Now, what if I move outside this ball? It turns out that the gravitational field due to the spherical distribution produces the same gravitational field as if all the mass were concentrated to a single point at the center of the sphere. This is pretty cool, as it allows us to easily calculate the gravitational field from the Earth using only the distance from the center of the object, rather than 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 would need this information to figure out how far a person would have to dig to lose 20 pounds of their weight.
Let’s start with the Earth as a sphere of radius (R) and mass (m). For this first approximation, I will assume the density of the Earth is constant so that the mass per unit volume of the object on the surface (like a rock) is equal to the mass per volume of the object in the center (like magma). This isn’t really true – but it’s fine for this example.
Imagine we dig a hole and a person climbs into it some distance (r) from the center of the Earth. The only mass that matters to the gravitational field (and weight) is this sphere of radius (r). But remember, 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 interior of the Earth by saying that the ratio of its mass to the mass of the entire Earth is the same as the ratio of their volumes, because we assume the density levelness. With that, and a bit of math, we get the following expression:
Illustrated by: Rhett Allin
This says that the gravitational field inside the Earth is proportional to the person’s distance from the center. If you wanted to reduce their weight by 20 pounds (say, 20 out of 180 pounds), you would need to reduce the gravitational field by a factor of 20/180, or 11.1%. That means they would need to travel to a distance from the center of the Earth of 0.889 × R, which is a hole with a radius just 0.111 times the radius of the Earth. Simple, right?
Well, Earth has a radius of 6.38 million meters – about 4,000 miles – which means the hole would have to be 440 miles deep. In fact, it’s even deeper than that, because density of the Earth is not a constant. It ranges from about 3 grams per cubic centimeter at the surface up to about 13 g/cm3 in the core. This means you need to get closer to the center to lose 20 pounds of weight. Good luck with that. If you really want to lose weight, you’d better join a gym.
