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24-Ruby IV
ValeryOchkov24-Ruby IV
24-Ruby IV
April 4, 2022
Solved

Can we solve it symbolical?

  • April 4, 2022
  • 1 reply
  • 2226 views

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

F-Chain-Min.png

F-Chain-Min-Solution.png

Best answer by AlanStevens

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.

 

1 reply

A
AlanStevens19-Tanzanite
19-Tanzanite
April 9, 2022

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

 

g0.jpg

See the attached for the derivation.

 

Alan

 

    V
    24-Ruby IV
    ValeryOchkov24-Ruby IVAuthor
    24-Ruby IV
    April 9, 2022

    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?

     

    A
    AlanStevens19-TanzaniteAnswer
    19-Tanzanite
    April 10, 2022

    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.

     

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