Two stars of masses `m_(1)` and `m_(2)` distance r apart, revolve about their centre of mass. The period of revolution is :

Options

A. `2pisqrt(r^3/(2G(m_1+m_2)))`
B. `2pisqrt((r^3(m_1+m_2))/(2G(m_1m_2)))`
C. `2pisqrt((2r^3)/(G(m_1+m_2)))`
D. `2pisqrt((r^3)/(G(m_1+m_2)))`

Answer

To find the period of revolution of two stars of masses \( m_1 \) and \( m_2 \) that are a distance \( r \) apart and revolve around their center of mass, we can follow these steps:

### Step-by-Step Solution:

1. **Define the distances from the center of mass**:
Let \( R_1 \) be the distance of mass \( m_1 \) from the center of mass and \( R_2 \) be the distance of mass \( m_2 \) from the center of mass. According to the definition of the center of mass, we have:
\[ R_1 + R_2 = r \]

2. **Use the center of mass formula**:
The distances \( R_1 \) and \( R_2 \) can be expressed in terms of the masses:
\[ R_1 = \frac{m_2}{m_1 + m_2} r \quad \text{and} \quad R_2 = \frac{m_1}{m_1 + m_2} r \]

3. **Set up the gravitational force and centripetal force**:
The gravitational force between the two masses provides the necessary centripetal force for their circular motion. The gravitational force is given by:
\[ F = \frac{G m_1 m_2}{r^2} \]
The centripetal force acting on \( m_1 \) is:
\[ F_c = m_1 R_1 \omega^2 \]
And for \( m_2 \):
\[ F_c = m_2 R_2 \omega^2 \]

4. **Equate gravitational force to centripetal force**:
For mass \( m_1 \):
\[ \frac{G m_1 m_2}{r^2} = m_1 R_1 \omega^2 \]
Simplifying gives:
\[ \omega^2 = \frac{G m_2}{R_1 r^2} \]

For mass \( m_2 \):
\[ \frac{G m_1 m_2}{r^2} = m_2 R_2 \omega^2 \]
Simplifying gives:
\[ \omega^2 = \frac{G m_1}{R_2 r^2} \]

5. **Combine the equations**:
Since both expressions equal \( \omega^2 \), we can set them equal to each other:
\[ \frac{G m_2}{R_1} = \frac{G m_1}{R_2} \]
This implies:
\[ m_2 R_2 = m_1 R_1 \]

6. **Express \( \omega \)**:
The total angular frequency \( \omega \) can be expressed as:
\[ \omega^2 = \frac{G (m_1 + m_2)}{R^3} \]
where \( R = R_1 + R_2 = r \).

7. **Find the period \( T \)**:
The period \( T \) is related to angular frequency \( \omega \) by:
\[ T = \frac{2\pi}{\omega} \]
Substituting for \( \omega \):
\[ T = 2\pi \sqrt{\frac{R^3}{G(m_1 + m_2)}} \]

### Final Result:
Thus, the period of revolution \( T \) is given by:
\[ T = 2\pi \sqrt{\frac{r^3}{G(m_1 + m_2)}} \]