Angular frequency for Potential Energy Function

[rogeralms]

Homework Statement

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?

Homework Equations

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

F= -kx = -(gradient) U

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.

 

[mfb]

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.