Can we solve it symbolical?

[ValeryOchkov]

There is a chain on the wall. What should be its length S so that the force F is minimal.
Below is one of the numerical solutions of the problem.
Is it possible to find an analytical solution. Not 1.258 but one expression!
Attach file Mathcad 15

[AlanStevens]

Here's your symbolic solution Val (though I've used a different notation from your original):

See the attached for the derivation.

Alan

[ValeryOchkov]

Thanks, Alan!

If the equation coth(k) = k has not the symbolical solution than our problem about optimal chain has not the symbolical solution.

True?

[AlanStevens · Accepted solution]

The equation I supplied is about as far as one can reasonably go symbolically.  The equation tanh(k) = 1/k doesn't have a closed form analytical solution as far as I know, so further calculations must be done numerically.

[ValeryOchkov]

Is this one kay for the solving our problem?