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: (1/2)mv_bottom^2 = (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)