# Homework Help: Oscillator (not shm)

1. Feb 28, 2010

### iloveannaw

1. The problem statement, all variables and given/known data
For a certain oscillator the net force on the body with mass {m} is given by F_x = - cx^3.
a) What is the potential energy function for this oscillator if we take U = 0 at x = 0?
b) One-quarter of a period is the time for the body to move from x = 0 to x = A. Calculate this time and hence the period.

2. Relevant equations
a) $$U = \int F dx$$
b) $$E = \frac{1}{2}(mv^{2} + kx^{2})$$

$$T = \int\frac{dx}{v(x)}$$

3. The attempt at a solution

ok, the first part is pretty straight forward. just integrating w.r.t x yields $$\frac{1}{4}cx^{4}$$

$$E = constant = \frac{1}{4}cA^{4} = \frac{1}{2}mv^{2} + \frac{1}{4}cx^{4}$$

solve to get v as function of x: $$v(x) = \sqrt{\frac{c}{2m}(A^{4}- x^{4})}$$

then integrating of interval x = 0 to x = A:

$$T = 4 \int\frac{dx}{v(x)}$$

$$= 8A^{2}\sqrt{\frac{2m}{c}}$$