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Four particles of equal masses are moving round a circle of radius r due to their mutual gravitation attraction. Find the angular velocity.
Figure shows four particles moving in a circular path due to gravitational force of attraction .

F1 is the force acting on each particle due to attraction by other particle that is diametrically opposite .

magnitude of F1 is

where G is universal gravitationla constant , m is mass of each particle and r is radius of circular path.

F2 and F3 are the forces acting on each particl due to left side and right side neighbour.

Magnitude of F2 and F3 are equal .

F2 and F3 each make angle 45o with F1 .

Hence resultant of F2 and F3 is in the same direction of F1 .

Magnitude of net force F acting on each particle is

Above force F provides the centripetal force required for circular motion .

Centripetal force of circular motion equals gravitational force of attraction . Let ω be the angular speed of rotation of masses.

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