Now, what if I transfer outdoors this sphere? It seems that the gravitational subject resulting from a spherical distribution produces the identical gravitational subject as if all of the mass was concentrated right into a single level on the heart of the sphere. That is sort of good, because it permits us to simply calculate the gravitational subject from the Earth by simply utilizing the gap from the middle of the item, as an alternative of worrying about its precise dimension and its complete mass.
Now, we’ve another factor to think about: How does the gravitational subject (and due to this fact your weight) change as you get nearer to the middle of the Earth? We’ll want this data 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 identical mass per quantity because the stuff on the heart (like magma). This really is not true—however 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 subject (and weight) is that this sphere of radius (r). However bear in mind, the gravitational subject will depend on each the mass of the item and the gap from the sphere’s heart. We are able to discover the mass of this inside a part of the Earth by saying that the ratio of its mass to the mass of the entire Earth is identical because the ratio of their volumes, as a result of we assumed uniform density. With that, and a little bit little bit of math, we get the next expression:
Illustration: Rhett Allain
This says that the gravitational subject 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 (for instance 20 out of 180 kilos), you would wish to lower the gravitational subject by an element of 20/180, or 11.1 p.c. Which means 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 instances the radius of the Earth. Easy, proper?
Properly, 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 must get even nearer to the middle to get a 20 pound discount in weight. Good luck with that. When you actually wish to shed weight, you would be higher off simply becoming a member of a gymnasium.
