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Thank you for taking the time to read and help!

I've been thinking about what happens to the normal force when you're entering into uniform circular motion for the first time.

I'm assuming a point mass that is under the influence of some applied tangential force around the loop so that it maintains a constant velocity the whole time. (as if it had a motor that would throttle perfectly to maintain uniform velocity).

enter image description here

When already moving in a vertical loop over and over, the normal force reaches a maximum at the bottom and a minimum at the top as indicated above. And the normal force's curve is vertically shifted on this graph depending on the velocity around the loop.

(Note: the graph is meant to indicate normal force magnitude, so the curve cannot fall below zero. Hence there is a lower bound on the velocity, but not an upper bound).

Yet, before you enter the circle, while driving on level ground, the normal force is only bound by being equal and opposite to the force of gravity, meaning it's velocity independent.

So I end up with: enter image description here

So on one side, given a set mass, the graph is locked into being flat at a certain value, but on the other side, the graph starts at a maximum and then lowers, and that maximum can be arbitrarily raised by raising the velocity.

Obviously there can't be a jump discontinuity there, and the two curves melt into each other smoothly, but I'm having trouble imagining what occurs at that transition point.

My intuition and gut guess would be that it'd have something to do with the fact that you must "press on the accelerator" and apply more tangential force as you go into the circle to maintain speed... but I'm having trouble imagining this.

Thanks for your time!!! :D

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The jump discontinuity is correct. Before entering the loop the mass is not accelerating and after entering it is accelerating. Therefore the force must have changed.

There is no requirement that forces must be continuous. However, part of the discontinuity is due to treating the mass as a point. Other forms could produce a continuous force, particularly if the object is treated as non-rigid.

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    $\begingroup$ For example, if we're talking about a car, as you drive onto the loop, the wheels will start travelling upward, compressing the suspension, which will cause a gradual (still pretty quick ofc) increase in normal force on the car itself - until the system stabilizes at the curve you've shown. $\endgroup$ Commented Jan 17, 2021 at 6:49
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    $\begingroup$ Wow, so true! Just like if I punch a force sensor, there’d be a quick spike up and down. Haha, I don’t know what I was thinking, thanks! $\endgroup$ Commented Jan 17, 2021 at 23:04

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