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## Homework Statement

A casing of mass m can glide along a thin metal ring lying horizontally. It has an initial speed of v

_{0}. 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( v

_{0}

^{2}+ √(v

_{0}

^{2}+ g

^{2}r

^{2})/ g * r )

The task is to provide the deductive path to reach that answer.

## Homework Equations

Centripetal force: mv

^{2}/r

Gravitational force: mg

Kinetic energy: mv

^{2}/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 * ( mv

^{2}/r + mg )

Since Wf is equal to the kinetic energy we can write the following equation:

s = Kinetic energy / F = (mv

^{2}/2) / (μ * ( mv

^{2}/r + mg ))

This can be quite cleanly simplified into something that looks similar to the answer I am supposed to reach:

s = r/2μ * ( v

^{2}/ (gr + v

^{2}) )

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!

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