Is it possible to separate these two rings

Question

Let say there are two metallic rings of radius r1=1 meter and r2= 2 meter such that their centers coincides with each other at a common point “p”. (or two spheres, empty from inside). Consider theses two rings in space. Now apply Newton’s law of gravitation i.e. F=GMm/R^2 to those two masses and neglect all other local attractions.

As gravitational force of attraction between these two rings is infinity ( because center to center distance b/t masses R= zero), therefore how much force is required to separate aforementioned rings, infinity or less? If less then how ? Such an example is possible in our atmosphere ?

Answer 1

Yes it is possible.  The apparent paradox is resolved by recognizing that in your example the center of mass is not the correct way to view the problem.  Because these masses have finite volume, you have to calculate the vector force between each part of the inner ring and each part of the outer ring.  What you will find is that although the center of masses are at the same point and you might think the gravitational force is zero, in fact there is zero net force because each part of each ring pulls evenly on each part of each ring.  

This is the same problem as what happens if you drill a hole through the Earth so you can fall all the way through.  As you get to the center, the gravitational force goes to zero, not infinity, because each part of the Earth's mass pulls outward symmetrically.  

If you are talking about mass singularities, mass with no volume, black holes, then yes, the gravitational force goes to infinity at zero separation and you can't separate the objects.

edit:  Ring thickness doesn't matter so long as it is finite.  If the thickness is zero so that the density is infinite, then the system isn't stable anyway.  Whenever you have finite densities and closed shells or rings of masses inside one another, that configuration is gravitationally stable and does not take infinite force to separate.  Google "Dyson sphere" for a great sci-fi example of this concept.

edit 2:  Below is a worked example for a spherical shell.  You can work out the planar ring example from that, if you want to.