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Yet another Lagrangian problem. Motion in a cone

  1. Nov 16, 2011 #1
    Man I hate to make two post in one day but I am really stuck!

    1. The problem statement, all variables and given/known data
    A particle slides on the inside surface if a frictionless cone. The cone is fixed with its tip on the ground and its axis vertical. The half angle of the tip is α. Let r be the distance from the particle to the axis, and let θ be the angle around the cone.

    1. Find the equations of motion.
    2. If the particle moves in a circle of radius r_0, what is the frequenct of this motion ω?
    3. If the particle is perturbed slightly from the circular motion, what is the frequency of oscillations about the radius r_0?

    2. Relevant equations
    Lagrangian and Lagranges Equations

    3. The attempt at a solution
    If need be, I will draw a picture and upload it so the relations used are more obvious.
    First thing is first, is my Lagrangian correct? (I made an image in MathType, its easier for me.)
    [PLAIN]http://img197.imageshack.us/img197/9997/lag1.gif [Broken]

    I do not think this is correct because I am not getting oscillatory motion for theta when I solve!

    EDIT: Oh! I forgot to mention! 'l' is the distance on the side of the cone from the point to where the particle is. I do not know how to include this in the lagrangian!
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Nov 16, 2011 #2
    Okay so I realized I didn't include [itex]\dot{z}[/itex]! After including that, I have...
    [itex]L=\frac{1}{2}m \left( \frac{\dot{r}^2}{sin^2σ}+r^2 \dot{θ}^2 \right)-\frac{mgr}{tanα}[/itex]
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