How Far Would You Have to Tunnel Underground to Lose 20 Pounds?

Now, what if I transfer outdoors this sphere? It seems that the gravitational subject on account of a spherical distribution produces the identical gravitational subject 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 subject from the Earth by simply utilizing the gap from the middle of the article, as a substitute of worrying about its precise measurement and its complete mass.

Now, we’ve got yet one more factor to contemplate: How does the gravitational subject (and subsequently your weight) change as you get nearer to the middle of the Earth? We’ll want this data to learn how 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’ll 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 middle (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 is dependent upon each the mass of the article and the gap from the sphere’s middle. We will 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 identical because the ratio of their volumes, as a result of we assumed uniform density. With that, and a bit little bit of math, we get the next expression:

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 example 20 out of 180 kilos), you would want to lower the gravitational subject by an element of 20/180, or 11.1 p.c. Meaning they would want 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?

Nicely, 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. When you actually need to reduce weight, you would be higher off simply becoming a member of a gymnasium.