Question

A triple star ststem consists of two stars each of mass m in the same circular orbit about central star with mass `M=2xx10^(33)kg`. The two outer stars always lie at opposite ends of a diameter of their common circular orbit the radius of the circular orbit is `r=10^(11)` m and the orbital period each star is `1.6xx10^(7)s` [take `pi^(2)=10` and `G=(20)/(3)xx10^(-11)Mn^(2)kg^(-2)`]
Q. The mass m of the outer stars is:
A. `(16)/(15)xx10^(30)kg`
B. `(11)/(8)xx10^(33)kg`
C. `(15)/(16)xx10^(30)kg`
D. `(8)/(11)xx10^(30)kg`

Answer 1

Correct Answer - B
`F_(min)=` gravitational foce between two outer stars `=(Gm^(2))/(4r)`
`F_(mM)=` gravitational force between central stars and outer star `=(GmM)/(r^(2))`
For circular motion outer star, `(mv^(2))/(r)=F_(min)+F_(mM)thereforev^(2)=(G(m+4M))/(4r)`
`T=` period of orbital motion `=(2pir)/(v)thereforem=(16pi^(2)r^(3))/(GT^(2))-4M=((150)/(16)-8)10^(30)=(11)/(8)xx10^(30)kg`