The first two conditions make intuitive sense for rods/wires. In order to reach the top, the bead/particle must have a positive velocity at that point. However, I do not understand why the 3rd condition is necessary for a string. I can imagine a non-taut string still making a complete circle by using the momentum it already has from the way up. Could someone explain why this is the condition for a string, instead of the particle having a positive speed.
2026-03-28 00:25:43.1774657543
Rod vs string in vertical circular motion
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The moment that the tension of the string is less than the required centripetal force, the particle will start moving towards the "interior" of the circle. Thus, not following a circular motion any more. In the other examples, the rod or wire does not allow the particle/bead to move in the interior, so that's not an issue.
Btw, I think that this question should be asked in Physics SE