= Circular orbit perpendicular to a uniform rod
{title2=$v(r)=\sqrt{GM/a}[1+(r/a)^2]^{-1/4}$}
For a light test <mass> in the rod's perpendicular-bisector plane, axial <symmetry> makes the attraction radial. Equating $mv^2/r$ to the transverse <force> gives the title <speed>. Far outside the rod it is Keplerian, proportional to $r^{-1/2}$. Well inside the half-length but outside its physical thickness, the rod approximates an infinite line and the circular <speed> is nearly constant, $\sqrt{GM/a}$. Extrapolation to an orbit inside the material or a zero-radius orbit is outside this thin-rod model.
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