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Homework Statement
I know how to do this problem, I'm just having trouble actually doing it.
A particle moves in a force field described by,
[tex]F(r)=-k(ar+1)\frac{e^{-ar}}{r^2}[/tex]
1. Obtain the condition for a circular orbit of radius r0
2. Apply a perturbation to the circular orbit and find the condition between a and r0 for the orbit to be stable
3. Obtain the period of the small oscillation about the stable circular orbit
Homework Equations
From previous parts of the problem, I obtained the equation of motion to be,
[tex] m \ddot r - \frac{l^2}{mr^3} + k(ar+1)\frac{e^{-ar}}{r^2}=0[/tex]
The Attempt at a Solution
1. The condition for a circular orbit is
[tex]f_{eff}(r_0)=0 \implies k(ar_0+1)\frac{e^{-ar_0}}{r_0^2} = \frac{l^2}{mr_0^3}[/tex]
2. The way to do part 2 is to define [itex]r=r_0+ \rho[/itex], [itex]\ddot r = \ddot \rho[/itex], plug these into the equation of motion, then get the equation into the form
[tex]\ddot \rho + \omega^2 \rho = 0[/tex]
Then the condition for the circular orbit to be stable is [itex]\omega^2>0[/itex].
In this case, however, I'm having trouble getting the equation of motion into that form:
[tex]m \ddot \rho - \frac{l^2}{m(r_0+\rho)^3}+k(a(r_0+\rho)+1)\frac{e^{-a(r_0+\rho)}}{(r_0+\rho)^2}=0[/tex]Using [itex]\frac{1}{(r_0+\rho)^n} = \frac{1}{r_0^n} \left ( 1-\frac{n \rho}{r_0} \right ) [/itex] and the condition for a circular orbit in 1,
[tex]m \ddot \rho - \frac{l^2}{mr_0^3} + \frac{3l^2 \rho}{mr_0^4} +k(ar_0+a\rho+1)\frac{e^{-a(r_0+\rho)}}{(r_0+\rho)^2}=0[/tex]
...
[tex]m \ddot \rho + \frac{3l^2 \rho}{mr_0^4} + \frac{l^2}{mr_0^3} \left [ e^{-a \rho} \left ( 1 - \frac{2 \rho}{r_0} \right ) \left ( 1 + \frac{a \rho}{(ar_0+1)} \right ) -1 \right ] =0 [/tex]And if I multiply everything out and get rid of the brackets there are exponentials with rho, as well as terms with rho^2.
Any help or suggestions is appreciated.