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?

\(\sqrt{5gR}\)

Conservation of Energy : \(\frac{1}{2}mv^2 = \frac{1}{2}mv_t^2 + 2 mgR\)

\(\frac{1}{2}mv^2 = \frac{1}{2}m(\sqrt{gR})^2+ 2 mgR\)

\(\frac{1}{2}mv^2 = \frac{1}{2}mgR+ 2 mgR\)

\(\frac{1}{2}mv^2 = \frac{5}{2}mgR\)

v = \(\sqrt{5gR}\)