Calculate the potential

stunner5000pt

the trajectory of a particle moving in a cnetral potential V(r) is given by [itex] r = a + b \sin(\eta \phi) [/itex] where a b and eta are constants

Compute the potential V(r) in which the particel moves (for arbitrary a, b, and eta) and sketch [itex] V_{eff} (r) = V(r) + \frac{L^2}{2mr^2} [/itex] for cases a =b

sketching is not hte problem here.. i just need to find out V(r)
well i know that i need to compute the force f first and then integrate f w.r.t. r to get V(r)
find find [tex] u = \frac{1}{r} = \frac{1}{a + b \sin(\eta \phi)} [/tex]
then using [tex] f = -\frac{L^2 u^2}{m} (u'' + u) [/tex]
and [tex] u '' = \frac{2b^2 \eta^2 \cos^2 (\eta \phi)}{r^3} + \frac{b \eta^2 \sin(\eta \phi)}{r^2} [/tex] (phew!)

once i sub expressions for u'' and u i get
[tex] f(r) = - \frac{L^2}{mr^2} \left( \frac{2b^2 \eta^2 \cos^2 (\eta \phi)}{r^3} + \frac{b \eta^2 \sin(\eta \phi)}{r^2} \right)+ \frac{1}{r} [/tex]
is that fine? Can i simplify that any more in terms of r?

are the steps correct? Any problems with the derivative? Just need to know if i can simplify any further.
Your help is greatly appreciated!

Thank you