How to calculate gravity inside of a circle of mass? I've been reading about the Shell Theorem and how the gravitational force inside of a sphere is equal to zero. I was wondering if the same was true for a circle, but couldn't find any definitive answer as to how to calculate it and whether the net gravity would also be zero.
How do you calculate the gravity inside of a circle of mass?
 A: Let $M$ mass of ring, $R_1$ radius of ring and $R_2$ field point with gravitational acceleration
$$g ={\frac {Gm}{r^{2}}} = \oint_l Gdm/r^2 = \\
 \displaystyle{\frac{GM}{R_1^2}\int_0^{2\pi} \frac{d\phi}{\sin^2\phi + \left(\cos\phi - \frac{R_2}{R_1}\right)^2}} = \\
\displaystyle{\frac{GM}{R_1^2}\int_0^{2\pi} \frac{d\phi}{1 - 2\frac{R_2}{R_1}\cos\phi + \left(\frac{R_2}{R_1}\right)^2}} = \\
\displaystyle{\frac{GM}{R_1^2} \frac{2\pi}{\sqrt{\left(1+\frac{R_2}{R_1}^2\right)^2 - 4\left(\frac{R_2}{R_1}\right)^2}}}$$
A relative plot etc. at Wolfram Alpha  where $x = R_2/R_1$.
A: ok then as a thought experiment, inside a two dimensional circle considered to have constant mass the same net gravitational pull, for an object inside it, in the same plane, would be the same as inside a sphere, In a hollow circle having mass only around the outer edge (as a ring), net gravity anywhere inside would be zero, because even if the object were off center it would be pulled equally in all directions, because of the inverse square law of gravitational distance. in a solid circle, having constant mass all across it (as a plate), net gravity would be zero only in the center.
