 #1
 420
 14
 Homework Statement:
 A particle with angular momentum `describes the orbit ##r = a (1 + \cos\theta)##. Find the central force that produces this orbit. Find the period of the orbit. Determine the minimum energy that the particle must have to escape from this orbit.
 Relevant Equations:

##\frac{l^2}{m}\left(\frac{d^2u}{d\theta^2}+u\right)=\frac{\partial V}{\partial u}##
##t=\sqrt{\frac{m}{2}}\int_{r_0}^{r}\frac{dr'}{\sqrt{EV_{eff}}}##
I've already found the potential and force that produce the given orbit. my results were:
##V=\frac{al^2}{mr^3}##
##\vec{F}=\frac{3al^2}{mr^4}\hat{r}##
Now, I've been trying to find the period using the equation##\vec{F}=\frac{3al^2}{mr^4}\hat{r}##
##t=\sqrt{\frac{m}{2}}\int_{r_0}^{r}\frac{dr'}{\sqrt{EV_{eff}}}##
Using ##r_0=r_{min}=a## and ##r=r_{max}=2a##, and multiplying by 2 to make a full orbit, I end up with the integral##T=2\sqrt{\frac{m}{2}}\int_{a}^{2a}\frac{dr}{\sqrt{E+\frac{al^2}{mr^3}\frac{l^2}{2mr^2}}}##
The problem is that I have no idea on how to integrate this. Is there a trick or substitution to integrate it, or am I wrong somewhere?