# Homework Help: Find force lawa from orbit equation

1. Oct 13, 2009

### KBriggs

1. The problem statement, all variables and given/known data
The orbit of a particle moving on a central field is a circle passing through the origin, namely, $$r = r_0cos(\theta)$$. Show that the force law is inverse fifth power.

2. Relevant equations

$$\frac{d^2u}{d\theta^2} + u = \frac{-mF(u^{-1})}{L^2u^2}$$
$$u=r^{-1}$$

3. The attempt at a solution

I keep getting that it is inverse third power....

$$u = \frac{1}{r_0cos(\theta)}$$
then
$$\frac{d^2u}{d\theta^2} = u + \frac{tan^2(\theta)}{u}$$

so

$$F(u^{-1}) = \frac{-1}{m}\left(L^2u^2\left(u + \frac{tan^2(\theta)}{u}\right) + L^2u^2\right)$$

$$=\frac{-2L^2}{m}\left(u^3+utan(\theta)\right)$$

so $$f(r) = \frac{-2L^2}{m}\left(\frac{1}{r^3}+\frac{tan(\theta)}{r}\right)$$

Where am I going wrong?

Last edited: Oct 13, 2009