[QUE/ME-09002] ME-PROBLEM

Two particles each of mass $m$ interact via a central force as specified for three different systems.

1. Potential energy of the system is \( V_0 \Big(\dfrac{R}{r}\Big). \)
2. Potential energy of the system is \( V(r) = V_0 \Big(\dfrac{r^2}{R^2}\Big).\)
3. The force between the particles as function of separation is given by \( - V_0 \Big(\dfrac{R}{r^2} + \dfrac{15 R^3}{r^4} \Big).\)

Here \(r\) denotes the separation of the particles and \(V_0, R\) are some positive constants. Assuming that the angular momentum of the particle is \(L=\sqrt{mV_0}\)R. Here \(V_0\) and \(R\) are some positive constants. For the three cases find if a stable circular orbits exist. If yes, find the radius of the circular orbit and corresponding value of energy in terms of $V_0$ and $R$.