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Author Topic: How does an object behave in a theoretical tunnel through the Earth's centre?  (Read 18518 times)

SteveFish

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In the 1960's there was a physics paper about a train that coasted through tunnels from one place to another and I remember reading about this in, I think, the science section of the Magazine of Fantasy and Science Fiction. It took me a while to find the correct search term, which is "Gravity Train." What I remembered was that you could connect any two places on earth with a straight tunnel, and if friction could be eliminated, the train would coast from one location to another, and get this, the time of 42 minutes 12 seconds is exactly the same for every possible tunnel. The speed at the center for the straight through tunnel in this thread would be 28440 km/h. The idea was originally presented to Isaac Newton in a letter by Robert Hooke.

This information and some good links are in a Wikipedia article-
http://en.wikipedia.org/wiki/Gravity_train

A recent article covering the Math-
http://www.docstoc.com/docs/566538/Gravity-Train-Project

Steve
 

Offline yor_on

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Sweet stuff
There's one on hyperphysics too
But without the train :(

Now for the next one..

"How long would the tunnel be?
Assume that the Earth is spherical and recall that latitudes range from 0° at the Equator to 90° N at the North Pole. Consider Dubuque, Iowa (42.50° N latitude), and Guatemala City (14.62° N latitude). The two cities lie on approximately the same longitude. Do not neglect the curvature of the Earth in determining the following.

If one could burrow through the Earth and dig a straight-line tunnel from Dubuque to Guatemala City, how long would the tunnel be?

From the point of view of the digger, at what angle below the horizontal would the tunnel be directed?"

Well, ahh?
 

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Offline Foolosophy

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who edited the title of this thread?
 

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Offline yor_on

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I would like

"A journey to the middle of the Earth?"

Or is that one taken?
 

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Offline Foolosophy

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I would like

"A journey to the middle of the Earth?"

Or is that one taken?


I recall wording the title as "lets tunnel to the center of the earth" what happened to that thread title?
 

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Offline yor_on

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Went for some fresh air?

Never mind, it's not the wrapping, it's the content that matters :)
 

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Offline Foolosophy

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Went for some fresh air?

Never mind, it's not the wrapping, it's the content that matters :)
true

But editing an authors title without permission is interference
 

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Offline JP

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If you look at the bottom of the first post, it will tell you who last edited it.  Right now, it says me, since I just tested changing your title and changing it back to see if my name showed up as editor.  Prior to me editing, it said you last edited it.  Prior to that, I don't know!

It might be that your post title wasn't in the form of a question--it's forum policy to phrase titles as questions, so a mod might have changed it to try to be helpful.
 

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Offline Foolosophy

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If you look at the bottom of the first post, it will tell you who last edited it.  Right now, it says me, since I just tested changing your title and changing it back to see if my name showed up as editor.  Prior to me editing, it said you last edited it.  Prior to that, I don't know!

It might be that your post title wasn't in the form of a question--it's forum policy to phrase titles as questions, so a mod might have changed it to try to be helpful.

I did edit the title back to the original one.

/i didnt realise you can see who edited last

why are you allowed to edit my post?
« Last Edit: 09/12/2010 03:30:13 by Foolosophy »
 

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Offline Geezer

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why are you allowed to edit my post?

er, because he's a forum moderator.
 

Offline syhprum

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I re did the calculation using G instead g and obtained 4.427 meters per sec which is much too low, the caculation quoted by SteveFish gives a more sensible value of 7.9Km/s which is the value derived by Hooke.
 

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Offline Foolosophy

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er, because he's a forum moderator.

A moderator should request permission from the author in order to edit the title of a thread.

I accept a moderator deleting a thread topic based of forum rules or language, insults, racism etc.

But to change the wording of a thread title purely based on some subjective reason or that they have free time on their hands is pathetic to say the least.

 

Offline QuantumClue

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Here's how you do it:

Force = -m M G r-2,
where m is your mass, M is the mass contained within a spherical shell under your feet, at any given depth, and r is how far you are from the center.

You can compute M=4/3πr3ρ,
where ρ is the mass density of the earth (you assume it to be uniform, but whatever... this is a back-of-the-envelope computation). 

ρ=Me/(4/3π R3),

where Me is the earth's mass.

Plugging all this back in

F=-mMeGR-3r.

All that junk to the left of r is constant.  This is identical to the force exerted by a spring, F=-kx, where k is given by all that constant stuff.  The period of a spring is 2π(k/m)-1/2, so the period of your oscillations through the center of the earth is
(MeGR-3)1/2

I get ~250 seconds transit time.  I might have made a slight mistake, but your motion through the earth is just like a spring.


ah so this place uses sub and superscripts. Good to know.
 

Offline syhprum

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The results from this thread seems to confirm that that the mathematical skills of our correspondents falls short of a Hooke or a Newton, or maybe they just dont care !
 

SteveFish

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I can't be critical about the math skills of others. My course through the biological sciences required only minimal math.
 

Offline Geezer

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I must admit I was thinking along the same lines as Syhprum. I was having a good giggle about it yesterday when I was walking the dogs. Fortunately, I live out in the country, so I don't meet too many people.

Breaking news! I took another look at my simulator (see Yoron's thread in Chat regarding free software) and, in theory, it should be possible to simulate the spring model that JP suggested and produce a simulation complete with animation and more graphs than you can shake a stick at. Yoron even thinks it can handle the Coriolis forces, but I don't think I'm up to that.

However, this is going to take a bit of figuring out, and probably some help from everyone, so don't expect any sudden breakthroughs.
 

Offline JP

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The results from this thread seems to confirm that that the mathematical skills of our correspondents falls short of a Hooke or a Newton, or maybe they just dont care !

If I had the math skills of Newton, I'd treat you all to pizza with my Nobel prize money!
 

Offline QuantumClue

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The results from this thread seems to confirm that that the mathematical skills of our correspondents falls short of a Hooke or a Newton, or maybe they just dont care !

If I had the math skills of Newton, I'd treat you all to pizza with my Nobel prize money!
No offense to Newton, he was beyond his own years, but his maths skills today would be somewhat lacking.
 

Offline Geezer

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The results from this thread seems to confirm that that the mathematical skills of our correspondents falls short of a Hooke or a Newton, or maybe they just dont care !

If I had the math skills of Newton, I'd treat you all to pizza with my Nobel prize money!
No offense to Newton, he was beyond his own years, but his maths skills today would be somewhat lacking.

I'd put it slightly differently. If it wasn't for Newton, all our math skills might be even more lacking than they presently are. I say this because he invented a lot of the math that we take for granted today  :D
 

Offline Geezer

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Wait a minute! I think the only possible way to do this without bouncing off the walls is with a tunnel through the Earth's axis (pole to pole). In the equator to equator case, I was thinking the trajectory would be naturally sort of 'S' shaped, so there was a chance the locus of the trajectory of the body would always lie in the tunnel, and that may turn out be true for the first equator to equator transit.

But here's the problem that I see. In order for the rabbit to pop back down the hole again for the return trip, it not only has to reverse it's direction towards the centre of the Earth, which it will because of gravity, but it also has to reverse its "forward" velocity component that it acquired from the Earth in the first place. It's not going to. That component is still the same as it was initially and nothing has happened, or will happen, that is going to change it.

So, it might be possible to make one transit, but the return trip will come to an abrupt halt as soon as it starts. (At least I think it will!)

Perhaps this is what Steve tried to explain to me.

 

Offline syhprum

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There will be no problem it will zip up and down indefinitely the potential energy that it has due to its position relative to the centre of the Earth will all have been converted to kinetic energy by the time it passes thru the centre but this will be re converted to potential energy as it climbs up to the surface.
As it arrives at the surface its speed will be zero and all that is required is a parking brake so that the passengers can alight.
I think I see your problem it could not be allowed to zip up and down empty it must be braked at the end of each trip so that the effects from the Earths rotation do not build up.
Maybe the equator to equator tunnel needs to be somewhat curved to compensate for the Earths rotation.  
« Last Edit: 10/12/2010 14:43:39 by syhprum »
 

Offline jartza

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