  We present a study on planar equilibria of a terminally loaded elastic rod
wrapped around a rigid circular capstan. Both frictionless and frictional
contact between the rod and the capstan are considered. We identify three cases
of frictionless contact -- namely where the rod touches the capstan at one
point, along a continuous arc, and at two points. We show that, in contrast to
a fully flexible filament, an elastic rod of \emph{finite length} wrapped
around a capstan does not require friction to support unequal loads at its two
ends. Furthermore, we classify rod equilibria corresponding to the three
aforementioned cases in a limit where the length of the rod is much larger than
the radius of the capstan. In the same limit, we incorporate frictional
interaction between the rod and the capstan, and compute limiting equilibria of
the rod. Our solution to the frictional case fully generalizes the
\emph{classic capstan problem} to include the effects of finite thickness and
bending elasticity of a flexible filament wrapped around a circular capstan.
