What would happen if a hole was drilled thru the earth(like the drawing suggests) and an object was dropped down it
i am completely stumped
http://img409.imageshack.us/img409/5166/earthholers1.th.png (http://img409.imageshack.us/my.php?image=earthholers1.png)http://img409.imageshack.us/images/thpix.gif (http://g.imageshack.us/thpix.php)

Gravity Train Solution
Let x(t) be the coordinate along the tunnel with the origin placed in the
middle, r the distance from the train to the center of the Earth. Then
x = r cos ; (1)
where  is the angle between the direction to the middle of the tunnel, and
vertical direction.
The gravity force acting on the train is equal to
F = −
mM(r)r−2;
where m is the mass of the train, M(r) the mass of the part of the Earth
below the train, and
the gravitational constant. (This expression of the
force follows from two theorems of Newton, mentioned in the hints; the mass
of the part of the Earth above the train has no influence). Now we have
M(r) = (4=3)r3, where  is the density of the Earth. Substituting this
expression for M(r), we obtain
F = −4
3

mr:
We only need the component of this force, acting along the tunnel; the rest
is compensated by the rail reaction. The component along the tunnel is
F cos , where  means the same as in (1).
Thus the equation of motion (Newton's Second Law) and (1) give
mx
00
= F cos  = −4
3

mr cos  = −4
3

mx;
where we have chosen the minus sign from the evident physical considerations.
Thus our train behaves like a linear oscillator
x00 = −k2x; where k2 =
4
3

:
As we know, the solutions are periodic with half-period =k. Half-period is
of course the time of a one way trip. Surprisingly, this time does not depend
on the length of the tunnel. This means that a trip from West Lafayette to
Chicago has the same length as a trip to Paris, or to New Zealand! This
1
means that in addition to fuel eciency and saving the environment, we also
have substantial savings on printing the schedules:-)
Now let us nd the length of a one way trip. It is

k
=

q
(4=3)

:
Let us stop for a moment and think how amazing this formula is. It shows
that the time of travel depends only on density and the gravitational constant.
This means that on all planets made of the same material as the
Earth, the time will be the same!
I explained once how to calculate the denominator without going to a
library: the acceleration of gravity on the Earth surface is g = (4=3)
R,
where R is the radius of Earth, so (4=3)
= g=R = 9:8  (=2)  10−7,
because the circumference of Earth is 40 millions meters. Now we take a
calculator and come with the following answer: 2532 sec=42 min 12 sec. Not
very long, even for a trip to Chicago, not speaking of New Zealand.
The maximal speed of the train is also of some interest. It is reached in
the middle, when r = r0, and can be found from the Energy Conservation
Law:
v2
max =
4
3

(R2 − r2
0):
Thus
vmax =
s
gR − gr2
0
R
:
The largest speed will be reached of course by a train passing through the
center of the Earth, when r = 0, and vmax = 7900 m/sec. This is the so
called 1-st cosmic speed (why?!).
Remarks Thanks to Graham Light who corrected an error in the above
solution.
There is some evidence that these calculations were essentially known to
Hooke, who explained in a letter to Newton, how a body will move inside the
Earth if there is no resistance (Newton's rst guess about this was wrong,
though much later he claimed that he knew the Law of Gravity long before
the time of this correspondence. In XIX century a gravity Train Project was
seriously oered to Paris Academy of Sciences. Early in XX century this
2
was a popular topic in elementary mechanics textbooks. I learned the idea
from one such textbook, which I read in the middle 1960-s. In 1966 one
US physicist (Paul Cooper) rediscovered the Gravity Train and published a
paper in American Journal of Physics as a serious project of transportation
of the future. This was noticed by a journalist from the Time magazine, who
published an article (Time, 11 Feb., 1966, p. 42) representing the idea as a
science news. A heated discussion in American Journal of Physics followed,
but nobody mentioned that the idea is in fact about 400 years old.
3
There's a similar thread that was around not too long ago, with a link to the .pfd I just quoted.

yes but if we were to watch it happen what would we see?

i would see molten mantle squirting out of the tunnel due to pressure from the the inside of the earth.

but lets say it were possible, remember that if you were standing next to the hole, it would not seem like a vertical hole, rather a tunnel that slants into the earth (like a cave, tilt your head so one end of the tunnel is at the 12'oclock position). this is because to your perspective, you are always standing over the earths core. so it wont "drop" into the earth, but would have to slide or roll into the tunnel.

the object would be affected by gravity and therefore, assuming it can overcome any friction or able to roll freely, it would roll past the midpoint of the earth, then roll back as potential gravitation energy builds up again. the object would sort of move in simple harmonic motion (back and forth). if the situation is realistic, and there were friction, it would eventually come to rest in the midpoint of the tunnel

*Throws shovel*

Try it. =]

this would be incredible to witness
we should get a large drill and try it!

It would scrape it's way down the eastern wall of the tunnel, because the deeper it got, the slower the rotational speed of the earth is, and the object would continualy decelerate.

If the tunnel had an atmosphere, then it would hit terminal velocity, and continualy slow down as the atmosphere got denser and terminal velocity got slower.

In a vacuum, and frictionless environment on the tunnel walls, it would pop out the other side of the earth (assumming the same altitude) reaching a maximum height equal to the height it was dropped from, and fall back in the hole, in a neverending cycle.

It's a pretty interesting idea, if you made a hole from one point in the crust to another do you think it would take the same time for the object to reach the other side regardless of distance (as it would accelerate and decelerate slower)

It's a pretty interesting idea, if you made a hole from one point in the crust to another do you think it would take the same time for the object to reach the other side regardless of distance (as it would accelerate and decelerate slower)

Assuming that there is no friction, it would take 42 min 12 sec.