# A rope in tension between Earth and Moon

• 8BitTRex
In summary: 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... not just one.
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?)

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?

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?

Khashishi said:
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?

CthlhuLies said:
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.

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?

CthlhuLies said:
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.

8BitTRex said:
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

Last edited:
CthlhuLies said:
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.

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.

dauto said:
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.

Last edited:
8BitTRex said:
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?

dauto said:
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?

8BitTRex said:
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.

CthlhuLies said:
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?

CthlhuLies said:
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.

dauto said:
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?

dauto said:

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.

8BitTRex said:
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.

8BitTRex said:
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

dauto said:
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.

8BitTRex said:
Would it always point radially between moon and COM of Earth or would it be deflected by some force?

It would be deflected.

8BitTRex said:
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.

PeterDonis said:
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?

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.

bahamagreen said:
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.

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.

Last edited by a moderator:
PeterDonis said:
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.

PeterDonis said:
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?

A.T. said:
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.

yuiop said:
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?

A.T. said:
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.

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.

A.T. said:
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.

mfb said:
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.

8BitTRex said:
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.

Nugatory said:
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?

Replies
3
Views
793
Replies
4
Views
1K
Replies
25
Views
1K
Replies
5
Views
4K
Replies
5
Views
1K
Replies
6
Views
2K
Replies
12
Views
2K
Replies
21
Views
2K
Replies
7
Views
7K
Replies
51
Views
6K