Approximate spring potential energy U(x) for small oscillations

[StationaryPanic]

Homework Statement

"Take a PE function U(x), which has an equilibrium point at x=0, and provides a restoring force in that region, and show that a Taylor expansion around that area can be approximated by a SHO PE function for small x."

Homework Equations

U=.5kx^2...x = (A^2)*sin^2(wo(t)-delta)
also typical taylor series expansion...MacLaurin expansion for x=0 i think?

The Attempt at a Solution

I've made several attempts...I don't think i have a general concept of how to proceed with this problem. Expansion of .5kx^-2 already ended up 0 + k*0 + k...integrating this k that remains twice gives kx^2 which aside from missing the .5 term seems to be validating our goal of the problem: showing that force and potential approximations both lead to the same basic form at small amplitudes. I just don't think I have any concept of how to expand a function using a series, which is sad considering I have already passed Mechanics II and am retaking for GPA reasons and shooting myself in the foot somewhat. Cheers

 

[JorisL]

So you have a general potential $U(x)$.
What is the Taylor series for such a function (the sum-expression)?

Next you use what's given, equilibrium means that the first derivative is 0.
Remember, you look at a small region around $x=0$. What can you say about $x^2$ compared to $x^4$?

Try to understand this assignment really well, it is common all around physics.
At least to get a feel for your system, this can help an awful lot.