Gravitational potential of a thin ring

[DerekDnl]

Hello, I am 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.)

I think I have to use the law of cosines. But why would I need to do an expansion?

 

[LowlyPion]

Welcome to PF.

Consider that wherever you are inside the ring you basically only need to be concerned about the component of the gravitational force that is in the radial direction. By symmetry anything normal to the radius will be canceled out left to right won't it?

So for any point a distance r away from the center all you need to do is develop an expression that describes, for each element about the ring the gravitational component that projects to the radius your mass is on.