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A particle on the edge of an inclined spinning disc

  1. Dec 27, 2015 #1
    1. The problem statement, all variables and given/known data
    A particle is fixed to the edge of a disc of negligible mass making an angle θ0 with the ground in a uniform gravitational field, and is free to rotate about the center of the disc. I need to find the equations of motion.

    2. Relevant equations
    The definition of the Lagrangian in spherical coordinates,

    ##\frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2 + r^2sin^2{\theta}\dot{\phi}^2) - U(r, \phi, \theta)##

    where ##\theta## is given to be the angle of altitude and ##\phi## is the azimuthal angle.

    3. The attempt at a solution

    It was easy to find the Lagrangian to be

    ##L = \frac{R^2}{2}m(\dot{\theta}^2 + sin^2{\theta}\dot{\phi}^2) + mgrcos{\theta}##

    It is from here that I do not know how to proceed. Wouldn't using that Lagrangian to find the equations of motion for ##\phi## and ##\theta## just get me the equations of motion for a particle constrained to move?

    The first thing I've thought to try is to treat the particle as though it is moving on the surface of a sphere, fixed to move along the great circle produced by the plane passing through the center of the sphere and making an angle θ0 with its equator, then I could use the equation of the great circle as a constraint relating ##\phi## and ##\theta##, but I'm struggling to find the parametric equations for a great circle.

    The other thing I tried was to use cylindrical coordinates and impose the constraint that the height ##z## is some function of ##\phi##. My guess is that this function would be something of the form ##z = Rcos{\theta_0}cos{\phi}##, then use ## f = z - Rcos{\theta_0}cos{\phi} = 0## as my equation of constraint. This gets me the equations

    ##\ddot{\phi} + \frac{\lambda}{R}cos{\theta_0}sin{\phi} = 0##
    ##\ddot{z} - g + \frac{\lambda}{m} = 0##

    It is from there that I do not know how to proceed.
     
  2. jcsd
  3. Dec 27, 2015 #2
    How you write the Lagrangian for horizontal disk on external field by angle ##\theta_0##?
    This Lagrangian produce the same equation of motion? Must be?
     
  4. Dec 27, 2015 #3

    haruspex

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    Are you taking the z axis as vertical? Doesn't that make phi and its derivatives a bit tricky?
    Have you tried using cylindrical coordinates but taking the z axis perpendicular to the disk?
     
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