Gravitational Field From Ring Mass

[edgepflow]

Most introductory physics textbooks derive the following formula for the gravitational field for a point mass on the on the axis of the ring:

$g_{x}$ = -$\frac{Gmx}{(x^2 + a^2)^{1.5} }$

where,

m = total mass of ring
x = distance from point mass to ring center
a = radius of ring

Is there a more general treatment for this ring when the point mass is not on the axis?

 

[elegysix]

yes, and for any other system as well. To solve it for an axis you need to have symmetry like the ring, or the math won't be pretty. Basically integrate dF (dF = GM/r^2 dm , dm=ρdV) over the volume to find F(r)