Hello, Im not sure how to set up this integral. Its a little more advanced than I am used too. Any ideas?

 

Consider a thin ring of radius a and mass M. A mass m is placed in the plane of the ring (not in the center!). Determine the gravitational potential for r < a. Find a position of equilibrium and determine whether it is stable or unstable. (Hint: Consider a small displacement from the equilibrium position and do an expansion.)

You need to write down an equation for r for an arbitrary point inside the circle, and then integrate it for all points around the circle.

 

Pick your point a distance a from the center, and draw a line to an arbitrary point on the circumference, and also draw a line from the center to that same point on the circumference. That will be your triangle. Apply the law of cosines.