Two particles of equal mass go around a circle of radius R under the action of their mutual gravitational attraction. The speed v of each particle is

When two particles of equal mass revolve about a common centre of mass under their mutual gravitational attraction, the balance between the gravitational force and the centripetal force for circular motion dictates their motion. Since masses are equal, the centre of mass lies at the midpoint on the line joining the two particles. Each particle revolves about the centre of mass in a circular path of radius R/2.

The gravitational force between the particles is what furnishes them with the required centripetal force. The gravitational force depends upon the masses of the particles and the distance separating them, which is given by R. On the other hand, centripetal force depends upon the mass of each particle, the orbital speed v, and the radius of their circular path.

From the equality of gravitational and centripetal forces, it follows that the orbital velocity of each particle is as follows: v = 1/2 √(Gm/R) It follows that speed of each particle is proportional to the square root of the constant of gravitation G and mass of the particle m and inversely proportional to the square root of R. The final expression for the speed is given by v = 1/2 √(Gm/R), which shows the relationship between mass, radius, and speed in this two-body system.