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

Answer

4

Explanation

Let `m_(1)` and `m_(2)` are the masses of A and B respectively. And assume that centre of mass is at distances x, and x, from the stars as shown in figure.

`x_(1)=(m_(2)d)/(m_(1)+m_(2))`
`x_(2)=(m_(1)d)/(m_(1)+m_(2))`
Moment of inertia of the system can be written as follows:
`I=m_(1)x_(1)^(2)+m_(2)x_(2)^(2)`
Substituting values we get: `I=(m_(1)m_(2))/(m_(1)+m_(2))d^(2)`
Angular momentum of the system:
`L_("system")=Iomega=[(m_(1)m_(2))/(m_(1)+m_(2))d^(2)]oemga`
Angular momentum of B with respect to centre of mass can be written as follows:
`L_(B)=m_(2)x_(2)^(2)omega=m_(2)((m_(1)d)/(m_(1)+m_(2)))^(2)omega`
`=(m_(2)m_(1)^(2))/((m_(1)+m_(2))^(2))d^(2)omega`
Ratio: `(L_("system"))/(L_(B))=(m_(1)+m_(2))/(m_(1))=(2.2M_(s)+11M_(s))/(2.2M_(s))=6`
Hence answer is 6.