Wrong, Wrong, Wrong!

A certain incident has been nibbling away at my consciousness for ages like a kind of psychic termite. It happened in my freshman physics class when the teacher was introducing Newton's law of gravitation. In order to protect the reputation of said teacher, I shall refer to him here as Professor X.

Professor X presented the equation for finding the force exerted on an object by the earth's gravity. It's a classic, lovely little equation...

M_e = the mass of the earth, about 1.3 x 10²⁵ lbs or 5.97 x 10²⁴ kilograms. (Do we remember our metric conversions and our scientific notation?)

m = the mass, in kilograms, of some object.

r = how far that object is from the center of the earth.

G = the universal gravitation constant, a sort of cosmic fudge factor, equal to

6.673 x 10^-11 N•m²/kg² 

(the "N" stands for "Newtons", the unit of measure for force; read as "Newton meters squared per kilograms squared").

A student asked a question: "So, if r = 0, the force is infinite?"

Knowing that we had been taught in our math classes that anytime you divide a number by 0 the answer is ∞, Professor X said, "Yes, if r is 0, F_g is infinitely large."

There are a few reasons why the professor might have responded to the student's question with such an incredibly wrong answer:

1) The student who asked the question was kind of annoying and may have been asking a smart-alecky question which, naturally, demanded a smart-alecky answer;

2) The professor may have simply been joking, oblivious to the fact that some people in the class might have taken him seriously; or, what I think is the most likely answer,

3) The professor wanted to get beyond gravitation and move on to the next subject without introducing additional, complicated material which would have blown the minds of unprepared freshman physics students.

Physicists are sometimes taken to task by other scientists for playing fast and loose with mathematics;  assumptions and leaps of logic that -- even as they are proven true beyond a reasonable doubt by experiment after experiment -- leave mathematicians flabbergasted by their audacity and seem a lot like cheating. It happens often enough that, sometimes, physicists get caught with their panties down and major flaws are found in their reasoning and their lovely little equations. The F_g equation works. Mostly. For the purposes of an undergraduate physics student, at least, it works just fine. But then you start to look at the assumptions being made and you realize the equation comes with some strings attached.

The big assumption being made in regard to the gravity equation and the earth is that you are dealing with a point mass, a planet-sized spherical mass of uniform density where all its gravity is coming from a single point in the center, which is kind of a funny idea when you think about it. If you're talking about objects near or above the surface of the earth, you can get decent approximations of F_g from the equation. But when you start to dig down, things get complicated.

Every part of the earth, from the crust to the core, has mass. Start boring a hole toward the center of the earth and eventually, you'll find yourself with a mass of earth-stuff above you that's large enough to exert a significant gravitational force. (How far down do you have to go? That's a future post.) The gravity you felt would be the sum of countless sources of gravitational attraction acting on you.

The total force of gravity acting on you deep below the earth's surface would be the total F_g from below you and minus the total F_g from above you (from the sides, it would mostly cancel out... more on that later). This is because of something called the superposition of forces. The total force of gravity you would experience would still be drawing you toward the center of the earth, but F_g (from beneath you) would be less than if you were on the surface.

Since you experience the sum of all the gravitational forces acting on you, two equal forces attracting you from opposite directions cancel each other out. Thanks to the principle of superposition, we know that if you were at the center of the earth, for all the mass in any direction exerting a gravitational force on you, there is an equal mass on the other side of you canceling it out. Newton's formula isn't wrong, it just needs to be applied in a slightly different way. F_g at the center of the earth, where r = 0, isn't ∞ like Professor X said, it's actually the sum total of all the "gravities" acting on you which add up to 0.

But it's not as if we could actually test this. It's about 6367 km (3956 miles) to the center of the earth and there's a helluva lot of stuff to dig through to get there, not to mention the fact that the middle of our planet is crazy hot (estimated at around 4000° C /7230° F). As any geophysicist will tell you, gravity gets complicated when you go underground.

Picture yourself standing pretty much anywhere on the earth's surface. We can describe the force of gravity that keeps you from flying off into space with a vector, an arrow pointing downward that represents both the direction of the force and how strong it is.

A force like the force due to gravity, is often represented in diagrams by a simple arrow which indicates the direction in which the force acts.

When you're underground, the force of gravity you experience as a result of being surrounded by earth can be described by a three-dimensional vector field that represents the total gravitational attraction from all the sources of mass that surround you. Vector fields are something a freshman physics student would probably only know about if they had taken a multivariable calculus class (which would be one or two semesters in the future for most students). There is no such thing as a quick lesson on vector fields... re-read this paragraph and you'll realize I really haven't told you anything about them except that they exist, so I suppose Professor X can be forgiven for not wanting to bring nasty old vector fields into our pleasant discussion of gravity law. I guess I can let him off the hook... for this at least.

Sources

• Geodynamics: Applications of Continuum Physics to Geological Problems by Donald L. Turcotte & Gerald Schubert. ©1982, John Wiley & Sons, Inc.
• Geophysical Methods in Geology, 2nd Ed. by P.V. Sharma. ©1986, Elsevier Science Publishing Co., Inc.