For a ball on a string in a vertical circle of radius R:

At the top: Minimum speed for string to remain taut: v_top = sqrt(gR) (tension = 0 at minimum). Below this, the ball falls off the circle.

At the bottom: Using energy conservation: v_bottom = sqrt(5gR) minimum. Tension at bottom: T = mg + mv^2/R (maximum tension, string most likely to break here).

At angle theta from bottom: v^2 = v_bottom^2 - 2gR(1-cos(theta)). General tension: T = mv^2/R + mg*cos(theta) (for angle from top) or varies based on geometry.

Ball sliding on hemisphere: Leaves surface when N = 0, at angle theta = cos^{-1}(2/3) from vertical (from the top).

Key difference: String can only pull (T >= 0). Track/rod can push AND pull, so minimum speed at top = 0 for a rod.