Can the Law of Universal Gravitation Override Earth's Gravity?

[c.evans]

We just started on a chapter about gravity in my physics class. I'm curious about this: If two rings of different radius and masses of, say, 3 kg and 2 kg, are placed inside each other so that their centers of mass are extremely close (i.e. 1.0 x 10^-9 m) together, would the gravitational force between the two objects be enough to overcome the gravitational force of earth, so that the smaller ring would float in midair within the larger ring? When I put these values into the equation for Newton's Law of Universal Gravitation, the force between the two rings came out as 400.2 N, while the force of the weight of the smaller ring came out as 19.6 N. Is this possible, or am I using the equation the wrong way?

 

[Drakkith]

Nope. First, the gravity on Earth is much stronger than anything your rings would have. I don't know the exact amount, but your number seems absurdly hi for some reason.

Also, the rings would never float in your example anyways. Placing a ring inside another ring wouldn't do anything. The inside ring would feel no net force in any direction to the larger ring.