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.

 

[thomate1]

I would suggest approaching this problem by first understanding the concept of gravitational potential. Gravitational potential is a measure of the potential energy of a mass at a particular point in a gravitational field. In this case, the thin ring can be thought of as a collection of point masses, each with its own gravitational potential. The net gravitational potential at a point is the sum of the individual potentials of all the masses.

To set up the integral, you can use the expression for gravitational potential due to a point mass, which is GmM/r, where G is the gravitational constant, m is the mass of the object, M is the mass of the ring, and r is the distance between the object and the ring. Since the ring is thin, we can assume that the distance between the object and each point mass on the ring is the same, which is the radius of the ring, a.

Therefore, the gravitational potential at a point with distance r < a from the ring can be expressed as GmM/a. As for finding the equilibrium position, we can use the concept of net force being equal to zero at equilibrium. This means that the gravitational force on the object due to the ring is balanced by the force of another object (such as the Earth) at that point. To determine whether it is stable or unstable, we can use the small displacement method and see if the net force on the object increases or decreases with the displacement.

In summary, to solve this problem, you can use the expression for gravitational potential due to a point mass, consider the thin ring as a collection of point masses, and use the concept of equilibrium and small displacement to determine the stability of the equilibrium position. I hope this helps. Good luck!