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When working through a physics problem, I realized there's a fundamental difference between when an object is spinning in a circle and is attached to some rigid object such as a beam fixed to an axle vs a non-rigid object such as a rope.

When looking at the minimum velocity at the top to continue moving in a circle, with an object such as a rope, the minimum velocity is $\sqrt{rg}$ to get around the top. But with an object fixed to a rigid object, the velocity at the top can be zero and it will continue to move in a circle. I feel like it has something to do with the rigid object providing a normal force that counteracts gravity which a rope doesn't, but I'm not sure how this provides the difference between minimum clearing velocity at the top.

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With a rigid tether, the question is whether the object starts with enough kinetic energy to make up the potential difference and reach the top. The rod is allowed to pull or push.

The rope can only pull and that causes this problem: some of the upwards kinetic energy possessed when the object passes the height of the circle's center has to be converted into sideways kinetic energy to stay on course. Tension can do this. However, in this quadrant of the circle, sideways energy can't be converted back into upwards kinetic energy or into gains in potential energy without a pusing force that the rope can't provide.

Try thinking of the problem backwards: start at the top and release the object. With a rod, it will swing outward because the rod pushes it. With the rope, it will fall straight down unless you start by throwing it sideways.

Basically, without the ability to push, you can't use all of the sideways motion to climb the top of the hill, so you can waste some energy.

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