Finding the frequency of very small oscillations

[Bjarni]

Homework Statement

Find the frequency of very small oscillations around the equilibrium distance

Relevant Equations

U(r) = E_0[(a/r)^12- (a/r)^6]
r_equilibrium = 2^(1/6)*a

So I'm working on this home assignment that has numerous segments. Firstly, I was asked to find the equilibrium distance between two particles in a potential well described by U(r).

I did that by setting U'(r) = 0 and came out with r_equilibrium = 2^(1/6)*a.

Now, I'm being asked to find the frequency of very small oscillations around r_equilibrium and I'm honestly lost. I think I only need a small push in the general direction of the solution because as of now I don't really know where to start.

Thanks in advance.

 

[mfb]

For very small oscillations you can approximate the potential by a parabola.

 

[Bjarni]

Thanks.

I used a Taylor-expansion and set F = -k(r-r_0) = -dU/dr.

Got k = 9*2^(2/3)*E_0/a^2

and since ω = sqrt(k/μ), I ended up with ω = (3*2^(1/3)/a)*sqrt(E_0/μ) which I feel pretty good about.

Sorry for my lack of LaTex skills..