Falling through the Earth

Date: 12/31/2001 at 17:14:02
From: Daren Henning
Subject: Falling through the Earth

My friends and I are having an argument about gravity. I posed a 
question that I read in a book: If a hole is drilled through the 
center of the earth to the other side and you jump in, what will 
happen?  

What I read was that you would continue falling through the earth 
until you reached the other side; then you would exit the hole, slow 
down, and be drawn back into the hole until you exited the hole on the 
other side - and this would again be repeated.  

A friend says that you would not exit the hole and would eventually 
stop in the middle of the earth weightless and floating, because 
gravity would be pulling equally on you from all sides, sort of a 
weightless existence. He also says the closer you get to the center of 
the earth the lighter you will be, since gravity is now pulling on you 
from two differant directions.  

I disagree with him and am inclined to believe what I read in the 
book, but this book was written a long time ago, so I am not sure.  
Could you please help end this argument, or point me to a place that 
could?

Thanks a lot,
Daren D. Henning

Date: 12/31/2001 at 18:25:31
From: Doctor Tom
Subject: Re: Falling through the Earth

If the earth is spinning, then it depends on where the hole is 
drilled. If it's from equator to equator, you'd bounce against the 
walls as you went down, since your velocity on the surface is
1000 mi/hr (approximately), and halfway down the tunnel would only be 
turning at 500 mi/hr. If it were from pole to pole, or if the earth 
weren't spinning, and the earth were a perfect sphere, and there were 
no wind resistance, you would keep gaining speed until you passed 
through the center, and then would lose speed as you came up the other 
side, and would stop going up just as you got to the surface at the 
opposite end. Then you would drop through again, and cycle that way 
forever, just rising to the surface at each end.

Now of course with air resistance, you would slow down on each cycle, 
and eventually would stop right in the center because your path would 
be constantly opposed by friction with the air.

Here's an amazing additional fact (assuming that the earth is not 
spinning and there is no air resistance). If you drill a hole at an 
angle, and slide along a frictionless track back and forth, exactly 
the same thing would happen. You'd cycle forever, coming just to the 
surface at each end each time. But the time for a complete cycle is
exactly the same for every straight-line path - approximately a 
90-minute round-trip. This is also the time that a satellite in orbit 
would take to do one loop around the earth right at the surface (or a 
fraction of an inch above it) with no air resistance and a perfectly 
spherical earth and circular orbit.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/31/2001 at 19:12:24
From: Daren Henning
Subject: Falling through the Earth

Thanks a lot, Dr. Tom!

Just one little clarification if you have the time. What would happen 
if you climbed down to the center of the earth? At that point would 
you really be weightless and able to float? Also, let's say you made a 
spiral staircase through the hole, would you continue to spiral the 
same all the way through the hole or would the spiral change at the 
center? In the answer you gave me with air resistance, you said that 
you would eventually stop in the middle; would you be standing on the 
wall, or would you float? Also, your answer said with the angle and 
sliding it would take 90 minutes - does this mean at any angle through 
the planet?

Thanks again -you've answered a lot for me!

Daren D. Henning

Date: 12/31/2001 at 19:31:19
From: Doctor Tom
Subject: Re: Falling through the Earth

Yes. Right at the center you would float weightless. It's based on the 
following (somewhat amazing) physical principle:

If you have a thin spherical shell of uniform material, the
gravitational field due to the material is exactly zero ANYWHERE 
inside the shell, and outside the shell, the field is exactly the same 
as if all the matter in the shell were concentrated at a point at the 
center of the sphere. Thus when you're half-way down, you can think of 
the entire earth as being made of a gazillion thin shells, one inside 
the other. The force due to all the outer shells is zero (zero force 
from each one) and the force due to the shells closer to the center
than you are each is as if all its mass is at the center of the earth.  
Thus the force on you is exactly as if all the matter of the inner 
shells were at a point at the center of the earth.

Thus a guy at the center of the earth would have only shells outside, 
each contributing zero force for a grand total of zero.

Of course you don't even need to know any physics to get this. If 
there is a net force in some direction, rotate the earth around by 180 
degrees and the force will have to go in the opposite direction, but a 
180-degree turn leaves it exactly the same by the symmetry of the 
sphere, so the force has to be equal to its negative, or, in other 
words, zero.

It turns out also to be true of electric charge, but that's another 
story. If you have a spherical conductor with a charge on it, the 
field anywhere inside is zero and the field outside is exactly the 
same as if all the charge on the shell were concentrated at the center 
of the sphere.

And finally, yes, the holes bored in the earth could be at any angle.  
All that is required is that they be straight line paths.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/31/2001 at 19:42:38
From: Daren Henning
Subject: Falling through the Earth

Thank you again, all VERY interesting! So by what you have told me the 
spiral staircase in the hole would have to reverse at the center of 
the hole because that is where gravity would change directions, and 
you could not walk down to the other side of the earth, and the deeper 
you walked down the hole, the lighter you would become. Yikes! Please 
correct me if I am wrong.

Much appreciated,
Daren D. Henning

Date: 12/31/2001 at 20:00:38
From: Doctor Tom
Subject: Re: Falling through the Earth

Exactly right, but of course near the center the gravitational
force would be very close to zero.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/