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Catenary problem

  1. Jul 29, 2017 #1
    1. The problem statement, all variables and given/known data

    Suppose we have a rope of length L and total mass M. Suppose we x its ends at points
    (xA; yA) and (xB; yB). We want to determine the shape the rope makes, hanging under the
    influence of gravity. The rope is motionless, with a shape parametrised by y(x) or equivalently,
    x(y), where x denotes the horizontal coordinate and y the vertical one. We are looking for the
    shape which minimises the potential energy of the rope.


    Image below

    gU615OS.png

    2. Relevant equations

    I'm guessing

    ds = sqrt ( dx^2 + dy^2) can be used.

    3. The attempt at a solution

    Integrate ds over s, and thus it is...

    integral ds = S [ from Yb to Ya]

    Xb and Xa would be zero as the horizontal length does not change.

    As you can see...I'm a bit confused. I don't know how to parametise dx and dy, or can I just use a polar coordinate system?
     
  2. jcsd
  3. Jul 30, 2017 #2

    Orodruin

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    This just gives you the length of the rope, which you know and should impose as a constraint. You need to find an integral that describes the potential energy and minimize it under that constraint.
     
  4. Jul 31, 2017 #3
    The integral for the potential energy was given as eq(1) of the problem statement. All that is necessary is to minimize that. This is a well known problem, written up in countless places in the literature.

    Your guess,

    was already incorporated in writing eq(1).
     
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