71The gravitational field of a ring on its axis
Let us consider a ring of mass M in the plane perpendicular to the plane of the paper. We want to find the gravitational field on its axis at a distance x.
Consider a differential length of the ring of mass dm.
dE = Fdmz2
gravitational-field-of-ring-on-its-axis
The Y-components of the fields due to diametrically opposite elements cancel each other. Thus, the X-components add up.
E = ∫Gdmz2 cosα = Gdmz2 cosα ∫dm = GMz2 cosα = GMx(a2+x2)3/2
