# Homework Help: Small Oscillations on a Parabola problem.

1. Nov 23, 2009

### jameson2

1. The problem statement, all variables and given/known data
Find the frequency of oscillations of a particle (mass m) which is free to move along the parabola y= -ax^2 + 2ax - a, and is attached to an ideal spring whose other end is fixed at (1,l) A force F is required to extend the spring to length l. a can be any real number.

2. Relevant equations
Lagrangian eqation, which can be used to get equation of motion.
Potential energy U=0.5kx^2
Stable equilibrium is at a point where the first derivative of U is 0 and second derivative is positive.

3. The attempt at a solution

First I need to get an equation for the potential energy to complete the lagrangian:
this requires the extension of the spring, which is what I'm having trouble with. Letting r be the extension: I tried just using Pythagoras' theorem and got the following
(l+r)^2=(x-1)^2 + (l-y)^2 and then neglecting the r^2 part due to the oscillations being small, Iihave this :
r=(1/2l)(x^2 -2x +1 -2ly +y^2)
then I think I need this in terms of x and not y, so I filled in the equation of the parabola for y, to get r in terms of x, and hence potential energy in terms of x.
However this gives me a really messy equation that I'm not going to bother typing out as it's clearly wrong. I don't see where I can change this aproach to soving the problem though, any ideas?