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Lagrangian mechanics: Kinetic energy of a bead sliding along a bent wire

  1. Jan 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Determine the kinetic energy of a bead of mass m which slides along a frictionless wire bent in the shape of a parabola of equation y = x2. The wire rotates at a constant angular velocity [tex]\omega[/tex] about the y-axis.

    2. Relevant equations

    T = [tex]\frac{1}{2}[/tex]m([tex]\dot{x}^2[/tex] + [tex]\dot{y}^2[/tex] + [tex]{x}^2[/tex][tex]\omega^2[/tex])

    3. The attempt at a solution

    The above equation represents my attempt to write down the kinetic energy of the system in an appropriate coordinate system. After this I eliminated [tex]\dot{y}[/tex] in favour of [tex]\dot{x}[/tex] using y = x2 and got:

    T = [tex]\frac{1}{2}[/tex]m([tex]\dot{x}^2[/tex] + 4[tex]{x}^2[/tex][tex]\dot{x}^2[/tex] + [tex]{x}^2[/tex][tex]\omega^2[/tex])

    Does this look right to anyone? The book (study guide) I'm using was unfortunately compiled by my University and no answers are supplied to end-of-chapter problems. This problem comes out of the first chapter of my study guide and all the problems there basically involves writing down a correct expression for the Lagrangian/Kinetic Energy.

    Thanks in advance for any help.
     
  2. jcsd
  3. Jan 8, 2009 #2

    turin

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    Homework Helper

    It looks right to me. I am, of course, assuming, as you have apparently done as well, that x is the distance from the y-axis, and not simply the Cartesian x-coordinate.
     
  4. Jan 8, 2009 #3
    Hi Turin,

    Thanks a lot for the help!

    Yes, x does represent the distance from the y-axis, but I'm wondering if it wouldn't have been better to use cylindrical coordinates, z and r for y and x respectively, instead?
     
  5. Jan 8, 2009 #4

    turin

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    Homework Helper

    I didn't even do anything, but, you're welcome. :)


    Of course, those are just letters, and what we have both implicitly assumed is that these ARE, in fact, cylindrical coordinates (in disguise), in the way that you have identified. I don't know why the author decided to use those letters as opposed to the standard \rho and z, but that was the author's decision, not ours.
     
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