Final answer:
To find the initial velocity for each particle and the time period of circular motion, we can consider the gravitational force between the particles. The initial velocity for each particle is 2*pi*sqrt(a/3*G*m) and the time period of circular motion is 2*pi*sqrt(a/3*G*m).
Step-by-step explanation:
To find the initial velocity for each particle and the time period of circular motion, we can start by considering the gravitational force acting between the particles. The force of gravity between two particles is given by the equation F = G(m1*m2)/r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the particles, and r is the distance between the particles.
In the given scenario, the particles are situated at the vertices of an equilateral triangle. Each particle is at a distance of 'a' from each other.
Since each particle moves in a circle while maintaining the original mutual separation a, we can consider the force of gravity between two particles as a centripetal force. The centripetal force required to maintain circular motion is given by the equation F = m*v^2/r, where m is the mass of the particle, v is its velocity, and r is the radius of the circle.
By equating the two forces, we can find the initial velocity of each particle as v = sqrt((G*m)/a). The time period of circular motion is given by the equation T = (2*pi*r)/v, where r is again the radius of the circle and v is the velocity of the particle.
Therefore, the initial velocity for each particle is 2*pi*sqrt(a/3*G*m) and the time period of circular motion is 2*pi*sqrt(a/3*G*m).