Binary stars rotate under mutual gravitational force at separation `2(G/(omega^(2)))^(1/3)`, where `omega` is the angular velocity of each of the star about centre of mass of the system. If difference between the mass of stars is `6` units. Find the ratio of mass of bigger star to smaller star.

Explanation

To solve the problem, we need to find the ratio of the mass of the bigger star (let's denote it as \( m_1 \)) to the mass of the smaller star (denote it as \( m_2 \)). We are given that the difference between the masses of the stars is 6 units, i.e.,

\[ m_1 - m_2 = 6 \]

We also know the relationship between the separation of the stars and their angular velocity, which is given by:

\[ R = 2 \left( \frac{G}{\omega^2} \right)^{1/3} \]

### Step 1: Express the distances from the center of mass
The center of mass \( R_{cm} \) of the two stars can be expressed in terms of their masses and distances from the center of mass:

\[ R_1 = \frac{m_2}{m_1 + m_2} R \]
\[ R_2 = \frac{m_1}{m_1 + m_2} R \]

Where \( R = R_1 + R_2 \).

### Step 2: Set up the gravitational force equation
The gravitational force \( F_g \) between the two stars is given by:

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

This gravitational force provides the necessary centripetal force for the circular motion of the stars, which can be expressed as:

\[ F_c = m_1 \omega^2 R_1 \]

### Step 3: Equate the forces
Setting the gravitational force equal to the centripetal force gives us:

\[ \frac{G m_1 m_2}{R^2} = m_1 \omega^2 R_1 \]

### Step 4: Substitute \( R_1 \)
Substituting \( R_1 \) from Step 1 into the equation gives:

\[ \frac{G m_1 m_2}{R^2} = m_1 \omega^2 \left( \frac{m_2}{m_1 + m_2} R \right) \]

### Step 5: Substitute \( R \)
Substituting \( R = 2 \left( \frac{G}{\omega^2} \right)^{1/3} \) into the equation:

\[ \frac{G m_1 m_2}{\left( 2 \left( \frac{G}{\omega^2} \right)^{1/3} \right)^2} = m_1 \omega^2 \left( \frac{m_2}{m_1 + m_2} \cdot 2 \left( \frac{G}{\omega^2} \right)^{1/3} \right) \]

### Step 6: Simplify the equation
After simplifying, we can find a relationship between \( m_1 \) and \( m_2 \).

### Step 7: Solve for the ratio
From the earlier steps, we can derive:

1. \( m_1 + m_2 = 8 \)
2. \( m_1 - m_2 = 6 \)

Now we can solve these two equations simultaneously.

Adding the two equations:

\[ (m_1 + m_2) + (m_1 - m_2) = 8 + 6 \]
\[ 2m_1 = 14 \implies m_1 = 7 \]

Substituting \( m_1 \) back into one of the equations:

\[ 7 - m_2 = 6 \implies m_2 = 1 \]

### Step 8: Find the ratio
Now we can find the ratio of the mass of the bigger star to the smaller star:

\[ \frac{m_1}{m_2} = \frac{7}{1} = 7 \]

Thus, the ratio of the mass of the bigger star to the smaller star is \( 7:1 \).