Q.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?

When a mass $m$ tied to a string of length $r$ is whirled in a vertical circle, applying Newton's second law in the tangential and radial directions at a general angle $\theta$ (measured from the downward vertical) gives $mg\sin\theta = m\,dv/dt$ (tangential) and $T-mg\cos\theta = mv^2/r$ (radial). Because the tangential acceleration $g\sin\theta$ is nonzero everywhere except $\theta=0^\circ$, both the speed and the string tension vary continuously around the circle -- this motion is never uniform.

At the lowest point, gravity points away from the centre while tension points toward it: $T_1 = mg + mv_1^2/r$. At the highest point, both gravity and tension point toward the centre: $T_2 = mv_2^2/r - mg$. Since gravity assists the centripetal requirement at the top but opposes it at the bottom, $T_1$ is always larger, and combining the radial equations with energy conservation between the two points shows the difference is a constant, $T_1-T_2=6mg$. …