A ball is tied to the end of a length of string. A student holds one end of the string and whirls it so the ball travels in a vertical circle. First she whirls the ball at a very fast speed, and then she very gradually lowers the speed. Where is the ball most likely to be when the string begins to sag?

The top of the circle

Explanation:

At the top of the circle, there are two forces pulling the ball towards the center of the circle: weight and tension.

∑F = ma

mg + T = mv²/r

As v goes down, T also goes down, until it eventually reaches 0.

At the bottom of the circle, weight is pulling down, or away from the center of the circle.  Sum of the forces:

∑F = ma

T − mg = mv²/r

As v goes down, T also goes down.  However, even at v=0, T cannot be less than mg.

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