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Block sliding down a moving plane whose angle of inclination changes

  1. Oct 11, 2013 #1
    1. The problem statement, all variables and given/known data

    A bar of soap of mass m is at rest on a frictionless square plate of side-length D rests on a horizontal table. At t=0 I start to raise one edge of the plate the plate pivots about the opposite edge. The angular velocity about the pivot is a constant ω Show that the equation of motion for the soap is of the form [itex]\ddot{x} - \omega ^{2} x = - g sin( \omega t )[/itex]. Solve this for x(t) given that . In this equation x is the distance from the pivot to the soap (i.e. measured up the slope).

    2. Relevant equations



    3. The attempt at a solution

    We will find that the potential energy is [itex]V = mgx sin( \theta )[/itex], and the kinetic energy is just [itex]T = \frac{1}{2} m \ddot{x}^{2}[/itex]. Then our Lagrangian is just [itex]L = \frac{1}{2} m \ddot{x}^{2} - mgx sin( \theta )[/itex] and since we know omega is constant, then θ = ωt, so our Lagrangian becomes [itex]L = \frac{1}{2} m \ddot{x}^{2} - mgx sin( \omega t)[/itex].

    But using the Euler-Lagrange equation yields [itex]\ddot{x} = -g sin( \theta )[/itex].

    Where did the [itex] - \omega ^{2} x[/itex] come from? i want to say that it has something to do with the fact that the angle is changing over time...but i can't see it right now. any help would be great. thanks in advance.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 11, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    You forgot the kinetic energy from motion perpendicular to the direction of x. As you know the motion of the slope, you can relate this to x and t.
     
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