For a ball on a string in vertical circular motion:

At the top, the minimum speed for the string to remain taut: mg = $mv_{top}^2$/R => v_{top}_{min} = sqrt(gR)

Using energy conservation from bottom to top: \frac{1}{2}$$mv_{bottom}^2 = \frac{1}{2}$$mv_{top}^2 + mg(2R) v_{bottom}_{min} = sqrt(5gR) (substituting v_{top}_{min} = sqrt(gR))

At various points (angle theta from bottom): $v^2$ = $v_{bottom}^2$ - 2gR(1 - cos(theta))

Tension at angle theta: T = $mv^2$/R - mg*cos(theta) + mg (general expression depends on geometry)

At bottom: $T_{bottom}$ = $mv_{bottom}^2$/R + mg (maximum tension) At top: $T_{top}$ = $mv_{top}^2$/R - mg (minimum tension)

Critical condition: $T_{top}$ >= 0 => $v_{top}$ >= sqrt(gR) => $v_{bottom}$ >= sqrt(5gR)