sim
...@germain.une.edu.au (Simant Dube) writes:
>I am not a physicist, but would appreciate if some physics
>person would post the answer to this problem:
> Suppose I dig up a vertical tunnel all the way through the
> earth so that it opens up on diametrically opposite place on
> the earth. If an object is then dropped into the tunnel, what will
> happen to it?
> - Simant Dube
Basically, you get simple harmonic motion, which is characterisitic of
a force proportional to distance, like a spring, where if you pull a
ball which is connected to a spring, the restoring force increases
linearly with the distance of the pull.
Since you said that you aren't a physicist, this should confuse you,
since you have heard that the gravitational force varies as 1/r^2, and one over r squared is certainly different from r!
Well, what happens is that as the object moves into the earth, there is now a "shell" of mass through which the object has penetrated. Since some of this "shell" is above the object, and some below, and some on the sides, you have to add up all the forces caused by that shell of mass.
Well, it turns out that for a 1/r^2 force law, the forces caused by the shell magically add up to exactly zero, so all you have to do is consider that the mass you "feel" is reduced by that shell. It turns out that the mass you "feel" (after subtracting the increasingly growing shell as you go deeper into the earth) varies as r^3 as you penetrate the earth. Now, r^3 times 1/r^2 = r, so you can see that the overall force goes as r.
Thus, it is simply a harmonic oscillator, and the result is that the
object accelerates to a maximum speed at the center, and proceeds to decelerate to zero as it reaches the antipodal point, and then turns around, oscillating.
Another interesting observation is that if the tube is cut through the
earth from, say, England to US, the time it takes for the object to fall through will remain the same as for a tube going through the center.
Anyway, Lewis Carroll knew about this, and made some "knowing" comment in "Alice in Wonderland", which I now forget.
Naresh Kannan
Cornell University, Applied & Engineering Physics Senior
AT&T Bell Laboratories, Technical Associate