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Name: Thomas Matthew F.
Status: student
Age: 16
Location: N/A
Country: N/A
Date: 3/19/2003


Question:
I searched the archives, but could not find the answer to my question about a zero-gravity shell at earth's core. I saw some answers concerning donut planets, and gravitons, but I still wonder: If one were to dig a hole directly through the center of the earth, how would one fall (if one were to jump into it)? Would one fall "down" then back out at the original dig site, or fall straight through, or fall to the center of the earth then remain suspended in true zero gravity at the exact center(assuming that temperatures and pressures would not be an issue)? If I missed the answer in the archives, can someone tell me what number it is, and under what subject (I thought Physics)?


Replies:
There is I am sure a detailed analysis of this proposal; however, in its simplest formulation you suggest the force of gravity on a mass 'm' a distance 'R' from the center of the earth, of mass 'M' is given by Newton's famous equation F = GmM/R^2, where 'G' is the universal gravitational constant. Implicit in this law is the concept of "point" masses. You are asking what happens when the "point mass" assumption no longer occurs. My thumbnail analysis is this: Suppose a rock on the surface of the earth falls into a well that extends from one surface to a corresponding point on the opposite side of the planet, e.g. the north and south poles. The potential energy is Ep= mgR where the m=GM. When the rock reaches the center of the earth it is being "pulled" in all directions equally so Ep=0 and the kinetic energy T=1/2(mv^2) reaches a maximum value. The rock continues to the opposite pole, comes to rest, and then reverses its fall back to the center of the earth. In a simple minded analysis this sinusoidal motion would continue indefinitely.

There is a "red flag" in this "simple" analysis and that is F=GmM/R^2 becomes infinite when R=0. This is a "warning" that our assumption of "point" masses is no longer appropriate and that a more detailed analysis, taking into account the forces of gravity on a point mass 'm' SURROUNDED by a spherical uniform mass "M". My gut reaction is we better look at the analysis very carefully because the instantaneous force "felt" by the falling rock may not average out in the simple way I assumed.

Vince Calder


Several answers in the Newton archives addressing your question are:

http://www.newton.dep.anl.gov/askasci/env99/env002.htm.
http://www.newton.dep.anl.gov/askasci/gen99/gen99159.htm
http://www.newton.dep.anl.gov/newton/askasci/1995/environ/env082.htm
http://www.newton.dep.anl.gov/askasci/phy99/phy99040.htm

a related question:
http://www.newton.dep.anl.gov/askasci/phy99/phy99439.htm

Richard E. Barrans Jr., Ph.D.
PG Research Foundation, Darien, Illinois


If you were to dig a hole straight through the earth, passing through its gravitational center (which would be at the geometrical center if the earth were symmetrical -- which it almost is), there would be zero gravity at the center. This is because for every portion of the earth pulling you towards it, there will be an equal force pulling you in the opposite direction due to a symmetrically placed other portion of the earth.

However, it you jump down the hole, you will pass through the center at a very high speed (close to 7 miles per second if you neglect air resistance and do not bump into the sides of the hole). Since the gravitational force there is zero, you will continue without slowing. As you leave the center, of course, the force of gravity on you will increase again, but will now be pointing in the opposite direction and increasing (linearly) in magnitude. If you continue to neglect air resistance and friction, you will continue on, though continually slowing, until you arrive at the surface at the other side of the earth. If you do not get out, you will commence the return trip. A round trip will take about an hour and a half, just about the time for a space station to go around the earth in a low altitude orbit.

It is interesting that if you drill a straight tunnel connecting any two points at the earth's surface at the same altitude, a frictionless wagon will travel between these two points in an hour and a half. This would make it possible to run fast trains which could connect any two points on the earth's surface with a 90 minute ride and at no use of energy. Of course, it is not easy to drill these tunnels and to evacuate them to avoid air resistance and to minimize friction, but the idea has been studied.

Best, Dick Plano Professor of Physics emeritus Rutgers



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