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Homework Help: Legrangian Eqn. of Motion

  1. Sep 1, 2007 #1
    Two blocks, each of mass M, are connected by an extensionless, uniform string of length l. One block is placed on a smooth horizontal surface, and the other block hangs over the side, the string passing over a frictionless pulley. Describe the motion of the system when the string has a mass m.

    By Hamilton's principle
    [tex]L = T - U[/tex]

    the kinetic energies will be
    [tex]T = 1/2 m \dot{x}^2 + 1/2 m \dot{y}^2[/tex]

    and if the potential is defined to be zero at the horizontal, the potential will be
    [tex]U = -Mgy + U_{string}[/tex]

    This is the part I need a quick help on. The x block has a zero potential because it stays along the horizontal where the zero potential is defined, and the hanging block will have a potential of -Mgy, and I know that the mass of the string contributing to the potential will increase until finally it reaches as the string moves down. So I was thinking that

    [tex]U_{string} = -\frac{m}{t}*g*y[/tex]

    That gives the mass per unit time for a given length y, which would also be

    [tex]U_{string} = -m g \dot{y}[/tex]

    But units don't work out correctly unless I divide U_string by t, which would create a discontinuity and not make any sense. I don't know why I am having so much trouble with such a simple prospect.
  2. jcsd
  3. Sep 2, 2007 #2
    Why did you use (m/t)? You should find the center of mass of the part of the string which hangs below the table. If the total mass is m, then the hanging part is
    and mass center is in the middle (y/2). This information should give you the potential energy of the string.
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