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A simple question about coordinates

  1. May 3, 2008 #1
    1. The problem statement, all variables and given/known data

    a bead is constrained to move on a wire: y=acosh(x/a).
    write the kinetic energy and the potential energy using the coordinates:
    1) x
    2) l - the length of a curve

    2. The attempt at a solution

    1) is simple:
    the kinetic energy in cartesian coordinates is 0.5([tex]\dot{x}[/tex]^2+[tex]\dot{y}[/tex]^2)
    and using y=acosh(x/a) I get the following expression
    0.5m[tex]\dot{x}[/tex]^2(cosh(x/a)^2
    the potential energy is mgh = mgy=mgcosh(x/a).
    2) l=integral(dl)=integral[0-->x]{sqrt[(1+(dy/dx)^2)]dx}=asinh(x/a).
    so the kinetic energy is 0.5m[tex]\dot{l}[/tex]^2=0.5m[tex]\dot{x}[/tex]^2(cosh(x/a)^2
    but what is the expression for the potential energy here? how do I express h?
     
  2. jcsd
  3. May 3, 2008 #2

    George Jones

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    Staff Emeritus
    Science Advisor
    Gold Member

    cosh^2(x/a) - sinh^2(x/a) = 1
     
  4. May 3, 2008 #3
    got it, thanks.
     
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