= Axial and transverse gravity of a uniform rod
{title2=$F_\parallel=GMm/(b^2-a^2),\quad F_\perp=GMm/[r\sqrt{r^2+a^2}]$}
Integrate Newtonian point-force elements along a uniform thin rod of total <mass> $M$ and half-length $a$. The first title formula applies on the axis outside the endpoint, $b>a$; the second applies on its perpendicular bisector. Both approach the point-mass result at large distance, but at equal distance their ratio is $1+3a^2/(2b^2)+O(a^4/b^4)$. A nonuniform linear <mass> profile cannot be replaced by these formulas using only total <mass> and <length>.
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