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Lagrangian problem: Ball oscillating in spherical bowl

  1. Mar 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider a solid sphere of radius r to be placed at the bottom of a spherical bowl radius R, after the ball is given a push it oscillates about the bottom. By using the Lagrangian approach find the period of oscillation.


    2. Relevant equations



    3. The attempt at a solution
    Ok so this is as far as i get, not sure if what im doing is correct:
    General coords (∅,x,y)

    T = 1/2*m*(xdot2+ydot2) + 1/2*I*∅dot2
    T = 1/2*m*xdot2+1/2*m*ydot2+(mr22)/5 (Since Inertia of Solid Sphere is = 2/5*m*r2)
    V = mgy

    L = 1/2*m*xdot2+1/2*m*ydot2+(mr22)/5 - mgy

    ∂L/∂∅dot = 2/5*m*r2*∅dot
    ∂L/∂∅ = 0

    ∂L/∂xdot = m*xdot
    ∂L/∂x = 0

    ∂L/∂ydot = m*ydot
    ∂L/∂y = -mg

    d(2/5*m*r2*∅dot)/dt = 0
    d(m*xdot)/dt = 0
    d(m*ydot)/dt = -mg

    Up to here im not sure whether i've been doing the right thing, and also, not quite sure how this helps me find the period of oscillation?

    EDIT: Sigh, I know my equations of motion are wrong because i should be getting something similar to the DE for an SHO. Could really use some help.
     
    Last edited: Mar 27, 2012
  2. jcsd
  3. Mar 28, 2012 #2
    There are some conditions you need to take into account: because bowl is spherical, you have a relation between x- and y-coordinates, and because the ball is rolling, there is a relation between phi dot and x dot and y dot.
     
  4. Mar 28, 2012 #3
    Ok, but did I use the correct coordinate system? Or should it be phi and theta, and once I have the constraints how do I encorporate them into the equation of motion?
     
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