A pair of stars rotates about a common centre of mass. One of the stars has a mass `M` and the other `m`. Their centres are a distance `d` apart, `d` being large compared to the size of either star. Derive an expression for the period of revolution of the stars about their common centre of mass. Compare their angular momenta and kinetic energies.

Answer

`(a) T=(2pid^(3//2))/(sqrt(3Gm)),(b) 2,(c) 2`

Explanation

They will rotate about their c.m.
position of centre of mass
`0=(2m(-x)+m(d-x))/(2m+m)impliesx=d/3`
(a) These force of atraction will provided required centripetal force
`((2m)v_(M)^(2))/(d//3)=(G2m.m)/(d^(2))impliesv_(M)=sqrt(R/3(GM)/d)`
`T=(2pi(d//3))/(v_(M))impliesT=(2pid)/3.sqrt((3d)/(GM))`
`T=2pisqrt((d^(3))/(3GM))impliesT=(2pid^(3//2))/(sqrt(3GM))`