Capstan

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We do not need a separate artice: the proof of the capstan equation can be incorporated into capstan equation. — RHaworth (Talk | contribs) 15:44, 30 January 2009 (UTC)Reply

Sorry about reverting you - I was trying to remove the example images and didn't mean to remove all your edits.--Kubigula (talk) 03:35, 31 January 2009 (UTC)Reply

The capstan equation derivation seem needlessly complex. Why not do it this way:

     Derivation of the Capstan Equation

The first step is to relate the radial or normal force F (Newtons/radian) of a rope wrapped around a capstan to the tension, T (Newtons), in the rope. This is classically done with a force balance method using vectors, trigonometry, and a little calculus. The answer can be found more simply with an energy argument as follows: Assume that the rope is inelastic (nonstretching) and the capstan is frictionless and compressible. Then the energy required to squeeze down the capstan radius R by an amount ∆R with one turn of rope equals the energy supplied by pulling the rope.

     		 2 π ∆R F = 2 π ∆R T

Cancel 2 π ∆R on each side leaving F = T

Let the coefficient of friction between the rope and capstan be k = friction force/radial force. So the frictional force over a wrap angle d phi is

k F d phi = k T d phi

The increase in tension dT over an angle d phi is the frictional force over that angle so

     	 	dT = k T d phi

Divide both sides by T gives

dT / T = k d phi

Integrating both sides gives

Ln T = k phi + C

Exponentiate both sides to give

T = exp(k phi + C) = exp( C) exp( k phi)

For phi = 0 T = Tin so

Tout = Tin exp(k phi) QED EE483597 (talk) 00:08, 2 March 2015 (UTC)Reply

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