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Constrained Lagrangian Problem

  1. Apr 5, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose we have a skateboard half-pipe. The half-pipe has a radius R. We take a bicycle wheel of radius ρ and let it roll into the half-pipe. The wheel rolls without slipping ( and doesn't fall over, of course). Let Θ be the angle from the vertical to the line connecting the center of the half-pipe to the center of the bicycle wheel, and ø the angle the bicycle wheel has rotated.

    (a) how are Θ,ø, R and ρ connected if the wheel rotates without slipping?

    (b) Find the Lagrangian for the bicycle wheel. Use your answer from the previous part to simplify the Lagrangian.


    2. Relevant equations

    L= T-U
    where L is the Lagrangian, T is the kinetic engery, and U is the potential energy



    3. The attempt at a solution

    I am very familiar with how to evaluate and set up lagrangians. However i am having trouble figuring out how to relate the variables for part a. I know the potential will be in the form of U=mgh and how to take partials to get the equations of motion.

    It may be that i am confused about the angle Θ and where it is defined. something just isn't clicking. Any point in the right direction would be greatly appreciated. mostly i just need another mind to help me decipher how to do this. Once i get the relationships i should be good to continue on.


    Thanks so much for all the help,

    Eric
     
  2. jcsd
  3. Apr 6, 2014 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    Hi Eric, welcome to PF!

    Imagine that a line is painted across the wheel, and it is vertical when the wheel is at the bottom of the tube. φ is the angle that line encloses with the vertical as shown in the figure, and θ is the angle showing the position of the centre of the wheel with respect to the vertical.

    The wheel spins around its axis, and the centre of the wheel moves along a circle around the centre of the tube.

    ehild
     

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