Question

In a double star, two stars (one of mass m and the other of 2m) distant d apart rotate about their common centre of mass. Deduce an expression of the period of revolution. Show that the ratio of their angular momentum about the centre of mass is the same as the ratio of their kinetic energies

Answer 1

Since the gravitational force contributes to centripetal force of the rotation, consider mass M, we have

2GM^2/R^2=Mv^2/R

where R is the distance between the two masses.The root of the problem with differing angular velocities lies in that the center of gravity would not exhibit straight-line motion. To see this, let's call the origin the center of gravity at the starting time. At this starting time (which is arbitrary), the stars must have the origin along the line between them (or else the origin would not lie along the line containing their center of gravity, which would be a contradiction of my definition of origin). The origin must continue to lie along the line between them, and therefore they must rotate around the origin at the same rate and the angular velocities must be equal.