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Since each particle rotates around the circle of their center of mass , the centripetal acceleration to keep them in the orbit is provided by the mutual gravitational force.
Putting this knowledge into equations,
Centripetal force = Mv^2/R, considering the mass of each particle as 'M' and the rotation speed as 'v'.
Mutual force of gravitational attraction = GM^2/(2R)^2.
Thus, GM^2/4R^2 = Mv^2/R.
We get speed of each particle,
v =(GM/4R)^(1/2).
That's your answer.
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