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Homework Help: Frequency of small oscillations about equilibrium point.

  1. May 14, 2009 #1
    A particle of mass m moves in one dimension subject to the potential:
    Find the equilibrium point and the frequency of small oscillations about that point.

    I think I've found the equilibrium point 'a', but using the formula V'(a)=0, and i got the answer a=1.

    However, im completely stuck on finding the frequency, I've found the Lagrangian and hence the equation of motion, but then dont know what to do.

    any help much appreciated!! :)
  2. jcsd
  3. May 14, 2009 #2
    Ok, you don't need the equations of motion to find the frequency of small oscillations.

    Here's what you need to do:

    At x=1, there's a stable equilibrium. For very small oscillations, it will approximate a SHO. So all you need to do is fit a parabola to that point by:

    1. Putting the equilibrium point of the parabola at the (x=1, y=???) point.
    2. Making the first derivative of the parabola equal the derivative at the point (of course, this has already been done for you by using a parabola).
    3. Making the second derivative match the second derivative of your potential function there.

    So what's the equation of a parabola?

    y = ax^2 + bx + c
    y' = 2ax + b
    y'' = 2a

    So all you need to do is to make sure that y(1) = V(1), y'(1) = V'(1), and y''(1) = V''(1).

    Then simply realize that F = -kx by Hooke's law, and that F = -y'. So whatever the coefficient on the 'x' term in your y' happens to be, that equals k. From that and the mass m, it should a straightforward exercise to find everything else you could want.
  4. May 14, 2009 #3
    ive managed to work out what you've said but i dont understand why?
  5. May 14, 2009 #4
    i dont understand why you need to fit the parabola at that point, and i dont really get how you do it, do you equate the V's and y's to find the values of a, b and c? and how does this help?
    and i dont see how to use the info to find the oscillations :(
  6. May 14, 2009 #5
    forget it!! ive figured it out! temporary blank.

    thanks very much for your help.
  7. May 14, 2009 #6
    The idea is that for small displacements from a stable equilibrium, all oscillations look like simple harmonic oscillations.

    Simple harmonic oscillations are described by a parabolic potential well.

    So by finding the parabola which best approximates your potential function V at the point of stable equilibrium, you will find the SHO potential which best approximates the oscillatory behavior near the equilibrium.

    To find the best-fit parabola at your point, you have to make it match your V in its zeroth, first, and second derivatives; you have to find the 2nd order Taylor polynomial for the potential function V expanded around the point of equilibrium.

    The system of equations I suggested fits the parabola.

    Once you have the parabola for the SHO, it's easy to use Hooke's law to find k, and once you have k and m, you can easily find everything else.
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