The hollow earth

There are some crackpots out there that have written books claiming that the earth is hollow and there a bunch of flying saucers inside it. I see. Well lets discuss what the force of gravity is like in this situation.

You have the forces from all the atoms in the shell tugging at you. If you're in the center, its pretty clear that they all cancel and you get zero. It turns out that the force inside is exactly zero even away from the center! As I mentioned before this can be understood most easily using Gauss's law, so I won't try to prove it here.

Let's use this fact to figure out what the force of gravity is as we journey to the center of the earth. Let's consider the earth to be of uniform density. Call the radius of the earth $R$. Let's say we're at a distance $r < R$ from the earth's center. Then we can think about the earth as being composed of a little sphere of radius $r$ lying beneath you, and a hollow spherical cavity above you. The hollow spherical cavity is just a bunch of spherical shells and doesn't apply any force to you. So you're just left with the little sphere of radius $r$.

As we said before, for when you're outside a sphere, the force of gravity is the same as if all the mass were concentrated at the center. Call the mass of the sphere of radius $r$ $M(r)$. Call your mass $m$. So the force on you is

$$F ~=~ {G M(r) m\over r^2}$$ (1.36)

What's $M(r)$? It's just the density $\rho$ times the volume. So

$$M(r) ~=~ \rho V ~=~ {M_e\over {4\over 3}\pi R^3 }{4\over 3}\pi r^3 ~=~ M_e {r^3\over R^3}$$ (1.37)

So using this we have that

$$F ~=~ {G M_e (r^3/R^3) m\over r^2} ~=~ {GM_em\over R^3} r$$ (1.38)

So the force varies linearly with distance, just like one gigantic spring!

If the earth wasn't so darned hot inside, it would suggest a neat means of transportation. Just dig really deep holes in the earth that go all the way through, and jump in! Ignoring air resistance, you could jump in at San Francisco, fall for a while, and end up in Shanghai!