Consider a ring of mass “M” as shown below. What is the gravitational potential at a point “p” at a distance of “r” from the centre of the ring?

-(G*M)/(a²+r²)^1/2

Consider an element od mass “dm” of the ring at a distance “z” from the point “p”.

The gravitational potential (dV) due to this elemental mass at point “p” is;

dV = -(G*dm)/z

Integrating dV for the whole ring of mass “M”, we get;

V = -(G*M)/z

From Pythagorean theorem, we get;

z = (a²+r²)^1/2

Therefore; V = -(G*M)/(a²+r²)^1/2.