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Angular frequency for Potential Energy Function

  1. Mar 8, 2014 #1
    1. The problem statement, all variables and given/known data

    A mass moves along the x axis with potential energy
    U(x)= - U0 a^2 / (a^2 + x^2). What is the angular frequency assuming the oscillation is small enough to be harmonic?



    2. Relevant equations

    w^2 = k/m with w as the angular frequency

    F= -kx = -(gradient) U



    3. The attempt at a solution

    Since this is one-dimensional we take the derivative of U with respect to x.

    I get -(gradient) U = -2 U0 a^2 x / (a^2 + x^2)^2

    Therefore k= 2 U0 a^2 / (a^2 + x^2)^2

    The correct answer does not have an x term in it.

    w (omega) = k/m = (2 U0 / m a^2) ^ (1/2)

    Is there a binomial expansion that would essentially eliminate the x term in the denominator?

    Thanks for any help.
     
  2. jcsd
  3. Mar 9, 2014 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    The oscillation happens around x=0, so you can approximate the gradient there and neglect the x^2. That is exactly the approximation required by the problem statement.
     
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