \( m_2 \) to the CM.

4. **Setting up the Distances**:
Since the total distance between the two masses is \( r \), we have:
\[ r_1 + r_2 = r \]

5. **Relating Distances to Masses**:
The distances \( r_1 \) and \( r_2 \) can be expressed in terms of the masses:
\[ r_1 = \frac{m_2}{m_1} r_2 \]
Substituting this into the equation for total distance gives:
\[ \frac{m_2}{m_1} r_2 + r_2 = r \]
Factoring out \( r_2 \):
\[ r_2 \left( \frac{m_2}{m_1} + 1 \right) = r \]

6. **Solving for \( r_2 \)**:
Rearranging the equation to solve for \( r_2 \):
\[ r_2 = \frac{r}{\frac{m_2}{m_1} + 1} = \frac{r}{\frac{m_2 + m_1}{m_1}} = \frac{m_1 r}{m_1 + m_2} \]

7. **Conclusion**:
The radius of the circular path of mass \( m_2 \) is:
\[ r_2 = \frac{m_1 r}{m_1 + m_2} \]

### Final Answer:
The radius of the circular path of mass \( m_2 \) is \( \frac{m_1 r}{m_1 + m_2} \).