Results 1 to 4 of 4

Math Help - oscillation of a ring on parabola

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    oscillation of a ring on parabola

    Attached Thumbnails Attached Thumbnails oscillation of a ring on parabola-12345.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2008
    Posts
    321

    no idea with (ii)

    no idea with ii
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by szpengchao View Post
    one ring is hang on a parabola with equation y= x^2/4a
    ignore friction. the ring is left at a height h from rest.

    find its period of oscilation.
    Start by getting the equations of motion using Lagranges equation. Here is a reference that saves me the trouble of doing it: (Example 3.2)

    Schaum's Outline of Theory and ... - Google Book Search
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,854
    Thanks
    321
    Awards
    1
    Quote Originally Posted by szpengchao View Post
    one ring is hang on a parabola with equation y= x^2/4a
    ignore friction. the ring is left at a height h from rest.

    find its period of oscilation.
    Or you could attack the thing head-on.

    Newton's Law in the two coordinate directions says
    m \frac{d^2x}{dt^2} = -\frac{x}{\sqrt{x^2 + a^2}}

    m \frac{d^2y}{dt^2} = -mg - \frac{a}{\sqrt{x^2 + a^2}}

    That system looks horrible to solve, but the thing is that I can't come up with a good generalized coordinate for this problem. The best I can offer is to put an angular variable at the focus of the parabola.

    However, things aren't all bleak. All we need is the coefficients of the expansion:
    \frac{d^2x}{dt^2} = A + Bx + Cx^2 + ~...

    Presumably if h is small then A = 0, the coefficients of the powers of x higher than 1 are small, and B is negative. In that case
    \frac{d^2x}{dt^2} \approx Bx
    which is the harmonic oscillator equation. I leave it to you to check that these conditions come out. (I haven't checked.)

    -Dan

    Edit: After seeing the original posting of this problem it is clear that the problem is requiring an exact, not approximate solution. The only way I can think of to get this is to use the Lagrangian technique. But again, I can't think of a good generalized coordinate here.
    Last edited by topsquark; May 11th 2008 at 07:37 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. oscillation behaviour
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: December 27th 2010, 02:13 PM
  2. Oscillation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: June 3rd 2010, 02:03 PM
  3. oscillation
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: May 30th 2010, 12:31 PM
  4. Oscillation(What is it?)
    Posted in the Calculus Forum
    Replies: 0
    Last Post: May 24th 2010, 10:11 AM
  5. pde a chain oscillation
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: March 16th 2010, 05:31 AM

Search Tags


/mathhelpforum @mathhelpforum