- Two particles of equal mass m are moving in a circular path with radius R.
- The particles are under the influence of their mutual gravitational attraction.
- We need to find the speed v of each particle.


- The force of gravity between two objects can be calculated using the equation F = (G * m1 * m2) / r^2, where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between them.
- In this case, the particles have the same mass m and the distance between them is 2R (since they are on opposite sides of the circle).
- The force between the particles is attractive and acts towards the center of the circle.


- The gravitational force provides the centripetal force that keeps the particles moving in a circular path.
- Centripetal force (Fc) is given by the equation Fc = mv^2 / R, where m is the mass of the particle, v is its velocity, and R is the radius of the circle.
- Equating the gravitational force with the centripetal force, we have (G * m^2) / (2R^2) = mv^2 / R.
- Simplifying the equation, we get v^2 = (G * m) / (2R).
- Taking the square root of both sides, we get v = √(G * m) / √(2R).


- We can further simplify the equation by rearranging it as v = √(G * m) / √(2) * √R.
- Since G * m / √2 is a constant, let's denote it as a new constant C.
- Therefore, v = C / √R.


- The speed v of each particle is given by v = C / √R, where C is a constant.
- Since we are asked to find the answer in terms of G, m, and R, we can substitute the value of C as C = √(G * m) / √2.
- Substituting the value of C in the equation, we get v = (√(G * m) / √2) / √R.
- Simplifying further, we get v = 1/2 * √(G * m / R).
- Hence, the speed of each particle is 1/2 * √(G * m / R).