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Classical mechanics

  1. Feb 14, 2009 #1
    Hello all, I'm having trouble with the following problem:

    Pb: A chain with constant density and infinite length is slipping down from the table without friction. Determine the position of the tip of the chain at time t.

    I know there are a few ways to approaching this problem, namely from newtons equations, or lagranges equations, but I am quite rusty with this, so any suggestions would help a lot, thanks.
     
  2. jcsd
  3. Feb 14, 2009 #2
    Hi,

    do you know the Euler–Lagrange equations ?
    You have to find an expression for the potential and kinetic energy,
    the difference is the Lagrange-function. Put this function in the Langrange equations
    and you get a second order diff.-equation.

    kind regards
     
  4. Feb 14, 2009 #3

    tiny-tim

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    Hi johnson12! :smile:

    Infinite length? … presumably only in one direction? :wink:

    Use conservation of energy.
     
  5. Feb 14, 2009 #4
    are you referring to this equation:

    [tex]\frac{\partial L}{\partial x_{i}}[/tex] - [tex]\frac{d}{dt}[/tex][tex]\frac{\partial L}{\partial \dot{x_{i}}}[/tex] = 0, i=1,2,3

    where L = T - U is the lagrange function,

    how can I use this to model my problem?
    (ps. im apologize if my physics is wrong, unfortunately im a math major).
     
  6. Feb 14, 2009 #5
    Yes, but we need only one variable x_1 = y for example. Now you have to find an expression
    for the kinetic energy T which is simple and for the potential energy U which is simple
    as well. U depends of course at your point of reference. Make a sketch.
     
  7. Feb 15, 2009 #6
    I get that [tex]T(\dot{x}) = \frac{1}{2}m \dot{x}^{2}[/tex]
    [tex] U(x) = mgx [/tex],

    [tex]\frac{\partial L}{\partial x}= - m g [/tex]
    [tex]\frac{d}{dt} \frac{\partial L}{\partial \dot{x}}= m\ddot{x}[/tex]
    Lagranges equation implies [tex]m\ddot{x} + mg = 0[/tex]

    but I'm a little confused, if the chain is of infinite length, would it then have infinite mass?

    so how can the chain move?
     
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