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 \( m_2 \) when two particles of masses \( m_1 \) and \( m_2 \) are moving in circular paths under their gravitational attraction, we can follow these steps:

### Step 1: Understand the system
We have two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \). The center of mass of the system will be closer to the heavier mass \( m_1 \) since \( m_1 > m_2 \).

### Step 2: Define the distances
Let:
- \( r_1 \) be the distance from \( m_1 \) to the center of mass.
- \( r_2 \) be the distance from \( m_2 \) to the center of mass.

Since the total distance between the two masses is \( r \), we can write:
\[ r_1 + r_2 = r \tag{1} \]

### Step 3: Center of mass equation
The center of mass \( x_{cm} \) for the two masses can be expressed as:
\[ x_{cm} = \frac{m_1 \cdot (-r_1) + m_2 \cdot r_2}{m_1 + m_2} \]
Since the center of mass is at the origin (0,0), we have:
\[ 0 = -m_1 r_1 + m_2 r_2 \tag{2} \]

### Step 4: Rearranging the center of mass equation
From equation (2), we can rearrange it to find a relationship between \( r_1 \) and \( r_2 \):
\[ m_1 r_1 = m_2 r_2 \implies r_1 = \frac{m_2}{m_1} r_2 \tag{3} \]

### Step 5: Substitute \( r_1 \) in equation (1)
Now, substitute equation (3) into equation (1):
\[ \frac{m_2}{m_1} r_2 + r_2 = r \]
Factoring out \( r_2 \) gives:
\[ r_2 \left( \frac{m_2}{m_1} + 1 \right) = r \]

### Step 6: Solve for \( r_2 \)
Now, solve for \( r_2 \):
\[ r_2 = \frac{r}{\frac{m_2}{m_1} + 1} = \frac{r}{\frac{m_2 + m_1}{m_1}} = \frac{m_1 r}{m_1 + m_2} \]

### Conclusion
Thus, the radius of the circular path of mass \( m_2 \) is:
\[ r_2 = \frac{m_1 r}{m_1 + m_2} \]