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Lagrangian problem

  1. Mar 11, 2007 #1
    1. The problem statement, all variables and given/known data

    Mass m is connected to a string and is being whirled in a circle in a horizontal plane of a table. The string passes through a hole in the center of the circle and is being pulled with a constant velocity V starting at t=0 so the radius decreases. Initially the mass is at a distance r0 from the hole and is revolving with angular speed [tex]\omega_0[/tex]. Use Lagranges equations and Lagrange multipliers to find then tension in the string as a function of time.

    2. Relevant equations

    I have attempted to write down the Lagrangian with the lagrange multipliers but i am not sure it is correct.
    what i have is that the speed of the mass is given by:
    [tex]v=r\omega = r\dot{\theta}[/tex] and i have that r = r0 - Vt so [tex]v=(r_0-Vt)\dot{\theta}[/tex]
    Using this in 1/2mv^2 for the Lagrangian along with the constraint r-r0+Vt=0 my Lagrangian with lagrange multiplier was this:
    [tex]\frac{m}{2}(r_0^2-2r_0Vt + V^2t^2)\dot{\theta}^2 + \lambda(r-r_0+Vt)=L[/tex]
    and when doing the derivative [tex]\frac{\partial{L}}{\partial{\dot{r}}}[/tex] i have this is the partial with respect to V. is this correct? then i would solve for lambda and this would be my force of tension?
    Last edited: Mar 11, 2007
  2. jcsd
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