Final answer:
The speed of each particle moving in a circle under the influence of their mutual gravitational attraction is √(GM/r), with option (a) being the correct answer.
Step-by-step explanation:
The question pertains to gravitational forces and circular motion between two particles of equal mass moving in a circle under the influence of their mutual gravitational attraction. According to Newton's law of gravitation and the equation for the centripetal force that maintains circular motion, we can deduce the relative speed of each particle.
By using Newton’s universal law of gravitation, F = G(mM)/r2, and equating it to the centripetal force required for circular motion, F = mv2/r, where m is the mass of the particle and M is the same since the masses are identical, we cancel out the mass and obtain GM = v2. Solving for the speed v, we get v = √(GM/r).
Given the options, the correct answer is (a) √(GM/r). This speed ensures that both particles are able to mutually orbit around their common center of mass.