Bottomless pit

[equazcion]

Let's say there were such a thing as a bottomless pit, a hole that went from one point on Earth clear through its core to the other side. Let's say you fell into it. What would happen to your body?

Would you:

A) slingshot past the core and emerge on the other side, breaking Earth's gravitational field and floating off into space?

b) go past the core only at first, then rebound back in the opposite direction, go past the core again, and keep rebounding back, like a plucked guitar string until you finally stopped at the very center?

c) or would the gravity at the core be so strong that you would simply stop once you got there, or be torn apart, or something?

d) or would something entirely different happen?

This is all assuming that your body didn't burn up due to the extreme heat and that the molten rock wasn't pouring all over the place to stop you from going anywhere. It's just a hypothetical that pops into my head from time to time so I thought I'd try asking someone. Let me know what you think.

[Pie]

This question was flogged to death during the "early" years of usenet. Aaah, the days of my youth.

Originally Posted by sci.physics

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

[Pie]

equazcion
Thinks "pie" is a funny word.
Join Date: Dec 2006
Location: New York City, NY, United States of America, Earth
Posts: 55

What do you have against pie, by the way??

[equazcion]

I had no idea -- thanks for the paste I don't have anything against pie. Do you have something against the words you find funny?

[Pie]

From the self-same thread on sci.physics, a more practical take:

Originally Posted by Paul Kwiat

In a very related question, suppose we drop a (teflon-coated) cat down this same tunnel. At what point will thecat turn around (since cats supposedly know to land on their feet)?
Before the cat was teflon-coated, the best answer was given by Prof. Ray Weiss at MIT, who pointed out that the center of such a tunnel, even if goes right through the center of the earth, is not a stable trajectory, and that given the slightest perturbation, the cat would "fall" to one side of the tunnel, and smear itself in a spiral down the sides of the tube.

[equazcion]

That teflon-coated cat with its heat-resistant life-support system is especially remarkable because I'm pretty sure teflon coatings need to be baked-on. Thanks again

PS - For those who care, the answer was choice B.

[Pie]

Originally Posted by equazcion

I had no idea -- thanks for the paste I don't have anything against pie. Do you have something against the words you find funny?

I'm funny how, I mean funny like I'm a clown, I amuse you? I make you laugh, I'm here to fuckin' amuse you? What do you mean funny, funny how? How am I funny?</goodfellas>

[equazcion]

Rita Rudner funny. </Family Guy>

[regular.joe]

..but not "ha ha" funny. /something I heard once.

[equazcion]

..but not "ha ha" funny. /something I heard once.

That comes from many places but my favorite is Ralph Wiggum in "Kamp Krusty", describing Barney as the phony Krusty.

[Happy Monkey]

It would be b), but the oscillation would be damped by your terminal velocity, so you probably wouldn't make it anywhere near the other side of the Earth.

Unless you managed to make the hole not only lava-, but also air-free.

[glatt]

Now, what about the rotation of the Earth?

If you are on the equator, you are traveling about 24,000 miles (approx. circumference of planet) in 24 hours (time for one revolution). This is 1,000 miles an hour. So you jump down into the hole and are moving at 1,000 mph in the horizontal direction. It's cool though, because at its surface, the Earth is revolving at 1,000 mph too. You match speeds. But as you get deeper into the hole, you are still moving horizontally at 1,000 mph, but the Earth is moving at less than 1,000 miles per hour. You are going to hit the walls of the hole, and leave a bloody streak. Your hole needs to be from one Pole to the other, or it won't work.

[TheMercenary]

Damm good point. I had not thought about that. But don't forget about precession.

http://en.wikipedia.org/wiki/Precession

[equazcion]

I dunno about that. For a skydiver, for a example, it takes very little time once he's out of the plane to lose forward momentum and drop straight down vertically. It's possible that by the time the rotational speed has slowed to the point where your momentum would throw you against the walls of the hole, you'd already have lost most of that momentum.

Also take into consideration the plunger effect. The confined airtight space of the tunnel (aside from its two openings) would act as its own transport for anything contained within it. As the Earth moves, the air within the tunnel would move with it, to some extent carrying objects within it along.

There's also lateral gravity to consider. While the main thrust of it would be pulling you towards the core, the mass of Earth on each of your sides would also exert force, carrying you along as it spins.

I think between all these different factors, I think there's a good chance you could freefall straight down to the Earth's core without hitting the sides, assuming your initial jump was properly centered within the diameter of the tunnel.

[glatt]

I was assuming a vacuum, but if there is air, then as Happy Monkey says, you would be slowed to terminal velocity and would only make it several hundred feet past the core, and no where near the other side of the planet.