Now, what if I transfer outdoors this sphere? It seems that the gravitational area attributable to a spherical distribution produces the identical gravitational area as if all of the mass was concentrated right into a single level on the middle of the sphere. That is type of good, because it permits us to simply calculate the gravitational area from the Earth by simply utilizing the space from the middle of the article, as a substitute of worrying about its precise dimension and its whole mass.
Now, we’ve yet one more factor to think about: How does the gravitational area (and due to this fact your weight) change as you get nearer to the middle of the Earth? We’ll want this info to learn the way far an individual must tunnel to cut back their weight by 20 kilos.
Let’s begin with the Earth as a sphere of radius (R) and mass (m). On this first approximation, I will assume the Earth’s density is fixed in order that the mass per unit quantity of stuff on the floor (like rocks) is similar mass per quantity because the stuff on the middle (like magma). This really is not true—nevertheless it’s advantageous for this instance.
Think about we dig a gap, and an individual climbs down it to a distance (r) from the middle of the Earth. The one mass that issues for the gravitational area (and weight) is that this sphere of radius (r). However keep in mind, the gravitational area relies on each the mass of the article and the space from the sphere’s middle. We are able to discover the mass of this interior a part of the Earth by saying that the ratio of its mass to the mass of the entire Earth is similar because the ratio of their volumes, as a result of we assumed uniform density. With that, and slightly little bit of math, we get the next expression:
This says that the gravitational area contained in the Earth is proportional to the particular person’s distance from the middle. If you wish to lower their weight by 20 kilos (as an instance 20 out of 180 kilos), you would wish to lower the gravitational area by an element of 20/180, or 11.1 p.c. Meaning they would wish to maneuver to a distance from the middle of the Earth of 0.889 × R, which is a gap that is simply 0.111 occasions the radius of the Earth. Easy, proper?
Effectively, the Earth has a radius of 6.38 million meters—about 4,000 miles—which suggests the opening must be 440 miles deep. Really, it is even deeper than that, as a result of the density of the Earth is not fixed. It ranges from about 3 grams per cubic centimeter on the floor as much as round 13 g/cm3 within the core. This implies you’d have to get even nearer to the middle to get a 20 pound discount in weight. Good luck with that. If you happen to actually need to drop a few pounds, you would be higher off simply becoming a member of a gymnasium.