Let's pretend I am given a potential energy function and nothing else. I need to find the effective spring constant for oscillation about the equilibrium point using a taylor series expansion. I can't find an example or explanation anywhere on how to do this. the potential energy function I have has a stable equilibrium point.
forgive me for not posting the exact problem. I am hoping someone can walk me through an example or point me to one, that way I can learn how to do it, then apply that to my homework rather than just doing the homework problem first.
thankyou in advance for the trouble.
taylor series expansion is
and it goes on continually to the desired precision
as for the potential energy function for this system, I am hoping that one with a stable equilibrium can be invented just for the purpose of explaining to me how to do it. Then I can attempt to apply that to my homework problem. if not I guess I can change the function I am given by setting all the constants equal to 1, but I'd really prefer to have to think about the problem rather than just plugging and chugging it into an algorithm created by an explanation. If you understand what i mean. but if I must, I will.
The Attempt at a Solution
1. I took the first derivative of my potential energy function.
2. I set it equal to 0 and solved for my independent variable. This gave me the equilibrium point (and the answer to part a)
3. I plugged it into a taylor series expansion, but I am uncomfortable with this because it makes no sense as to why that would give me the effective spring constant. Spring constant is Force / mass. I feel like there is probably an equation that I am supposed to plug the equilibrium value into that will give me my spring constant. and then I plug that into the taylor series.
Please advise me, and I appreciate the advice.