What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

To just complete the circle, at the highest point, tension is zero and the gravitational force provides the necessary centripetal force. Hence, mv²top/R = mg (i)
Applying conservation of energy at the top and bottom points, 1/2mv²bottom = mg × 2R + 1/2mv²top (ii)
Substituting mv²top from (i) in (ii) and solving, vbottom = √5gR