= Gravitational field of a finite uniform line segment
{title2=$g_r=-GM/[r\sqrt{r^2+a^2}]$}
Spread <mass> $M$ uniformly along $-a<z<a$ on an axis, with <linear mass density> $\lambda=M/(2a)$. In its perpendicular bisector plane, axial field components cancel, and the radial component is
$$
g_r=-G\lambda r\int_{-a}^a\frac{dz}{(r^2+z^2)^{3/2}}=-\frac{GM}{r\sqrt{r^2+a^2}}.
$$
This is an exact <line mass> model and a thin-rod approximation for a physical cylinder. A centred <circular orbit> in this symmetry plane has $v^2=GM/\sqrt{R^2+a^2}$. Thus $v\to\sqrt{GM/a}=\sqrt{2G\lambda}$ for $R/a\to0$, and $v\sim\sqrt{GM/R}$ for $R/a\to\infty$. For a finite-width rod, the first limit is useful only while its transverse size remains much smaller than $R$.
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