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Understanding the Minimum Speed to Keep Carriage on Tracks in a Loop
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[QUOTE="etotheipi, post: 6308984"] You can think of it as writing the equations for the constrained circular motion, and then finding the critical values for when the constraints can no longer be satisfied. So at an arbitrary point on the loop, with ##\theta## measured from the downward vertical, you might say$$N - mg\cos{\theta} = \frac{mv^{2}}{r}$$We also impose the constraint that ##N\geq 0##, if we assume the rollercoaster is not bound to the track! Of course, if we had a situation like a bead on a wire, we might indeed permit negative values of ##N## because the normal force could act in both directions! But I'm assuming that isn't the case here. As the rollercoaster goes up the loop, the normal force will decrease and decrease until eventually it will reach zero. That is, the radial component of the weight force becomes sufficiently large to fulfil the entire requirement for the centripetal force. If it carried on for an infinitesimally small amount of time, ##N## would become negative (as if to pull the rollercoaster back toward the track) - but that's not allowed! The result is that the motion must become unconstrained. You want this to happen at the top of the loop, so you can set ##(N, \theta) = (0, \pi)## in the above equation. [/QUOTE]
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Understanding the Minimum Speed to Keep Carriage on Tracks in a Loop
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