Earth Hole

[GamerColyn117]

In church the other day my friend and I were thinking of something profound. Taking physics for a long time I thought I would find a logical answer. We were thinking about something. If there was a hole trough the earth and someone jumped in, what would happen? Would they fall through and land on the other side? Would they fall through and continue into the sky towards space on the other side? Would they fall through and get halfway and be stuck in the middle of the magnetic pull (like the earth has gravity going one way for each degree of it and they would be stuck in the middle of them all)? I chose the latter of the choices due to an "educated guess". Does anyone have any input on our ideas or have ideas of their own? Please comment and help me figure this out. Thanks.

[ydoaPs]

We were thinking about something. If there was a hole trough the earth and someone jumped in, what would happen?

 

It depends where it was drilled. You'd either run into the sides immediately, or have some sort of harmonic motion until you run into the sides or equalize in the centre.

[GamerColyn117]

It depends where it was drilled. You'd either run into the sides immediately, or have some sort of harmonic motion until you run into the sides or equalize in the centre.

 

Our idea uses the exact center of the earth and the person not being able to melt/burn to death from the core.

[ydoaPs]

It still depends on where it was drilled. Through the equator? Pole to pole? Some other diagonal?

[Delta1212]

I'm assuming you're just asking about gravitational effects, so I'm going to answer in that vein.

 

The gravitational pull will decrease as you get closer to the center of the earth. This is because, as you "pass" more of the earth's mass on the way down, there is less and less mass below you and more above you canceling out its effect. At the very center of the earth, you would experience no gravitational pull, because for any direction you chose, there would be an equal amount of mass in the opposite direction canceling out the pull.

 

You would, however, still have all the momentum you built up from falling on the way down, so you would shoot past the center and begin to feel gravity increasing its pull on you back toward the center. You'd make it most of the way up the other side, but not all the way, before gravity managed to cancel out your momentum and you began moving back down to the center.

 

You'd wind up going back and forth along the tunnel going less and less far along the way back up at each end until you finally found yourself floating in the center in what would effectively be a zero gravity environment.

 

If you tried moving in either direction along the tunnel, however, gravity would slowly begin to reappear and pull you back to the center.

[elfmotat]

Assuming the hole was drilled right down the center of the Earth, there's no air resistance, rotational effects, etc. you would oscillate in simple harmonic motion (the same motion that a mass hanging on a bobbing spring would follow) with a period (time it would take to fall all the way through and come back to your initial position) of about:

 

[math]T=\sqrt{\frac{3\pi}{G\rho}}[/math]

 

, where ρ is the average density of the Earth. If you plug in the numbers, you get about 84 minutes. So it would take about 42 minutes to reach the other side. Interestingly enough, this is the same period that a satellite orbiting very close to the Earth's surface would have.

 

 

Your maximum speed (speed at the very center of the Earth) would be:

 

[math]v_{max}=2R\sqrt{\frac{\pi G\rho}{3}}[/math]

 

, where R is the radius of Earth. This works out to be ~ 17,700 mph.

Edited February 22, 2012 by elfmotat

[D H]

Assuming the hole was drilled right down the center of the Earth, there's no air resistance, rotational effects, etc. you would oscillate in simple harmonic motion (the same motion that a mass hanging on a bobbing spring would follow) with a period (time it would take to fall all the way through and come back to your initial position) of about:

 

[math]T=\sqrt{\frac{3\pi}{G\rho}}[/math]

 

, where ρ is the average density of the Earth. If you plug in the numbers, you get about 84 minutes.

That is assuming a uniform density Earth. That's a nice simplifying assumption for a freshman physics kind of problem, but not so good assumption of the real Earth.

 

This assumption results in gravitational acceleration dropping linearly with increasing depth. This is not what happens inside the Earth. The D" layer (the core/mantle boundary) is 2890 km below the surface. Per this simplifying assumption, gravitational acceleration at that depth would be a bit over half of surface gravity. That is off by a factor of two! Gravitational acceleration is 10.6823 m/s^2 at that depth.

 

A better model is that gravitational acceleration is constant at 10 m/s^2 from the surface down to halfway to the center of the Earth and then drops linearly from that point inward. This results in a 76.41 minute round trip as opposed to your 84 minutes. An even better model is to use the Preliminary Earth Reference Model (the source of that 10.6823 m/s^2 figure), but now you have to resort to numerical integration. I did this a while ago and got a 76.38 minute round trip.

 

 

See this post for references on the Preliminary Earth Reference Model.

[elfmotat]

That is assuming a uniform density Earth. That's a nice simplifying assumption for a freshman physics kind of problem, but not so good assumption of the real Earth.

 

This assumption results in gravitational acceleration dropping linearly with increasing depth. This is not what happens inside the Earth. The D" layer (the core/mantle boundary) is 2890 km below the surface. Per this simplifying assumption, gravitational acceleration at that depth would be a bit over half of surface gravity. That is off by a factor of two! Gravitational acceleration is 10.6823 m/s^2 at that depth.

 

A better model is that gravitational acceleration is constant at 10 m/s^2 from the surface down to halfway to the center of the Earth and then drops linearly from that point inward. This results in a 76.41 minute round trip as opposed to your 84 minutes. An even better model is to use the Preliminary Earth Reference Model (the source of that 10.6823 m/s^2 figure), but now you have to resort to numerical integration. I did this a while ago and got a 76.38 minute round trip.

 

 

See this post for references on the Preliminary Earth Reference Model.

 

Cool stuff, thanks for the info.

[JohnStu]

Yes the person would be dead though. From lack of oxygen and change of gravity. Did you know our digestive system need same gravity value to function properly, otherwise you cannot eat steak or digest protein well as the acid in stomach will flow around and mess up the whole thing.

Edited February 28, 2012 by JohnStu

[Cap'n Refsmmat]

Yes the person would be dead though. From lack of oxygen and change of gravity. Did you know our digestive system need same gravity value to function properly, otherwise you cannot eat steak or digest protein well as the acid in stomach will flow around and mess up the whole thing.

How do astronauts survive for six months in the International Space Station, then? They seem to digest food just fine.

[Narroo]

Robert Hooke and Isaac Newton Already thought about this.

 

If you removed friction, it would take 42 minutes to move from one end to the other, even if it doesn't pass through the earth's center.

Neato!

[space noob]

When you got to the center you would experienced zero gravity not to mention a lot if pressure that I don't think one could survive, you would get stuck

[zapatos]

When you got to the center you would experienced zero gravity not to mention a lot if pressure that I don't think one could survive, you would get stuck

Why so much pressure? Enough to kill you? And why would I get stuck?

[elfmotat]

When you got to the center you would experienced zero gravity not to mention a lot if pressure that I don't think one could survive, you would get stuck

 

Did you read any of the previous posts?

[questionposter]

I think due to possibly hitting the side of the tube as well as the Earth's shifting rotation and lack of perfectly equivalent gravity, an over-damped harmonic oscillator could model this problem and a person falling in would eventually come to a stop near the center. At first a person would approach the other side and then fall back, maybe even though to grab the ledge and get out, but otherwise they would just gradually lose momentum and come to a stop after swinging past the center over and over. Though I suppose it does also depend on the angle that the tube is placed. If it goes through the center of the Earth, then the oscillator can model it, but if it's at a angle and cuts through like only a quarter of the Earth, and you can probably just crawl through.

Edited May 8, 2012 by questionposter

[wucko]

this seems to be a FAQ i ve asked myself "how does gravity infact work" "would i be squeezed or stretched" http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/earthole.html

[pmb]

When you got to the center you would experienced zero gravity not to mention a lot if pressure that I don't think one could survive, you would get stuck

If the hole was depressureized then a person in freefall would not be able to tell if he was in free-fall in a gravitational field or at rest in an inertial frame of reference since, according to the Equivalence Principle, they'd be the same.

 

At best the person in free-fall could carry instrumentation in which was sensitive enough detect the gravitatidal forces in his frame. No person can detect such a small variation in the gravitational field.

Edited May 17, 2012 by pmb