Gravity At The Center Of The Earth

This may seem elementary to those of you educated in physics, but I'm not one of those, so I just have this humble question to ask:

Say there were a tube in your backyard that went all the way through the Earth to the polar opposite (somewhere in the Indian Ocean?). Now you jump into it and fall all the way through. What happens? Do you stop in the center of the Earth? Would you pass through the center and fall way out the other side and then repeat the process until the momentum wore out? If you eventually did stop in the center of the Earth, would you be very heavy? Or weightless? What else come into play as factors? Would it be fun? Are there any experiments that demonstrate what would happen?

OK, it was a few questions, can someone school me?

The Earth is not a uniform sphere, being denser in the middle, so the motion is only approximately Simple Harmonic motion, and the travel time is a bit lower than predicted by this simple model.

Math for the above

More links:
http://www.npl.washington.edu/av/altvw32.html
http://forum.physorg.com/index.php?showtop...ndpost&p=131138
The old topic on this forum: A hole through the earth and its effects?

This post has been edited by rpenner on Aug 28 2007, 05:41 PM

At the center of the earth, you'd be weightless. There's equal mass in all directions, and its gravitational pull cancels itself out.

Likewise, by a tricky implication of Calculus, the acceleration of gravity at any point on your descent is determined only by the total mass beneath your current distance from the center. In other words, as soon as you clear the first ten miles of descent, the outer ten miles' worth of earth's mass no longer contributes to the gravitational pull you experience.

Another weird effect, if the tube through the center of the earth is absolutely straight, then you'll be scraping along the east (?) side of the tube all the way down, due to the fact that your angular momentum at the surface must be conserved even as you fall, so you'd be going faster towards the east than the tube would rotate. (I tend to get those sorts of explanations backwards, but the phenomenon is real. Not sure about whether it's the east side or the west side.)

Of course, aerodynamic drag would slow your descent to whatever the terminal velocity would be at each point along the way, if you aren't falling through an evacuated tube. Terminal velocity is the speed where your weight is resisted by your aerodynamic drag, which is itself going to change as you descend. As you fall, there's less gravity pulling you, and the atmospheric density continues to increase, so while your terminal velocity at the surface of the earth is about 200feet/sec, that figure will continue to drop significantly as you get closer to the center of the earth.

Thanks for your replies, this question came to me while I was watching The Elegant Universe special on PBS. The part where it showed the Earth causing a bend or a groove in the fabric of space caused me to wonder where the bend would be and if it still was bent at the center of the earth, or any round object for that matter.

The 2D analogy of a weight on a rubber mat works here. The edge of the weight isi where the rubber slopes downwards the most (i.e. gravity is highest), but underneath the center of the weight it's as flat as if there was no weight at all.

For those that don't have the time to deal with GR, you can approximate curvature as Newtonian tidal forces. Inside any hollow, spherically symmetric body, Newton's shell theorem says the gravity is zero, so the differential gravity (tidal forces) is zero, and so the curvature is zero. This is true no matter how large the hole at the center is, so the curvature vanishes at the center of the earth.

It is a great question. I'm surprised no one yet has got into the nature of the air that would have filled the tube before the subject jumped. Would it not reach an incredibly high density as one fell closer the the center of the Earth? Would it not be pushed "down" by its own "wieght" from both ends? In fact, wouldn't the dynamics of this column of air be the overriding factor in the experience?

I'm no expert, but my guess is that the traveler(victim?) would soon pass out as the air became too dense. Further downward travel would slow and eventually crush the body well before the center of the planet. Travel would be slow; the body would take quite a while to get there, perhaps never, as it would reach neutral boyancy only a fraction of the way down.

But what would someone with knowledge speculate?

Joel

If you're going to take that into consideration you'd might as well consider the magma that would instantaneously fill the hole in. And if you could somehow stop that you'd have to come up with something extremely heat resistant to keep the traveler from burning to a crisp. But I highly doubt the earth has enough gravity to compress air to the point it would crush you. Twice as dense as the air we breathe is no denser than water.