Motion in a Vertical Circle

A body moving in a vertical circle -- unlike one moving in a horizontal circle at constant speed -- does not move at constant speed at all, because gravity now has a component along the direction of motion for most of the path, doing positive work as the body descends and negative work as it climbs. Finding the conditions under which such a body can complete a full vertical circle (a stone on a string, water in a whirled bucket, a car on a vertical loop track) requires combining Newton's second law, applied along the radial direction, with conservation of mechanical energy.

Condition at the top of the circle. At the topmost point of the circle, both the weight $mg$ and the string's tension $T$ (or, for water in a bucket, the normal reaction from the bucket's base) point straight downward -- that is, both point toward the centre of the circle, since the centre lies directly below the body at this instant. Newton's second law along the radial (centre-directed) direction gives

Since a string can only pull, never push, $T$ cannot become negative; the string stays taut only as long as $T \geq 0$. The critical (minimum-speed) condition occurs exactly when $T = 0$, i.e. when gravity alone supplies the entire centripetal force needed at the top:

If the body's actual speed at the top is less than this value, gravity alone would be more than enough to supply the required centripetal force, meaning the string would need to push to keep the body on the circular path -- which it cannot do -- so the string instead goes slack and the body departs from the circular path (falling as a projectile) before ever reaching the top.

Condition at the bottom of the circle. At the bottommost point, the tension $T$ points straight up (toward the centre, now directly above), while $mg$ still points straight down (away from the centre here), so Newton's second law gives $T - mg = mv_{\text{bottom}}^2/r$, i.e. $T = mg + mv_{\text{bottom}}^2/r$ -- always a positive quantity here, so the tension itself imposes no additional restriction at the bottom; the real constraint comes from relating $v_{\text{bottom}}$ to $v_{\text{top}}$ via energy conservation.

Relating the two speeds by energy conservation. The bottom and top of the circle are separated by a height difference of $2r$ (the circle's diameter). Applying conservation of mechanical energy between these two points (only gravity and the string's tension act, and tension, being always perpendicular to the motion, does zero work):

$\frac{1}{2}mv_{\text{bottom}}^2 = \frac{1}{2}mv_{\text{top}}^2 + mg(2r) \quad\Longrightarrow\quad v_{\text{bottom}}^2 = v_{\text{top}}^2 + 4gr$ …