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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 takeUnder the approximation that the earth is homogeneous, i.e.
about 42 minutes to make it to the other side. I always wanted to
know this value. Dos anyone know how this was found?
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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