Assume you are at a party where nerds gather and someone proposes
a tunnel drilled diametrically thru the earth, and asks you to assume
away all traces of an atmosphere and any asphericity of the earth.
She asks if a particle dropped into the tunnel will arrive before or
after another particle in a treetop orbit, assuming the tunnelling particle
is dropped just as the orbiting particle goes by; that is will the particle
that falls thru the earth take more or less than 42 minutes.
The other nerds know the answer is about 42 minutes. They are
wrong of course, for you now know that the correct answer, which I
finally got off my lazy ass and calculated, is 37 minutes. The reason
it takes less than 42 minutes is that the earth has a definite density
gradient, so that the core is much denser than the crust and mantle.
If you wish to appear astute before the other nerds, you might go
on to observe that a particle thru the center can be arranged to
take longer or shorter than the minimal orbit particle, but in order
to make it take the same time, the sphere needs to be of uniform
density. Once can arrange for the particle thru the center to take
longer than the orbitting particle by having a hollow sphere for
Regarding Damien Broderick's train on a straight frictionless track
on a non-diametric chord thru the earth, the train will arrive at its
destination, regardless of where, in slightly less than 42 minutes. spike
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