# Tension at highest point in vertical circular motion

Suppose a body is attached to a string, and is whirled around along a vertical circular path. At the highest point, it falls downward. Why do we say that the tension is 0? Tension and the force of gravity act downward so wouldn't they add up causing the string to slacken even if tension were positive? I understand that tension provides necessary centripetal force for the object to continue its circular motion, but centripetal force also acts downwards.

At the top of the circle, the net force must have a magnitude of $$m v^2/r$$. Gravity and tension both point inwards, so we must have $$T + mg = \frac{mv^2}{r} \quad \Rightarrow \quad T = \frac{mv^2}{r} - mg.$$ But if $$v$$ is small enough, the right-hand side of this last equation becomes negative. And tension can't be negative — that would mean that the string would be pushing the body outwards rather than pulling it inwards. So this means that the string must go slack before it gets to that point.