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A summer problem.
I guess the important thing to realise is that the size atoms is irrelevant to the problem. If we considered two atoms joined together to be a new "atom" then they would be twice as heavy, so the forces would be four times as strong, but there would be only half as many atoms, so there would be four times fewer pairs.
So the answer is just the integral as r and r' range over the interior of the earth of G ρ(r) ρ(r')/(r-r')^2, where ρ(r) is the density. We can assume constant density, but I still can't be bothered to do the integral.
The earth has m... (read more)