A rope in tension between Earth and Moon

[8BitTRex]

I attatch a rope of a constant density to the Moon and fasten it down to the surface. I set the other end in space right above Earth's atmosphere. The angular speed of the entire rope is brought up to the same angular speed as the Moon at time=0.

http://i.imgur.com/kHlVvnM.png

What would happen to the rope? Would it stay pointing radially between moon and COM of earth? Why or why not?

My gut says that there would be some force that would cause the rope to not point radially from some gravitational dragging effect but I would love some insight.

Assumptions:

no interaction from atmosphere (ideal vacuum)
mass of the rope does not effect the rotation of either planet
moon does not precess
moon in circular orbit(would it matter?)

 

[Khashishi]

Tension in the rope would add a small attractive force between the Earth and the moon. There's some minimum tension, because one end of the rope falls toward the Earth, and one end of the rope falls toward the moon. Since the Earth pulls harder, the end of the rope on the moon will get pulled off the surface by the rope's tension, unless it were attached to the moon somehow.

Are you interested in frame-dragging effects? Or just Newtonian gravity?

 

[CthlhuLies]

Wouldn't the rope tear? I mean Earth's gravitational pull is stronger are we assuming the rope is strong enough to withstand the tension? Also is it attached to the surface or just touching?

 

[8BitTRex]

Tension in the rope would add a small attractive force between the Earth and the moon. There's some minimum tension, because one end of the rope falls toward the Earth, and one end of the rope falls toward the moon. Since the Earth pulls harder, the end of the rope on the moon will get pulled off the surface by the rope's tension, unless it were attached to the moon somehow.

Are you interested in frame-dragging effects? Or just Newtonian gravity?

I said in the problem statement that I fastened 1 end of the rope to the moons surface.

I was curious what would happen to the rope. I wanted to assume that the rope had mass to feel the effect of gravity, but that the tension in the rope would not signifacantly alter the orbital trajectory of the idealized moon(no precession and circular orbit).

Would the rope always point toward the COM of Earth or is there some force that would deflect it?

 

[8BitTRex]

Wouldn't the rope tear? I mean Earth's gravitational pull is stronger are we assuming the rope is strong enough to withstand the tension? Also is it attached to the surface or just touching?

It is attached to the surface of the moon and the rope has infinite tensile strength.

 

[CthlhuLies]

Not sure how strong it would be but depending on the mass of the rope centrifugal force would just push it back to the moon. If the gravity of the Earth was less then the centrifugal then the rope would go back to the moon I believe, However if was more massive the gravitational force would overcome the centrifugal force correct?

 

[8BitTRex]

Not sure how strong it would be but depending on the mass of the rope centrifugal force would just push it back to the moon. If the gravity of the Earth was less then the centrifugal then the rope would go back to the moon I believe, However if was more massive the gravitational force would overcome the centrifugal force correct?

Yes, But... The gravitational force on the rope would always be greater than the centrifugal force on the rope. If this were not the case, the the moon would be pulled away from the Earth by its large orbital velocity.

 

[CthlhuLies]

Yes, But... The gravitational force on the rope would always be greater than the centrifugal force on the rope. If this were not the case, the the moon would be pulled away from the Earth by its large orbital velocity.

There is evidence that suggest that the moon is pulling away from out orbit though.
http://lro.gsfc.nasa.gov/moonfacts.html CTRL-F search for drift or 1.5

 

[8BitTRex]

There is evidence that suggest that the moon is pulling away from out orbit though.
http://lro.gsfc.nasa.gov/moonfacts.html CTRL-F search for drift or 1.5

I am making the assumption of a circular lunar orbit of constant radius over time.

 

[dauto]

The answer is very simple. Under the idealized conditions as described by the problem the rope would just stay there under incredible tension. You might find googling "space elevator" worthwhile.

 

[CthlhuLies]

The answer is very simple. Under the idealized conditions as described by the problem the rope would just stay there under incredible tension. You might find googling "space elevator" worthwhile.

This is different then a space elevator because it is under gravity by both the moon and earth. A space elevator uses the centrifugal force to overcome gravity pressing down on it, meaning the problem he is stating can only be achieved using the two celestial bodies in question however a space elevator can be attained with only one body.

 

[CthlhuLies]

I am making the assumption of a circular lunar orbit of constant radius over time.

Even so with the current velocity of the moon and the gravity of Earth It would head back to the moon due to centrifugal force, but are you assuming that there are the right conditions for the rope to overcome the centrifugal force and not drift due to centrifugal force?

 

[8BitTRex]

The answer is very simple. Under the idealized conditions as described by the problem the rope would just stay there under incredible tension. You might find googling "space elevator" worthwhile.

Since I have only taken classical mechanics, that is the answer that I arrived at. Are you certain that there is no effect from General Relativity that would cause the rope to deflect?

 

[dauto]

Since I have only taken classical mechanics, that is the answer that I arrived at. Are you certain that there is no effect from General Relativity that would cause the rope to deflect?

There is frame drag due to the rotation of the Earth. Too small to worry about.

 

[dauto]

Even so with the current velocity of the moon and the gravity of Earth It would head back to the moon due to centrifugal force, but are you assuming that there are the right conditions for the rope to overcome the centrifugal force and not drift due to centrifugal force?

You're forgetting about Earth's gravity

 

[8BitTRex]

Even so with the current velocity of the moon and the gravity of Earth It would head back to the moon due to centrifugal force, but are you assuming that there are the right conditions for the rope to overcome the centrifugal force and not drift due to centrifugal force?

The total centrifugal force on the rope is waaaaay weaker than the force from Earth's gravitational field.

 

[8BitTRex]

There is frame drag due to the rotation of the Earth. Too small to worry about.

Too small to worry about? But that is the whole point of my question. If there is an effect, however small, then the rope would not point radially.

So I guess the answer to my question is...

No, the rope would not point toward the COM of the Earth. Because there would be a small force(very small) from frame dragging, which would cause a non-radial force on the end of the rope. This force would cause a small deflection which would cause the rope to point somewhere other than the COM of Earth.

Is that correct?

 

[CthlhuLies]

You're forgetting about Earth's gravity

We would have to observer the rope, Earth's current gravity the moon is escaping, however would the rope escape the same way the moon is or since it is closer would it stay? It might stay but if that was true wouldn't the moon be staying as well. since the speed and distance is proportional to the speed and distance at both ends of the rope, would the stronger gravity at the end not attached to the moon be enough to keep it in orbit or would the tip come towards the moon like the moon is, that depends on the length of the rope I guess we assume that the length is long enough for gravity to overcome the centrifugal force.

 

[CthlhuLies]

Too small to worry about? But that is the whole point of my question. If there is an effect, however small, then the rope would not point radially.

So I guess the answer to my question is...

No, the rope would not point toward the COM of the Earth. Because there would be a small force(very small) from frame dragging, which would cause a non-radial force on the end of the rope. This force would cause a small deflection which would cause the rope to point somewhere other than the COM of Earth.

Is that correct?

Also there is the fact that there is a small push on the rope from the sun, the rope would act as a small solar sail.

 

[dauto]

Too small to worry about? But that is the whole point of my question. If there is an effect, however small, then the rope would not point radially.

So I guess the answer to my question is...

No, the rope would not point toward the COM of the Earth. Because there would be a small force(very small) from frame dragging, which would cause a non-radial force on the end of the rope. This force would cause a small deflection which would cause the rope to point somewhere other than the COM of Earth.

Is that correct?

Yes, I think so.

If that's the point of the question than you should've asked it at the relativity forum

 

[8BitTRex]

Yes, I think so.

If that's the point of the question than you should've asked it at the relativity forum

Thanks I'll do that.

 

[PeterDonis]

Would it always point radially between moon and COM of Earth or would it be deflected by some force?

It would be deflected.

My gut says that there would be a force from frame dragging

There would be a very small force due to frame dragging, caused by the Earth's rotation, but this would be much too small to measure.

However, you've ignored a much bigger force: since you've forced the rope to have the same angular velocity everywhere at time t = 0, different parts of the rope, at different altitudes, will have different *linear* velocities, so there will be stress in the tangential direction (i.e., perpendicular to the rope) between any two neighboring pieces of the rope at time t = 0. In Newtonian terms, this is called the Coriolis force. Since the lower end of the rope (the end closest to the Earth) is free to move, it will do so in response to the Coriolis force, deflecting the rope.

 

[8BitTRex]

It would be deflected.
There would be a very small force due to frame dragging, caused by the Earth's rotation, but this would be much too small to measure.

However, you've ignored a much bigger force: since you've forced the rope to have the same angular velocity everywhere at time t = 0, different parts of the rope, at different altitudes, will have different *linear* velocities, so there will be stress in the tangential direction (i.e., perpendicular to the rope) between any two neighboring pieces of the rope at time t = 0. In Newtonian terms, this is called the Coriolis force. Since the lower end of the rope (the end closest to the Earth) is free to move, it will do so in response to the Coriolis force, deflecting the rope.

Which way would the stress deflect the rope? And would it oscillate?

 

[bahamagreen]

Here is how I'm thinking about it...

Average lunar distance LD = 238,900 miles
Rope's center of mass = 119,450 miles from Earth
Lagrange point L1 = 200,900 miles from the Earth

84% of the rope's mass is on the Earth side of L1
(good reason to fasten it down to the Moon's surface)

Orbital period of L1 = the Moon's orbital period
Rope between L1 and Earth will try to progressively decrease orbial period with approach to Earth, which it can't do, so it will curve "forward" of the line between the Earth and Moon.

Rope between L1 and Moon will try to increase orbital period with approach to Moon, which it can't do, so it would tend to make the rope fall "behind" the line between the Earth and Moon; but the fixed end fastened to the Moon surface is holding that to form a tight curve.

Once the rope finds its static shape, I think with respect to your diagram, the rope will look like a "sword"... pointing left and down, the long part on the Earth side of L1 will be like the blade and curve gradually "up" as it approaches the Earth, and the small section between L1 and the Moon will curve "down" in a half loop like the knuckle guard on the hilt.

 

[8BitTRex]

Once the rope finds its static shape, I think with respect to your diagram, the rope will look like a "sword"... pointing left and down, the long part on the Earth side of L1 will be like the blade and curve gradually "up" as it approaches the Earth, and the small section between L1 and the Moon will curve "down" in a half loop like the knuckle guard on the hilt.

If I understand your odd description correctly, is it like this picture?
http://i.imgur.com/Y08NsU7.png

I'm having a hard time picturing it.

 

[yuiop]

NASA conducted an experiment with a satellite tethered to a shuttle by a 20 kilometre cable that might be of interest in the context of your question. See this video that describes the dynamics of the tethered system You might want to skip forward 16 minutes into the video for the most relevant part.

 

[A.T.]

you've forced the rope to have the same angular velocity everywhere at time t = 0,

So at t=0 the entire rope is at rest in the rotating rest frame of Moon and Earth's center.

Since the lower end of the rope (the end closest to the Earth) is free to move, it will do so in response to the Coriolis force, deflecting the rope.

Coriolis forces exist only in rotating frames, and act only on objects which move relative to that rotating frame.
In the rotating rest frame of Moon and Earth's center the entire rope is at rest, so there are no Coriolis forces on the rope in this frame.

Which rotating frame are you considering to get Coriolis forces on the rope, perpendicular to the rope?

 

[yuiop]

Coriolis forces exist only in rotating frames, and act only on objects which move relative to that rotating frame.
In the rotating rest frame of Moon and Earth's center the entire rope is at rest, so there are no Coriolis forces on the rope in this frame.

This is demonstrated in the video I linked above. The Coriolis force that creates a torque on the combined shuttle and tethered satellite system, only appears when the satellite is reeled in or out.

 

[A.T.]

The Coriolis force that creates a torque on the combined shuttle and tethered satellite system, only appears when the satellite is reeled in or out.

Yes obviously, if the rope moves in the rotating frame, there are Coriolis forces. But the OP assumes the rope is initially at rest in the rotating frame. Why would there be any Coriolis forces, that deflect the rope?

 

[yuiop]

Yes obviously, if the rope moves in the rotating frame, there are Coriolis forces. But the OP assumes the rope is initially at rest in the rotating frame. Why would there be any Coriolis forces, that deflect the rope?

I was agreeing with your argument.

 

[mfb]

If we start with the extended rope as described, neglect the influence of the sun and assume a circular orbit for earth, all forces are radial and the rope stays as it is, orbiting Earth once per ~27 days.

Relativistic effects are negligible here, so I'm not sure if this is the right forum.

 

[PeterDonis]

Coriolis forces exist only in rotating frames, and act only on objects which move relative to that rotating frame.

You're right, Coriolis force is the wrong term for what I was thinking of. I was thinking that the initial condition of the rope, with different tangential velocities (linear velocities, not angular velocities) at different heights, would prevent the rope from remaining in purely radial alignment. But I need to think about this some more.

 

[8BitTRex]

If we start with the extended rope as described, neglect the influence of the sun and assume a circular orbit for earth, all forces are radial and the rope stays as it is, orbiting Earth once per ~27 days.

Relativistic effects are negligible here, so I'm not sure if this is the right forum.

Are they negligible(veeeery small) or are they non existant? I was wondering if there was any force that would deflect the rope.

 

[Nugatory]

Are they negligible(veeeery small) or are they non existant? I was wondering if there was any force that would deflect the rope.

Relativstic effects may be "negligible", but never "non-existent" except in the special case of a closed system within which all relative velocities are zero, no the case here.

However, the problem you're describing (dangle an ideal rope from the moon towards the earth, figure what shape it will assume) is purely classical in the sense that the relativistic effects are negligible.

I expect that you could solve it by considering the forces at work on any infinitesimal section of the rope (tension in two directions, gravitation in a third). In steady state, that section of the rope will experience zero tangential acceleration, whatever radial acceleration is consistent with a circular trajectory, and the distance along the rope from that section to the moon will be constant - this is an ideal rope of zero thickness, non-zero mass, infinite tensile yield strength and infinite modulus of elasticity.

I also expect that the solution will be somewhat hairy , but I don't need to solve it to know that the solution is not going to be the rope hanging straight up and down - that follows just from considering the forces on the section at the free end of the rope.

 

[8BitTRex]

I also expect that the solution will be somewhat hairy , but I don't need to solve it to know that the solution is not going to be the rope hanging straight up and down - that follows just from considering the forces on the section at the free end of the rope.

What non-radial force are you thinking about, in this Classical and ideal problem?

 

[dauto]

If we start with the extended rope as described, neglect the influence of the sun and assume a circular orbit for earth, all forces are radial and the rope stays as it is, orbiting Earth once per ~27 days.

Relativistic effects are negligible here, so I'm not sure if this is the right forum.

That question started elsewhere and I told the author to post it here because after I stated that relativistic effects were negligible the author said the relativistic effects were the whole point of the question. Those effects, negligible as they may be, are exactly what (s)he is trying to understand.

 

[PAllen]

So, we can rephrase the OP as: given a classical solution, what corrections would there be due to GR, and what would their order of magnitude be?

There are at least two corrections: to the inverse square law, and due to Earth rotation (frame dragging). My guess would be GR corrections to inverse square law (though exceedingly tiny) would be larger than those due to frame dragging. But I have computed nothing in making this statement.

 

[A.T.]

My guess would be GR corrections to inverse square law (though exceedingly tiny) would be larger than those due to frame dragging.

Would the corrections to inverse square law deflect the rope perpendicularly to the rope, or just affect the stresses in the rope.

 

[Nugatory]

No, the rope would not point toward the COM of the Earth. Because there would be a small force(very small) from frame dragging, which would cause a non-radial force on the end of the rope. This force would cause a small deflection which would cause the rope to point somewhere other than the COM of Earth.

Is that correct?

If I understand you properly, one end of the rope is tethered at the moon and the other end is free and reaching all the way down to the top of the Earth's atmosphere. If that's the problem, the rope will NOT hang straight up and down, and this has nothing to do with any relativistic effects - it's a pure classical problem.

 

[mfb]

Are they negligible(veeeery small) or are they non existant? I was wondering if there was any force that would deflect the rope.

I guess frame-dragging gives a small tangential deviation. The deviations from the inverse square law are still radial forces.

However, keep in mind this is a purely hypothetical setup. In our solar system, nearly everything else would lead to larger deviations - the non-circular orbit of moon, the thin upper atmosphere of earth, the non-spherical mass distribution of earth, gravitational forces from the sun, sunlight pressure, gravitational effects from some other planets, maybe even non-uniform thermal radiation from the cable, ...

@Nugatory: The ideal, non-relativistic cable has an analytic solution for the tension.

 

[8BitTRex]

If that's the problem, the rope will NOT hang straight up and down, and this has nothing to do with any relativistic effects - it's a pure classical problem.

Can you explain what would cause it to not hang straight? What non-radial force component is there in this problem? I can't think of anything that would cause the rope to deflect.

 

[PeterDonis]

I guess frame-dragging gives a small tangential deviation.

Only if there is radial motion. If each small piece of the rope remains at a constant radius, all frame-dragging will do is slightly change the radial proper acceleration required to hold the piece of the rope at its current radius. This was the topic of a thread on PF some time ago, I'll see if I can dig up the link.

 

[mfb]

Only if there is radial motion. If each small piece of the rope remains at a constant radius, all frame-dragging will do is slightly change the radial proper acceleration required to hold the piece of the rope at its current radius. This was the topic of a thread on PF some time ago, I'll see if I can dig up the link.

Ah okay.

 

[8BitTRex]

I guess frame-dragging gives a small tangential deviation. The deviations from the inverse square law are still radial forces.

However, keep in mind this is a purely hypothetical setup. In our solar system, nearly everything else would lead to larger deviations - the non-circular orbit of moon, the thin upper atmosphere of earth, the non-spherical mass distribution of earth, gravitational forces from the sun, sunlight pressure, gravitational effects from some other planets, maybe even non-uniform thermal radiation from the cable, ...

yeah I understand that. I thought it was an interesting problem, however unrealistic, to develop my intuition, thanks for the reply.

 

[PAllen]

Would the corrections to inverse square law deflect the rope perpendicularly to the rope, or just affect the stresses in the rope.

I'm thinking the perfect, non-stationary, two body system is not stable in GR (whatever initial non-stationary initial conditions you assume, there cannot be exact stability in GR). GW would cause orbiital decay, thus change of angular speed. This would produce tangential effects on the rope. The GW themselves also would produce tangential effects. All of this would be mind-bogglingly small, but that seems to be the intent of the question - to think about a perfect setup in an isolated universe.

 

[bahamagreen]

If I understand your odd description correctly, is it like this picture?
http://i.imgur.com/Y08NsU7.png

I'm having a hard time picturing it.

You have the curve lagging behind; I think it would lead forward (just rotate the drawing 180 degrees around the line between the Earth and Moon)... the leading curve is on the same side as the direction of rotation in your picture.

There is a small curve between the Moon and L1 in the lagging direction.

In general, I think the dominant influence on the rope at any point along its length is the difference between the orbital period of a free mass at that altitude versus the constraint suffered by the rope to be at that altitude and limited to the orbital velocity of the Moon.

Below the Lagrange point L1, the rope wants to orbit the Earth progressively faster with decreasing altitude, and will try to do so, forming a curve in front of the Earth-Moon line.

 

[8BitTRex]

In general, I think the dominant influence on the rope at any point along its length is the difference between the orbital period of a free mass at that altitude versus the constraint suffered by the rope to be at that altitude and limited to the orbital velocity of the Moon.

Why would the orbital period of a free mass create a force on the suspended rope? If angular momentum is conserved, wouldn't forces be radial? What would cause the tangential influence you are discussing?

 

[PeterDonis]

there would be a small force(very small) from frame dragging, which would cause a non-radial force on the end of the rope. This force would cause a small deflection which would cause the rope to point somewhere other than the COM of Earth.

Is that correct?

No. Your initial condition specified no radial motion, and in that case frame dragging does not add any tangential motion; it only changes the radial force required to hold a piece of the rope at constant altitude. See my previous post in response to mfb.

 

[8BitTRex]

No. Your initial condition specified no radial motion, and in that case frame dragging does not add any tangential motion; it only changes the radial force required to hold a piece of the rope at constant altitude. See my previous post in response to mfb.

To clarify, because the answers seem to vary...

The rope would ALWAYS point radially (in my ideal problem), classically and relativistically, because there is no tangential force that would act on it??


Is that right?

 

[PeterDonis]

The rope would ALWAYS point radially (in my ideal problem), classically and relativistically, because there is no tangential force that would act on it??

I think there's general agreement that, if the classical solution is indeed an equilibrium with the rope oriented purely radially and at rest in the rotating frame, there are no relativistic effects that would introduce tangential motion.

I'm not sure there's general agreement that the classical solution is in fact an equilibrium with the rope oriented purely radially and at rest in the rotating frame.