- #1

Davidllerenav

- 424

- 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{E-V_{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{E-V_{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?