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Catenary spanned

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data
    A cable is spanned between two points at the left touching the ground and at the right smoothly spanned over a large roller. Assume that there's no friction present. The tension left at the bottom is H and the tension T at the top is such that T>H. The cable has a weight per unit length q and is spanned over an height h and is in static equibrilium.

    Prove that:

    T = H +q*h

    http://img220.imageshack.us/img220/537/96775335.png [Broken]



    2. Relevant equations



    3. The attempt at a solution
    I thought that the roller would give reaction forces perpendicular to the cable but other students told me that it works like a pulley and since there's the assumption that there's no friction I can disregard these forces, right? But I still don't know how to prove that formula.
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Mar 21, 2012 #2
    Hint:

    Calculus and Analytic Geometry by Thomas.....
     
  4. Mar 22, 2012 #3
    Can you guide my in the right direction with the info of the book? Or give me the right chapter of the book?
     
    Last edited: Mar 22, 2012
  5. Mar 22, 2012 #4
    Chapter 10, third edition. The title of the chapter is Hyperbolic Functions.
     
  6. Mar 23, 2012 #5
    Got it but it probably doesn't work if it's a beam instead of a cable, right? What happens with the expression if it is a beam?
     
  7. Mar 23, 2012 #6
    How do you "smoothly span" a beam over a roller? A beam is a different animal.
     
  8. Mar 27, 2012 #7
    I mean no friction suppose that at the ends there's no shear force or bending moment (or at least negliglible) is the formula still valid?
     
  9. Mar 27, 2012 #8
    The reason why the cable pulls on both ends is because it cannot support itself. It has no resistance to bending stress - actually a cable does to a slight extent. If a beam could be bent over the frictionless pulleys such that it is horizontal where it touches them, it would remain there if undisturbed.

    Think of a wheelbarrow with a load in it and a good set of bearings on its wheel. If you raise it, it does not want to run away from you. Yet, it is tilted upward.
     
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