A tunnel through the earth.....

[LordOfTheGeeks]

I am not sure that this fits into this section of the forum, but its something i was thinking about last night while trying to fall asleep. This is hypothetical and not achieveable with current technologies, but if you were to dig a tunnel straight down so it would cross the exact center of the earth, then you completely removed all particles from the tunnel so its a complete vacumm. Then you proceed to drop a ball with a mass of, say 1 kg down it, would it forever fall from one side of the earth to the other. My reasoning is that at the surface of the earth the ball has a certain G potential relitive to the center of the earth, so when you drop it into the tunnel all of that would be converted to kinetic energy on the journey towards the center, then back into G potential as it reachs the surface again on the opposite side of the earth, then it would journy back to your side and so on. It would never lose any energy and will continue to "fall" though the earth forever. Is there anything that would stop the ball and how does this "perpetual" motion relate to the laws of thermodynamics.

[hhEb09'1]

Welcome to BAUT, LordOfTheGeeks!

That's an interesting question, that we've discussed here before. It turns out that the period of such oscillation is the same as if the ball were actually orbitting the earth--so that probably points to the answer of your last question. Thermodynamically, you can treat it like any other orbitting body in the solar system, cycling around.

[LordOfTheGeeks]

In this case then would gravity waves play a part, as far as i understand all orbiting bodies loose some energy (though very small amounts) through gravity waves.

[NEOWatcher]

Is there anything that would stop the ball and how does this "perpetual" motion relate to the laws of thermodynamics.

Well; hhEb09'1 got it in a nutshell. I tried to help by finding a good thread, but I can't seem to get a good search string to find it.

I just wanted to add, that the movement of the Earth is going to be another factor in considering the thought experiment.

[LordOfTheGeeks]

Well; hhEb09'1 got it in a nutshell. I tried to help by finding a good thread, but I can't seem to get a good search string to find it.

I just wanted to add, that the movement of the Earth is going to be another factor in considering the thought experiment.

I did think that the rotation on the earth could cause some problems, but what if inorder to eliminate that variable the tunnel is drilled through the rotational pole of the earth

[NEOWatcher]

I did think that the rotation on the earth could cause some problems, but what if inorder to eliminate that variable the tunnel is drilled through the rotational pole of the earth

At a high level, probably.
But; digging deep (pun intended), you probably will need to have a non-straight, or very wide tunnel to account for the slight variations in the Earth's non-uniform gravity, and for any relativistic effects of a moving gravity field.

[Saluki]

Digging through the pole would eliminate the rotational portion of problem, but not the motion of the Earth around the Sun, its wobbles caused by the moon, the motion of the Solar system through the galaxy, the motion of the galaxy in our local group, etc.

[LordOfTheGeeks]

Digging through the pole would eliminate the rotational portion of problem, but not the motion of the Earth around the Sun, its wobbles caused by the moon, the motion of the Solar system through the galaxy, the motion of the galaxy in our local group, etc.

It would be interasting to know how much of an effect these would actually have, also things like local density variations and diffrence in radius at diffrent point around the earth would also play a part.

[Argos]

But; digging deep (pun intended), you probably will need to have a non-straight, or very wide tunnel to account for the slight variations in the Earth's non-uniform gravity.

***WARNING***

Your thought experiment has been hijacked by reality!

I think an ideal, uniform, body is implicit in the thought experiment.

[NEOWatcher]

...I think an ideal, uniform, body is implicit in the thought experiment.

Look up one post (#8). The OP seems interested in the scale of those effects.

[cosmocrazy]

I would assume that the Earth's geographical centre and its actual centre of gravity are not quite the same thing. As already been suggested the mass distribution and density would be non-uniform so the hole would have to pass very accurately straight through the centre of gravity, which might not be a straight line between the 2 poles of rotation.

[LordOfTheGeeks]

I would assume that the Earth's geographical centre and its actual centre of gravity are not quite the same thing. As already been suggested the mass distribution and density would be non-uniform so the hole would have to pass very accurately straight through the centre of gravity, which might not be a straight line between the 2 poles of rotation.

isnt the pole of rotation a function of the center of mass... i am not sure but it seems logical that it would cross through the center of mass of the earth, i could be mistaken

[antoniseb]

isnt the pole of rotation a function of the center of mass... i am not sure but it seems logical that it would cross through the center of mass of the earth, i could be mistaken

The axis of rotation of a solid body does go through its center of gravity, but that doesn't mean that the axis goes through the center of the sphere, if the sphere is non-uniform. Normally we wouldn't worry about the non-uniformity of the Earth when talking about a theoretical tunnel through the center, but since you're asking about tiny effects, such as the emission of gravitational waves, and n-body problem issues, then you can't assume...

[hhEb09'1]

The axis of rotation of a solid body does go through its center of gravity, but that doesn't mean that the axis goes through the center of the sphere, if the sphere is non-uniform. Normally we wouldn't worry about the non-uniformity of the Earth when talking about a theoretical tunnel through the center, but since you're asking about tiny effects, such as the emission of gravitational waves, and n-body problem issues, then you can't assume...

No, not in this case. The surface (the "geoid") is an equipotential surface, so its center (the center of the sphere) is the same as the center of mass. So, nothing to worry about there.

[Gandalf223]

What about the moon? The earth and moon are a system (ignoring the effects of other planets which are pretty small) and the center of mass is not static. It moves within the earth as the moon orbits. There is no "point" where gravity is constant inside the earth, or on its surface, is there?

[Grey]

Here are two places where we've discussed this before, but it sounds like you're mostly interested in losses to the system. As some folks have pointed out, there will be some very slight losses from some effects. Friction would have been the largest damping effect, but you're assumng that is completely eliminated. Effects like gravitational radiation will be small enough that you probably couldn't measure them, except over a very long period of time.

The laws of thermodynamics are fine with that kind of "perpetual motion", just like they're fine with the Earth remaining in essentially the same orbit (after losses to friction and other effects) for billions of years.

[neilzero]

Having the axis of Earth's rotation inside the tunnel does solve the largest curved path problem. large diameter tunnel helps as the errors will mostly cancel long term, but not forever. Making the ball and tunnel wall perfectly elastic may reduce the effect of a rare bounce off the wall to negligible. There are minuscule tidal forces at work, so the length of the path will shorten by a few (many?) picometers each oscillation.
The wall temperature will be very hot producing a strong infrared photon flux, unless perfect vacuum also removes photons and gravitons. The latter possibly makes the energy loss to gravity waves negligible.
Each round trip takes about 2.7 hours as the average gravity field is about 0.6 g over the length of the tunnel? Neil

[hhEb09'1]

The wall temperature will be very hot producing a strong infrared photon flux, unless perfect vacuum also removes photons and gravitons. The latter possibly makes the energy loss to gravity waves negligible. l

Like a Hoover?

I'm not sure how that would be set up, even in a perfect world. Could you go into some detail there?

[Hop_David]

I am not sure that this fits into this section of the forum, but its something i was thinking about last night while trying to fall asleep. This is hypothetical and not achieveable with current technologies, but if you were to dig a tunnel straight down so it would cross the exact center of the earth, then you completely removed all particles from the tunnel so its a complete vacumm. Then you proceed to drop a ball with a mass of, say 1 kg down it, would it forever fall from one side of the earth to the other. My reasoning is that at the surface of the earth the ball has a certain G potential relitive to the center of the earth, so when you drop it into the tunnel all of that would be converted to kinetic energy on the journey towards the center, then back into G potential as it reachs the surface again on the opposite side of the earth, then it would journy back to your side and so on. It would never lose any energy and will continue to "fall" though the earth forever. Is there anything that would stop the ball and how does this "perpetual" motion relate to the laws of thermodynamics.

Here's a pic I did of this awhile back:


Such a tunnel from earth's north pole to south pole is an idle fantasy.

But not so with small or medium sized asteroids. You could travel from the asteroid's north pole to the south pole just by jumping in a hole. An object's orbital period (which is same as the tunnel round trip time) is dependent of the central body's density.

[hhEb09'1]

An object's orbital period (which is same as the tunnel round trip time) is dependent of the central body's density.

Isn't the orbital period independent of the density, and the round trip dependent?

[Gandalf223]

Isn't the orbital period independent of the density, and the round trip dependent?

Not in a vacuum.

[hhEb09'1]

Not in a vacuum.

That doesn't make sense to me. It certainly seems to make a difference at the end points of the distributions--say all the mass was concentrated at a point in the middle (the object would accelerate quickly to the center) or if it was concentrated in a shell (the object would quit accelerating as soon as it entered the shell, and take a long time to traverse the interior).

[grant hutchison]

I believe there's an unstated assumption. If the mass distribution of the sphere is uniform, the orbital period and tunnel oscillation period are equal and dependent only on the sphere's density.
If the mass distribution of the sphere is allowed to vary radially, then the orbital period is unaffected, but the tunnel oscillation period changes. For instance, if all the mass is concentrated at the centre of the sphere, the oscillation period falls to 1/sqrt(2) times the orbital period.

Grant Hutchison

[hhEb09'1]

Ah! thanks Grant, I knew there'd be something I was missing.

So, Hop_David's statement that I responded to earlier is referring to the fact that the period of the orbit of a sphere (at the surface of the sphere) does not depend upon its mass or radius, but upon its density.

In other words, if we had a one meter diameter titanium ball (with a little steel, so it matches the density of earth) in a little sheltered room of the bay of the shuttle, and started a tiny marble rolling on its surface, it would orbit the ball in about the same time as the shuttle orbits the earth.

[Hop_David]

Ah! thanks Grant, I knew there'd be something I was missing.

So, Hop_David's statement that I responded to earlier is referring to the fact that the period of the orbit of a sphere (at the surface of the sphere) does not depend upon its mass or radius, but upon its density.

In other words, if we had a one meter diameter titanium ball (with a little steel, so it matches the density of earth) in a little sheltered room of the bay of the shuttle, and started a tiny marble rolling on its surface, it would orbit the ball in about the same time as the shuttle orbits the earth.

Right. Sorry about the omission. In the idealized model I was assuming a sphere and uniform density. Practically speaking an asteroid can be nonspherical and is likely to have mascons.

In the illustration I posted, the black vector is the acceleration vector for the orbiting body. I broke it into two components, the red component running north and south. With the body falling through the tunnel, it only has one acceleration component which is the same as the orbiting body's north south component.

[Hungry4info]

Dropping a ball through such a tube going through the planet wouldn't quite be similar to that of an orbit.

In a circular orbit, the gravitational attraction between the two bodies is constant. Dropping a ball through the planet will have different effects. At the centre of Earth, all of Earth's mass is uniformly distributed around the second body, so all gravitational effects will cancel out.

I agree the ball should oscillate back and forth, but I don't think it would be quite as much as is being postulated here.

[Hop_David]

Dropping a ball through such a tube going through the planet wouldn't quite be similar to that of an orbit.

In a circular orbit, the gravitational attraction between the two bodies is constant. Dropping a ball through the planet will have different effects. At the centre of Earth, all of Earth's mass is uniformly distributed around the second body, so all gravitational effects will cancel out.

For convenience I'm reposting this image. The black vector is the orbiting body's acceleration vector. I broke this vector into two perpendicular components, the red component running north and south.

As I just mentioned, this idealized model assumes a sphere of uniform density.

With the tunnel object the blue component is zero since it's canceled out by equal masses pulling from the left and to the right. But the red component remains.

If you examine the tunnel object's acceleration vector, you will notice it goes to zero when the object reaches the center. At that point "all gravitational effects cancel out" just as you say.

[TheThorn]

Turns out it would take 42 minutes to go from one end of that tunnel to the other. The interesting part is that if you put the tunnel through the earth off center, and had frictionless rails for the "ball" to travel on, it would take 42 minutes to get to the other end, regardless of the angle. A tunnel from New York to Hong Kong, or London, or Philadelphia - 42 minutes for each trip.

see: http://www.time.com/time/magazine/ar...842469,00.html

[Hop_David]

Turns out it would take 42 minutes to go from one end of that tunnel to the other. The interesting part is that if you put the tunnel through the earth off center, and had frictionless rails for the "ball" to travel on, it would take 42 minutes to get to the other end, regardless of the angle. A tunnel from New York to Hong Kong, or London, or Philadelphia - 42 minutes for each trip.

see: http://www.time.com/time/magazine/ar...842469,00.html

For an earth orbit at 0 km altitude I get an 84.5 minute orbital period (not considering air friction, of course). Which corresponds to two tunnel trips: from north to south and then south to north. His results match mine pretty closely.

For some reason I've always imagined tunnels that don't lie on a line along the sphere's diameter to be brachiostones. I can't remember why.

On an asteroid, heat and pressure don't put a limit on how deep you can tunnel (well, maybe the bigger asteroids like Ceres might be hard to tunnel clear through). I would imagine tunneling to reach asteroidal resources would be a natural activity.

Straight line chords between various spots on a spherical surface suggest some pretty pictures.
For example, imagine 12 subway hubs on the surface, each corresponding to the face center of a Platonic dodecahedron:

The top tunnel system will take passengers to a next door neighbor hub.
The bottom tunnel system will take passengers to a neighboring hub "two doors down".

[joema]

Of course a tunnel straight through the earth's center isn't possible with today's technology. However magnetically-levitated trains which run in evacuated tunnels have been discussed for many years. Scientific American, August 1965, had an article on this: http://en.wikipedia.org/wiki/Gravity-Vacuum_Transit

It's theoretically possible to build one which accelerates to over 10,000 mph, following the earth's curvature but underground. At cruising speed passengers would be weightless -- similar to an orbiting vehicle but underground.

Such a vehicle could take passengers from New York to Los Angeles in about 20 minutes, likewise from New York to London.

Here's a more recent article in Popular Science about a similar but slower (4,000 mph) concept: http://www.popsci.com/scitech/articl...tlantic-maglev

As opposed to a tunnel straight through the earth (which is impossible) a hyper-speed underground maglev is merely expensive.