1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lagrange equations of a spinning parabola

  1. Apr 16, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity ω about its vertical axis. Use cylindrical polar coordinates and let the equation of the parabola be ##z = kρ^{2}##. Write down the lagrangian in terms of ρ as the generalized coordinate. Find the equation of motion.

    2. Relevant equations

    $$L = T-U = \frac{1}{2}mv^2-mgy$$

    3. The attempt at a solution

    I just need a hint on how to set up the kinetic energy of the parabola, maybe someone can explain it in a different way that would push me in the right direction.

    Also, sorry if this isn't considered advanced physics. This is a classical mechanics class (upper level at my school) and it seems like a gray area for me.
    Last edited by a moderator: Apr 16, 2013
  2. jcsd
  3. Apr 16, 2013 #2
    you need to change your coordinates to suit the problem better
  4. Apr 16, 2013 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    A straightforward method is to write down expressions for x(t), y(t), and z(t) and then differentiate them with respect to time. For example, ##x(t) = \rho\cos \phi##, so
    $$\dot{x}(t) = \dot{\rho}\cos \phi - \rho\sin\phi \,\dot{\phi}.$$ When you have all three, plug them into ##v^2 = \dot{x}(t)^2 + \dot{y}(t)^2 + \dot{z}(t)^2##.

    By the way, I think the potential term in the Lagrangian should be mgz, not mgy.

    It's in the right place.
  5. Apr 16, 2013 #4
    Ah, I suppose I should add in the major fact that they want it in terms of the generalized coordinate rho. I can try and find/make a picture of the graph if it helps.

    and yes, you are correct, the potential would be mgz.
  6. Apr 16, 2013 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    The expression you end up with will be in terms of ##\rho##.
  7. Apr 16, 2013 #6
    Everything worked out properly! Thanks a bunch for your help.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted