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Introductory Physics Homework Help
Friction on a casing on a metal ring
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[QUOTE="Victarion, post: 4514395, member: 488987"] [h2]Homework Statement [/h2] A casing of mass m can glide along a thin metal ring lying horizontally. It has an initial speed of v[SUB]0[/SUB]. 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[SUB]0[/SUB][SUP]2[/SUP] + √(v[SUB]0[/SUB][SUP]2[/SUP] + g[SUP]2[/SUP]r[SUP]2[/SUP])/ g * r ) The task is to provide the deductive path to reach that answer. [h2]Homework Equations[/h2] Centripetal force: mv[SUP]2[/SUP]/r Gravitational force: mg Kinetic energy: mv[SUP]2[/SUP]/2 [h2]The Attempt at a Solution[/h2] 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[SUP]2[/SUP]/r + mg ) Since Wf is equal to the kinetic energy we can write the following equation: s = Kinetic energy / F = (mv[SUP]2[/SUP]/2) / (μ * ( mv[SUP]2[/SUP]/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[SUP]2[/sup]/ (gr + v[SUP]2[/sup]) ) 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! [/QUOTE]
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Friction on a casing on a metal ring
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