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Author Topic: How does an object behave in a theoretical tunnel through the Earth's centre?  (Read 18521 times)

Offline Foolosophy

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Let's assume that is is practically possible to dig a tunnel through the center of the Earth.

How will a body behave if it fell into the tunnel from rest?

« Last Edit: 09/12/2010 22:31:15 by chris »


 

Offline QuantumClue

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You would fall towards the center at an increasing rate, slow down as you pass the center and maybe reach the other end due to your kinetic energy resulting in your speed... You would fall back if you did not grab hold of anything.
 

Offline JP

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This gets asked here a lot and the answer is what QC said--you would oscillate back and forth around the center.  Without air resistance you would keep going forever, popping up to the surface on one side, falling back to the other side and so on.

Looking at the picture I have another question: if the hole were narrow enough, would you hit the wall since you'd be rotating along with the earth's surface when you jumped in?
 

Offline QuantumClue

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That I don't have an answer to. I would imagine, the fall would not be too bumpy.
 

Offline yor_on

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I will now make the sound..

 Boiiiing -----> <---Boiiing---> <--Boiing--> <-boing etc.

And then a lonely "Allloh, anyone here?"
 

Offline syhprum

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Your transit time down to the other side and back is the same as the orbit time of a ground level satellite about 84 minutes.
This subject has been discussed many times on this forum.
 

Offline Foolosophy

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Your transit time down to the other side and back is the same as the orbit time of a ground level satellite about 84 minutes.
This subject has been discussed many times on this forum.
82.7 minutes to be precise
 

Offline JP

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I thought about it a bit, and I suspect you'd smack into the wall on the way down.  When you're on the surface, you're traveling in a circle at a constant velocity given by the earth's rotation.  Further down, the velocity that you'd have to move to stay in the hole would be smaller than at the surface, but since I don't think you have any way of "braking" yourself, you'd probably smack into the wall...
 

Offline maffsolo

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If gravity decreases as you descend to the center, your weight become less, your rate of decent will approach zero.
Also the affects like being in a space craft orbiting the earth.
 You will not hit the sides of the hole walls, just like the astronauts are not pinned to the aft section of the bulkhead during orbit.
What about how a plumb-bob works.
You will decelerate until you reach the center and stop and float indefinitely, or maybe when you reach the point where the earths radius, revolution per second  rate, matches the rate of descent.
« Last Edit: 07/12/2010 06:35:15 by maffsolo »
 

Offline JP

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I'm not sure I understand fully.  The tangential velocity of any section of the earth varies with radius, so it's highest at the surface and lowest near the center.  For you not to hit the walls, wouldn't your tangential velocity have to decrease as you fell?  I feel like I'm missing something obvious here.
 

Offline yor_on

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Maff?

If I'm free falling towards the middle my velocity should diminish like you think, I think :)But I would expect that you still would have a momentum taking you past that middle point? And I think JP is right, the earth is like a carousel, rotating. The fastest rotation at its surface just like the rim of that carousel. If you laid a path 'in' that carousel, from its rim to its center, and then let a ball roll to the center, would you expect it to keep to the middle of your path with the carousel spinning?
 

Offline Foolosophy

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....is the path or trajectory parabolic?
 

Offline yor_on

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In which frame :)
 

Offline yor_on

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 No, I don't think so? The trajectory should slow down as gravity becomes less but it should work from the middle of the 'shaft' to the wall, shouldn't it?

What would one see if one was a ant on the wall? (With binoculars)

Yes, ants have eyes too.
==

small ones.

The eyes I mean.
Blue?

Coriolis force too? Would it be noticeable?

« Last Edit: 07/12/2010 20:21:07 by yor_on »
 

Offline maffsolo

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Maff?

If I'm free falling towards the middle my velocity should diminish like you think, I think :)But I would expect that you still would have a momentum taking you past that middle point? And I think JP is right, the earth is like a carousel, rotating. The fastest rotation at its surface just like the rim of that carousel. If you laid a path 'in' that carousel, from its rim to its center, and then let a ball roll to the center, would you expect it to keep to the middle of your path with the carousel spinning?

yep I am having trouble wrapping my head around it.
I am visualizing revolution of a small wheel compared to a large wheel.
I see the small wheel spinning more frequently and big wheel slowly spinning
I know I am looking at this incorrectly, the same wheel speed slows to zero as you approach the absolute center, as if the hole was a wedge or rather cone shape... Thanks JP..


Since we are looking at a hole shaft with  parallel sides, that do not converge as the descend to the center!  The revolving speed is constant, wouldn't it take the same time to travel across the diameter of the hole, no matter the depth?  So wouldn't the rotational velocity in this case be the same with respect to the hole? Until the radius of the hole matches the distance to the center.

----
As far as having inertia and the momentum while approaching the center of the earth, acceleration becomes less and less by per second squared, your weight becomes less and less, so inertia will become less and less it will be like putting the breaks on
« Last Edit: 08/12/2010 01:50:37 by maffsolo »
 

SteveFish

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If the tunnel were between the exact north and south pole, and all air was removed from the tunnel, it would be as Quantum Que and JP's first posts just after the 0P. If the tunnel were from equator to equator, as an object fell it would have to decelerate in a direction parallel to the surface from 1,000mph at the surface to 0 at the center, and then accelerate back to surface speed again as it approached the opposite surface of the earth. I believe that this is a pure example of Coriolis force in action. This situation would require some sort of frictionless slide, such as maglev, for the object to make it all the way back to the surface.
« Last Edit: 08/12/2010 00:39:03 by SteveFish »
 

Offline Geezer

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If the tunnel were between the exact north and south pole, and all air was removed from the tunnel, it would be as Quantum Que and JP's first posts just after the 0P. If the tunnel were from equator to equator, as an object fell it would have to decelerate in a direction parallel to the surface from 1,000mph at the surface to 0 at the center, and then accelerate back to surface speed again as it approached the opposite surface of the earth. I believe that this is a pure example of Coriolis force in action. This situation would require some sort of frictionless slide, such as maglev, for the object to make it all the way back to the surface.

Good point about the pole to pole trip Steve.

In the equator case, does anyone know of a shortcut to compute the trajectory? It's a bit tricky as the accelerating force is not constant. Based on absolutely nothing at all, I have a hunch that the trajectory will coincide with the center of the shaft as the Earth rotates, but I'm trying to avoid doing the calculus required to find out.
 

Offline Foolosophy

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No, I don't think so? The trajectory should slow down as gravity becomes less but it should work from the middle of the 'shaft' to the wall, shouldn't it?



I am refering to its trajectory through space, wouldnt it be some sort of parabolic curve?

And if the entry point to the tunnel is not directly along the earths axis the trajectory would be 3D helical in nature too? Wouldnt it?
 

Offline Geezer

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I am refering to its trajectory through space, wouldnt it be some sort of parabolic curve?

And if the entry point to the tunnel is not directly along the earths axis the trajectory would be 3D helical in nature too? Wouldnt it?

Yes. As Steve pointed out, along the axis of rotation, the trajectory would be a straight line.

Equator to equator will be a bit more complicated due to the variation in gravity. The Coriolis forces will cancel, so the trajectory will be in a single plane. Can you figure out an equation for the trajectory that takes account of the reduction in gravitational attraction?

It's beyond me.
 

SteveFish

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Just to be clear. The pole to pole tunnel completely avoids the Coriolis force problem because surface speed due to rotation is 0. Speed of falling to the center will increase all the way but the rate of acceleration of this speed will taper off as the center is approached. The trip back out will be a mirror image of this.

On the equator to equator tunnel, Coriolis force will be maximal because the speed due to the earth's rotation is maximum there (about 1K mph I think) and it must decrease to 0 at the center and increase back to the original speed as it approaches the opposite surface. If the tunnel is somewhere between the pole and equator the surface speed from the earth's rotation will be something between the equator and poles depending upon latitude, but the center is obviously still 0.

The Coriolis force is a problem because it would be applied to the object free falling in the tunnel by the walls of the tunnel.

 

Offline Geezer

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On the equator to equator tunnel, Coriolis force will be maximal because the speed due to the earth's rotation is maximum there

(

That's true, but precisely at the Equator, the Coriolis force will consist of two components. Due to symmetry, these components will exactly cancel each other.
 

Offline syhprum

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There seems to be some confusion between acceleration and velocity ones acceleration would be at its highest at the Earths surface but the velocity would be at it highest as you passed thru the center, (you would have enough momentum to carry you up to the surface).
 

Offline Geezer

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There seems to be some confusion between acceleration and velocity ones acceleration would be at its highest at the Earths surface but the velocity would be at it highest as you passed thru the center, (you would have enough momentum to carry you up to the surface).

Yes. I entirely agree with you.
 

Offline Foolosophy

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I am refering to its trajectory through space, wouldnt it be some sort of parabolic curve?

And if the entry point to the tunnel is not directly along the earths axis the trajectory would be 3D helical in nature too? Wouldnt it?

Yes. As Steve pointed out, along the axis of rotation, the trajectory would be a straight line.

Equator to equator will be a bit more complicated due to the variation in gravity. The Coriolis forces will cancel, so the trajectory will be in a single plane. Can you figure out an equation for the trajectory that takes account of the reduction in gravitational attraction?

It's beyond me.

If an obeserver outside the earths boundaries tracked the bodies trajectory through the tunnel, that tracjectory would be dependent on where the tunnel is drilled.

THe earth is spinning through on its axis and moving in an orbit around the sun

The trajectory is not linear from the observers point of view - very complex when the tunnel isnt through the earths axis.

Its a complex spiral towards the earths center in some cases
 

Offline JP

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You're right.  It depends where you drill the hole.  I guess that's probably the answer to my question about hitting the walls.  You clearly won't hit them if you drill it between the poles.

I don't think it's necessary to worry about orbiting the sun unless you really want to complicate the problem.  For this problem, it should be a very good approximation that the earth is stationary, but rotating.
 

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