Gravity and Gauss's law: Inside the Earth

Gauss's law for gravity can be used primarily to show what happens if the earth or other really large objects had different shapes.  We would find that spheres and point masses have 1/r^2 behavior, cylinders have 1/r behavior, and flat objects would have a uniform, constant force of gravity.

But Gauss's law can be used to figure out what happens inside objects as well.  There are two really fun cases to consider: i) a hollow earth, and ii) a solid earth with uniform density.  These situations are normally considered for solid charged objects (nonconductors) in electricity and magnetism units, but the same math and concepts can be used for gravity, too.

The key to Gauss's law is that the strength of the gravitational field at some radius from the center of the earth depends on how much mass is inside that radius.  If the earth is hollow, this means that once you drill a hole into the earth and jump in, because there is no mass inside then there would be no gravity inside!  You would be truly weightless and would simply float around inside a hollow earth!!

Because the earth is not hollow, we will not have a chance to test that prediction from Gauss's law.  But a solid earth can be analyzed.  If we assume the earth has a uniform density, and we were to drill a tunnel all the way through the center of the earth to the other side, what would happen if you jumped into that tunnel?

Check out the video to see more details, but the answer is you would oscillate back and forth through the earth as if you were attached to a spring!  The motion would turn out to be the same as that of something in simple harmonic motion!