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



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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