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A tabletop mass $m$ joined through a hole by a massless inextensible string of length $l$ to a hanging mass $M$ has radial inertia $M+m$ and conserved angular momentum $J=m{r}^2\dot{\theta }$. The radial energy is $\frac{1}{2}(M+m){\dot{r}}^2+U_J(r)$, with $0<r<l$. Its effective potential diverges at the hole for $J\ne 0$ and has a unique minimum at $r_0^3=J^2/(mMg)$. Since $U_J^{\prime \prime }(r_0)=3Mg/r_0$, small radial oscillations at fixed $J$ have squared angular frequency $3Mg/[(M+m)r_0]$.

For a cold corotating field line anchored in a circular point-mass orbit, the effective potential has meridional Hessian ${\omega }^2\mathrm{diag}(-3,1)$. Along a straight line inclined by $\alpha$ to the vertical its quadratic curvature is ${\omega }^2(1-4{\sin }^2\alpha )$. Negative curvature gives the strict local launching criterion. Equality is marginal and global escape requires additional information.