Two stars of mass `m_1` and `m_2` are parts of a binary system. The radii of their orbits are `r_1` and `r_2` respectively, measured from the C.M. of the system. The magnitude of gravitational force `m_1` exerts on `m_2` is

Options

A. `(m_1m_2G)/((r_1+r_2)^2)`
B. `(m_1G)/((r_1+r_2)^2)`
C. `(m_2G)/((r_1+r_2)^2)`
D. `(m_1+m_2)/((r_1+r_2)^2)`

Explanation

To find the magnitude of the gravitational force \( F \) that mass \( m_1 \) exerts on mass \( m_2 \) in a binary star system, we can use Newton's law of universal gravitation. The formula for the gravitational force between two masses is given by:

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

where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)),
- \( m_1 \) and \( m_2 \) are the masses of the two stars,
- \( d \) is the distance between the two masses.

### Step-by-Step Solution:

1. **Identify the Distance Between the Masses**:
The distance \( d \) between the two stars can be expressed in terms of their respective orbital radii \( r_1 \) and \( r_2 \). Since \( r_1 \) and \( r_2 \) are measured from the center of mass (C.M.) of the system, the total distance \( d \) between the two stars is:
\[ d = r_1 + r_2 \]

2. **Substitute the Distance into the Gravitational Force Formula**:
Now, we can substitute \( d \) into the gravitational force formula:
\[ F = \frac{G \cdot m_1 \cdot m_2}{(r_1 + r_2)^2} \]

3. **Final Expression**:
Thus, the magnitude of the gravitational force that mass \( m_1 \) exerts on mass \( m_2 \) is:
\[ F = \frac{G \cdot m_1 \cdot m_2}{(r_1 + r_2)^2} \]