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) = m$a_t$ (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).