To complete a vertical loop of radius $R$, a body must maintain a minimum velocity at the highest point such that the tension in the string (or the normal force) does not become zero . At this critical highest point, the minimum required velocity is $v_{top} = \sqrt{gR}$ . According to the law of conservation of mechanical energy, the total energy at the lowest point (entry) must equal the total energy at the highest point: $E_{bottom} = E_{top}$ . This relationship is expressed as: $\frac{1}{2}mv_{min}^2 = \frac{1}{2}mv_{top}^2 + mg(2R)$. By substituting $v_{top}^2 = gR$, the equation becomes $\frac{1}{2}mv_{min}^2 = \frac{1}{2}mgR + 2mgR = \frac{5}{2}mgR$. Solving for $v_{min}$ yields $v_{min} = \sqrt{5gR}$ .