A double star is a system of two stars of masses `m` and `2m`, rotating about their centre of mass only under their mutual gravitational attraction. If `r` is the separation between these two stars then their time period of rotation about their centre of mass will be proportional to

Options

A. `T=2pi sqrt((L^(3))/(2mG))`
B. `T?=2pi sqrt((L^(3))/(3mG))`
C. `T=2pi sqrt((L^(3))/(4mG))`
D. `T=2pi sqrt((L^(3))/(mG))`

Answer

B

Explanation

The two starts are at A and B. The system rotates about their centre of mass C. It divides the distence AB = L, in the ratio of 1 : 2.

`because 2m xx (L)/(3)=m xx (2L)/(3)`
Thus `AC=(L)/(3)` and `BC=(2L)/(3)`.
The gravitational force of attraction between them is
`F=(G(2m)(m))/(L^(2))` and this provides the C.P. force for the mass m' to rotate in a circular of radius `(2L)/(3)`.
`therefore (G.2m^(2))/(L^(2))=mr omega^(2)=m. ((2L)/(3))omega^(2)`
`because omega^(2)=(3Gm)/(L^(3))therefore ((2pi)/(T))^(2)=(3Gm)/(L^(3))`
`therefore T = 2pi sqrt((L^(3))/(3Gm))`