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In 'nearly' circular motion, the radius is not constant. If the force is central the and the angular momentum is still conserved, we have a central force:

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where $r_0$ is equilibrium position.

For this motion, the orbital radius will 'oscillate' due to the centripetal and centrifugal force. However, in such motion, I am not sure how the angular frequency is given by:

enter image description here

whereby $Ω$ is the angular frequency due to the 'oscillating' radius, and ω is the angular frequency of the orbit itself.

Any help in explaining this concept and explaining how the angular frequency is derived will be appreciated.

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Conservation of energy $$ E = \frac{1}{2}m(\dot{r}^2 + r^2\omega^2) + U(r) = \frac{1}{2}m\dot{r}^2 + \frac{L^2}{2mr^2} + U(r) $$ yields the equation of motion $$ m\ddot{r} = \frac{L^2}{mr^3} - \frac{\partial U}{\partial r} = - \frac{\partial U_\text{eff}}{\partial r}. $$ For a circular orbit with radius $r_0$, we have $\ddot{r}\equiv 0$, so that $$\left.\frac{\partial U_\text{eff}}{\partial r}\right|_{r=r_0}=0\qquad \text{and}\qquad \frac{L^2}{mr_0^3} = \left.\frac{\partial U}{\partial r}\right|_{r=r_0}\quad. $$ For nearly circular stable orbits, we can write $r = r_0 +x$, with $x\ll r_0$, and the equation of motion becomes $$ m\ddot{x} = -\left.\frac{\partial U_\text{eff}}{\partial r}\right|_{r=r_0+x}\quad. $$ We can Taylor expand the right-hand side up to first order in $x$: $$ \left.\frac{\partial U_\text{eff}}{\partial r}\right|_{r=r_0+x} = \left.\frac{\partial U_\text{eff}}{\partial r}\right|_{r=r_0} + x\left.\frac{\partial^2 U_\text{eff}}{\partial r^2}\right|_{r=r_0} + \mathcal{O}(x^2) = x\left.\frac{\partial^2 U_\text{eff}}{\partial r^2}\right|_{r=r_0} + \mathcal{O}(x^2), $$ so we obtain $$ m\ddot{x} + x\left.\frac{\partial^2 U_\text{eff}}{\partial r^2}\right|_{r=r_0} = 0. $$ This is the equation of a harmonic oscillator, with frequency $$ \Omega = \sqrt{\frac{1}{m}\left.\frac{\partial^2 U_\text{eff}}{\partial r^2}\right|_{r=r_0}}\quad. $$ Your example involves a central force $F(r) = -Kr^n$. The rest is a straightforward calculation.

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