A casing of mass m can glide along a thin metal ring lying horizontally. It has an initial speed of v0. How long a distance s will it glide before stopping if its friction coefficient is μ?
The answer is given as
s = r/2μ * ln( v02 + √(v02 + g2r2)/ g * r )
The task is to provide the deductive path to reach that answer.
Centripetal force: mv2/r
Gravitational force: mg
Kinetic energy: mv2/2
The Attempt at a Solution
The casing will stop once all the kinetic energy has turned into friction work. The friction is a sum of that caused by the gravitational force and the centripetal force. The friction work Wf is thus
Wf = s * F = s * μ * ( mv2/r + mg )
Since Wf is equal to the kinetic energy we can write the following equation:
s = Kinetic energy / F = (mv2/2) / (μ * ( mv2/r + mg ))
This can be quite cleanly simplified into something that looks similar to the answer I am supposed to reach:
s = r/2μ * ( v2/ (gr + v2) )
But this is where I run into a brick wall. I can't figure out how to integrate it properly, all the similar equations I have seen integrated seem to include arctan somewhere in the answer.
I apologise any mistakes I made in the translation of the problem and in laying them out clearly, and thank you for your attention!