Now what if I move outside this sphere? It turns out that the gravitational field due to a spherical distribution produces the same gravitational field as if all the mass were concentrated at a single point in the center of the sphere. That’s pretty cool, because it lets us easily calculate the Earth’s gravitational field just using the distance from the center of the object, instead of worrying about its actual size and total mass.
Now we have one more thing to consider: how does the gravitational field (and therefore your weight) change as you get closer to the center of the Earth? We will need this information to know how far a person would have to tunnel to reduce their weight by 20 pounds.
Let’s start with the Earth as a sphere of radius (R) and mass (m). In this first approximation, I’m going to assume that the Earth’s density is constant, so that the mass per unit volume of matter at the surface (like rocks) is the same mass per volume as the matter at the center (like the magma). Not true, but that’s fine for this example.
Imagine that we dig a hole and a person descends into it at a distance (r) from the center of the Earth. The only mass that matters for the gravitational field (and weight) is this sphere of radius (r). But remember that the gravitational field depends on both the mass of the object and the distance from the center of the sphere. We can find the mass of this inner part of the Earth by saying that the ratio of its mass to the mass of the whole Earth is the same as the ratio of their volumes, because we have assumed a uniform density. With that, and a bit of math, we get the following expression:
This says that the gravitational field inside the Earth is proportional to the distance of the person from the center. If you want to reduce their weight by 20 pounds (say 20 out of 180 pounds), you will need to reduce the gravitational field by a factor of 20/180, or 11.1%. This means that they would have to move at a distance from the center of the Earth of 0.889×R, which is a hole that is only 0.111 times the radius of the Earth. Simple, right?
Well, the Earth has a radius of 6.38 million meters, or about 4,000 miles, which means the hole should be 440 miles deep. In fact, it’s even deeper than that, because the earth density is not constant. It varies from about 3 grams per cubic centimeter at the surface up to about 13 g/cm3 in the core. It means you will have to take revenge closer center to achieve a 20-pound weight reduction. Good luck with that. If you really want to lose weight, you better join a gym.