We consider the motion of individual two-dimensional vortices in general radially symmetric potentials in Bose-Einstein condensates. We find that although in the special case of the parabolic trap there is a logarithmic correction in the dependence of the precession frequency $omega$ on the chemical potential $mu$, this is no longer true for a general potential $V(r) propto r^p$. Our calculations suggest that for $p>2$, the precession frequency scales with $mu$ as $omega sim mu^{-2/p}$. This theoretical prediction is corroborated by numerical computations, both at the level of spectral (Bogolyubov-de Gennes) stability analysis by identifying the relevant precession mode dependence on $mu$, but also through direct numerical computations of the vortex evolution in the large $mu$, so-called Thomas-Fermi, limit. Additionally, the dependence of the precession frequency on the radius of an initially displaced from the center vortex is examined and the corresponding predictions are tested against numerical results.
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