Q:

Frequency of Oscillations about Circular Orbit

I'm trying to figure out the frequency of small oscillations about the basic circular path of a mass at the end of a spring, being spun around a table. I understand that the spring will stretch out a bit, and then it will have a stable circular orbit. Moreover, introducing a little bit of extra stretching will cause it to oscillate about this orbit, but I have no idea what approach to take. I've seen things like binomial expansions and the like to calculate small changes to orbits, but I can't even seems to get a good, functional equation to describe the set up. Can someone give me a little guidance?

A:

The potential energy is $U\left(r\right) = k \left(r - r_0\right)^2 / 2$, where $r_0$ is  the equilibrium position of the spring. Since this is a central potential (depends only on $r$), angular momentum $L$ is conserved, and we can use the concept of an effective potential energy (see equation 335 here)
$$
\begin{eqnarray}
U_{eff}\left(r\right) &=& \frac{L^2}{2 m r^2} + U\left(r\right) \\
&=& \frac{L^2}{2 m r^2} + \frac{1}{2} k \left(r - r_0\right)^2
\end{eqnarray}
$$
The equilibrium position $r_*$ is then given by
$$
0 = \frac{d U_{eff}}{dr} \Bigg|_{r=r_*} = -\frac{L^2}{mr_*^3} + k\left(r_* - r_0\right) \ \ \  \Rightarrow \ \ \ u^4 - u^3 = C,
$$
where $u = r_*/r_0$ and $C = L^2 / \left(m k r_0^4\right)$.
The angular frequency of small oscillations about $r=r_*$ is then
$$
\omega = \left(\frac{1}{m} \frac{d^2 U_{eff}}{dr^2} \Bigg|_{r=r_*} \right)^{1/2} = \left(\frac{k}{m} + \frac{3 L^2}{m^2 r_*^4}\right)^{1/2}
$$
Now it remains to solve for $u$ in $u^4 - u^3 = C$.

