Period of small oscillations in central potential

[Chen]

Hi,

A particle is subjected to a central potential of:

[tex]V(r) = -k\frac{e^{-\alpha r}}{r}[/tex]
Where [tex]k, \alpha[/tex] are known, positive constants.

If we make this problem one-dimensional, the effective potential of the particle is given by:
[tex]V_{eff}(r) = -k\frac{e^{-\alpha r}}{r} + \frac{l^2}{2 m r^2}[/tex]
Where the second term is the "centrifugal potential", [tex]l[/tex] is the absolute value of the angular momentum the particle has.

Now suppose that this effective potential has a minimum at [tex]r_0[/tex], which is known, so that if placed there the particle will have a circular motion.

The question is - what is the period of small oscillations (in the r-dimension) around the circular orbit?
The answer needn't depend on the energy of the particle or its angular momentum.

Thanks,
Chen

 

[quantumdude]

You know the rules: Post what you've got so far, and we'll help from there.

 

[Chen]

I've solved it myself. Thank you very very much.