[Phys-L] gravity train

[John Denker <jsd@av8n.com>]

Here's the answer to a slightly different question.
Optimal tunnel (as opposed to straight tunnel i.e. chord):
  http://www.physics.unlv.edu/~maxham/gravitytrain.pdf

I found this by googling:
  https://www.google.com/search?q=%22Through+the+Earth+in+Forty+Minutes%22
and that in turn leads to
  https://www.google.com/search?q=%22gravity+train%22

The math is messy enough to put it outside the scope of 
the introductory physics class.  However, it may still
be worth a glance, because:
 -- the question is simple
 -- the final answer is simple and interesting
 -- the concept of calculus of variations is 
  reasonably simple.

There is a famous rule of thumb that says if a messy
calculation produces a simple result, look for a way
to simplify the calculation.  One obvious simplification
is that if an oracle told you the solution you could
relatively easily verify that it is optimal, more
easily than finding the solution ab_initio.  I suspect 
that if a smart person thought about it additional 
simplifications could be found.


On 09/26/2014 06:27 AM, Folkerts, Timothy J wrote:
> It is probably worth noting that all of this assumes that the earth
> is a uniform density as well.  Estimates like this one
> (http://en.wikipedia.org/wiki/Gravity_of_Earth#mediaviewer/File:EarthGravityPREM.svg)
> suggest that the g is remarkably close to 10 m/s^2 almost half way
> down to the center of the earth because the density id much greater
> near the center than near the surface.

That's interesting.

It might be amusing to redo the calculations (for
straight tunnel and/or optimal tunnel) assuming
|g| proportional to r^0 rather than r^1.