For a mass on a string (length L):

At angle theta from the lowest point:

• Radial: T - mg*cos(theta) = mv^2/L
• Tangential: mgsin(theta) = ma_t (decelerating on the way up)

At the top (theta = 180): T_top = mv_top^2/L - mg At the bottom (theta = 0): T_bottom = mv_bottom^2/L + mg

Minimum conditions (for complete loop):

• At top: T_top >= 0 => v_top >= sqrt(gL)
• Using energy conservation: v_bottom >= sqrt(5gL)

T_bottom - T_top = 6mg (always, for any speed completing the loop)

For a mass on a rigid rod: The rod can push (compress), so T can be negative. Minimum speed at top = 0. Minimum speed at bottom = 2*sqrt(gL) = sqrt(4gL).