Re: Hollow Earth

[Rodney Dunning <dunnirb4@WFU.EDU>]

Bob Yeend wrote:

> I think this thread started with a question about whether or not an object
> dropped through a hole  bored from the north to the south pole would
> experience simple harmonic motion.
>
> That seemed reasonable to me, given the assumptions - no friction,
> temperature near core wouldn't melt the object, the earth is homogeneous . .
> .
>
> Someone, however, (perhaps Leigh?) posted that it would not, but didn't give
> an explanation. Please help with the explanation.

The "tunnel in the Earth" problem is an standard problem that shows up in first-
and second-year college mechanics courses. As others have pointed out, we make
simplifying assumptions (e.g., that the Earth is a nonrotating, uniform sphere)
so that we can solve the problem without spending a lot more time than it
deserves. The reason a ball would not execute simple harmonic motion is because
our simplifying assumptions wouldn't apply to a real tunnel. For an idealized
tunnel connecting the poles, the equation of motion is

r'' = - GMr/R^3

where r is the distance from the center. The minus sign indicates that the force
and displacement are in opposite directions. Of course, this is identical to
Hooke's Law (r'' = -Kr, where K = GM/R^3). See Sample Problem 14-5, Halliday,
Resnick & Walker, 1997, page 329.

--
Rodney Dunning
dunnirb4@wfu.edu
http://www.wfu.edu/users/dunnirb4