We study the motion in space of light nonlinearly elastic and viscoelastic rods with heavy rigid attachments. The rods, which can suffer flexure, extension, torsion, and shear, are described by a general geometrically exact theory. We pay particular attention to the leading term of the asymptotic expansion of the governing equations as the inertia of the rod goes to zero. When the rods are elastic and weightless, and when they have appropriate initial conditions, they move irregularly through a family of equilibrium states parametrized by time; the motion of the rigid body is governed by an interesting family of multivalued ordinary differential equations. These ordinary differential equations for a heavy mass point attached to an elastica undergoing planar motion are explicitly treated. These problems illuminate such phenomena as snap-buckling. On the other hand, when the rods are viscoelastic and weightless, the rigid body is typically not governed by ordinary differential equations, but, as we show, the motion of the system is well-defined for arbitrary initial conditions. This analysis relies critically on the careful use of our properly invariant constitutive hypotheses.