A binary star consists of two stars `A(mass 2.2 M_s)` and B `(mass 11 M_s)` where `M_s` is the mass of the sun, they are separted by distane d and are rotating about their center of mass, which is stationary. The ratio of the total angular momentum of the binary to the angular momentum of star B about the centre of mass is

Explanation

To solve the problem, we need to find the ratio of the total angular momentum of the binary star system to the angular momentum of star B about the center of mass. Let's break it down step by step.

### Step-by-Step Solution:

1. **Identify the Masses and Distances**:
- Let the mass of star A be \( m_A = 2.2 M_s \).
- Let the mass of star B be \( m_B = 11 M_s \).
- The total mass of the system is \( m_A + m_B = 2.2 M_s + 11 M_s = 13.2 M_s \).

2. **Determine the Center of Mass**:
- The center of mass (CM) of a two-body system is given by:
\[ R_{cm} = \frac{m_A r_A + m_B r_B}{m_A + m_B} \]
- Let \( r_A \) be the distance of star A from the center of mass and \( r_B \) be the distance of star B from the center of mass. By the definition of center of mass:
\[ m_A r_A = m_B r_B \]
- Thus, we can express the distances in terms of a ratio:
\[ \frac{r_A}{r_B} = \frac{m_B}{m_A} = \frac{11 M_s}{2.2 M_s} = 5 \]
- Therefore, \( r_A = 5 r_B \).

3. **Calculate the Moment of Inertia**:
- The moment of inertia \( I \) for a point mass is given by \( I = m r^2 \).
- For star A:
\[ I_A = m_A r_A^2 = (2.2 M_s)(5 r_B)^2 = 2.2 M_s \cdot 25 r_B^2 = 55 M_s r_B^2 \]
- For star B:
\[ I_B = m_B r_B^2 = (11 M_s)(r_B)^2 = 11 M_s r_B^2 \]

4. **Calculate the Total Moment of Inertia**:
- The total moment of inertia \( I_{total} \) of the system is:
\[ I_{total} = I_A + I_B = 55 M_s r_B^2 + 11 M_s r_B^2 = 66 M_s r_B^2 \]

5. **Calculate Angular Momentum**:
- The angular momentum \( L \) is given by \( L = I \omega \), where \( \omega \) is the angular velocity.
- The angular momentum of star B about the center of mass is:
\[ L_B = I_B \omega = (11 M_s r_B^2) \omega \]
- The total angular momentum of the binary system is:
\[ L_{total} = I_{total} \omega = (66 M_s r_B^2) \omega \]

6. **Find the Ratio**:
- The ratio of the total angular momentum of the binary system to the angular momentum of star B is:
\[ \text{Ratio} = \frac{L_{total}}{L_B} = \frac{66 M_s r_B^2 \omega}{11 M_s r_B^2 \omega} \]
- Simplifying this gives:
\[ \text{Ratio} = \frac{66}{11} = 6 \]

### Final Answer:
The ratio of the total angular momentum of the binary system to the angular momentum of star B about the center of mass is **6**.