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Re: [Phys-L] gravity + tunneling to the antipodes



It seems to me that the "linear density" line and the "constant density"
line should be switched. What am I missing?

P.S. Here's a question: how fast does the Earth have to spin such that
the dropped object doesn't make it through the tunnel but instead pops
back up out the hole?

Robert A. Cohen, Department of Physics, East Stroudsburg University
570.422.3428 rcohen@esu.edu http://www.esu.edu/~bbq


-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@phys-l.org] On Behalf Of brian
whatcott
Sent: Thursday, January 03, 2013 12:45 PM
To: phys-l@phys-l.org
Subject: Re: [Phys-L] gravity + tunneling to the antipodes

Hmmmm....depends how far along the axis from physicist's model to
reality you wish to stray....
constant density Earth, frictionless walls, airless hole
to
dense core Earth with increasing temperature to the core...

This is a helpful depiction..
http://en.wikipedia.org/wiki/File%3aEarthGravityPREM.jpg

Brian W

On 1/3/2013 7:24 AM, Chuck Britton wrote:
For this homogeneous earth - you can dig a straight tunnel between ANY
two points and ride a frictionless skateboard from one end to the other
in this same 42 minute time interval!!

(First made aware of this in a Martin Gardiner "Mathematical Games"
article in SciAm.)

On Jan 2, 2013, at 11:15 PM, John Denker wrote:

On 01/02/2013 08:09 PM, Anthony Lapinski wrote:

if you drop an object [through a tunnel in the earth] it would take
about 42 minutes to make it to the other side. I always wanted to
know this value. Dos anyone know how this was found?
Under the approximation that the earth is homogeneous, i.e.
the same density throughout, it's an easy calculation.
There's a theorem going back to Newton to the effect that:
a) if you're outside a spherical shell, its contribution
to the gravitational field is the same as if all its
mass were concentrated at the center.
b) If you're inside a spherical shell, it contributes
nothing to the gravitational field.

As a corollary of the above, the time for a round-trip oscillation in

a tunnel is the same as the period for a low earth orbit. This is a
famous and amusing result.

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