according to the concept of reduced mass when two bodies rotate about their common centre of mass such that a force of attraction acts between them and the dist.b/n the 2 bodies remains constant then their angular velocites are exactly same both in magnitude and direction .the results for angular velocity and time period that u obtain for single bodies can be obtained by using the reduced mass of  2 body system .To reduce the 2-body system to 1-body just consider 1 body of mass=m(reduced mass) and consider its rotation about a fixed point at a dist=dist between the 2 bodies.thus u get a simple circular motin whose radius is known and the centrifugal force is balanced by the force of attraction that was originally present between the bodies.