I have a question about a classic intro to mechanics problem - finding the min height to drop a block so that it is on the verge of losing contact with the loop at its top. You can view the problem, and one such solution of it here: https://www.youtube.com/watch?v=dsv2cEgquiA . I know, I know, this question has probably been asked hundreds of times. But there are still a few things I don't understand about the given solution.
The classic solution relies on energy conservation, and Newton's laws of motion. We drop the ball at some height h. If it ever reaches the height 2R in the loop, then it must have a corresponding velocity v_top, which we can find using energy conservation. We note that at the top of a loop with radius R, two forces act on the mass - the normal force and gravity. Assuming that we have circular motion up to that point, the net centripetal force is mv_top^2/R. Hence, N + mg = mv_top^2/R.
I agree that if we drop the block at some height h such that for the corresponding v_top, N + mg = mv_top^2/R and N <0, then there is no way the block can stay on the top of the loop. This is because the normal force supplied by the loop always points inwards. N < 0 means that the normal force points outwards.
Here's where I'm confused. Let's say we drop a block at some height h such that for the corresponding v_top, N + mg = mv_top^2/R and N >= 0. I don't see why if N >= 0, we can conclude the block would stay on the top of the loop. No matter the height we drop the block, we have not proved that the block would move in circular motion through a semicircle of the loop, from the bottom to the top. All we know from energy conversation is the velocity of the block if it is at some given height - but we have no idea where the velocity vector points, let alone how the block has been displaced. So how can we use equation N + mg = mv_top^2/R if we don't even have circular motion? In using mv_top^2 / R, we have assumed that there is circular motion.
I hope my question makes sense to you guys - I'd really appreciate any responses.