 Solution

For the initial radius $r_0=l/2$ to be circular, the radial equation requiresConsider a small radial perturbation at fixed conserved angular momentum $J$. Using $J^2/(m{r}_0^3)=Mg$ and $r=r_0+ϵ$ in the radial equation givesConsequently the linearized radial motion is simple harmonic motion:The positive curvature of the effective potential establishes radial stability. This is the small-amplitude limit, not an exact finite-amplitude sinusoid. A perturbation that changes $J$ shifts the circular radius; its radial oscillations are about the new radius to the same order. A radial displacement or radial impulse preserving $J$ gives the stated oscillations about $l/2$.