  We study the motion of a superfluid vortex in condensates having different
background density profiles, ranging from parabolic to uniform. The resulting
effective point-vortex model for a generic power-law potential $\propto r^k$
can be experimentally realized with recent advances in optical-trapping
techniques. Our analysis encompasses both empty-core and filled-core vortices.
In the latter case, the vortex acquires a mass due to the presence of
distinguishable atoms located in its core. The axisymmetry allows us to reduce
the coupled dynamical equations of motion to a single radial equation with an
effective potential $V_{\rm eff}$. In many cases, $V_{\rm eff}$ has a single
minimum, where the vortex precesses uniformly. The dynamics of the vortex and
the localized massive core arises from the dependence of the energy on the
radial position of the vortex and from the $r^k$ trap potential. We find that a
positive vortex with small mass orbits in the positive direction, but the sense
of precession can reverse as the core mass increases. Early experiments and
theoretical studies on two-component vortices found some qualitatively similar
behavior.
