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(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Calculate the potential

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**