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Gravitational Field at Planet Center
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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