= Axial gravitational field of a uniform ring
{title2=$g_z=-GMz/(a^2+z^2)^{3/2}$}
A uniform thin ring of radius $a$ and <mass> $M$ has axial <gravitational field> $g_z=-GMz/(a^2+z^2)^{3/2}$. Every ring element is the same distance from the axial point; transverse contributions cancel and axial projection supplies the factor $z/\sqrt{a^2+z^2}$. This also follows by differentiating the axial potential $\Phi=-GM/\sqrt{a^2+z^2}$. Near the centre $g_z\simeq-GMz/a^3$, whereas far away it approaches the point-mass field.
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