Gravitational Effects on a 1 Mile Length of Rope

[wolram]

If Earth could be considered the only body acting gravitationally on a length of rope, if the rope say a 1 mile length, is aligned in a straight line with the center of the Earth in some orbit, when the rope is free, will it tend to curve
to the shape of the orbit, or will it fall in on itself (end up in a heap).

 

[AlephZero]

The rope would curve was if the forces on different parts on the rope acted in different directions, and/or were of different of magnitudes.

Clearly if the rope was of a similar length to the distance between the Earth and the rope, this would be the case.

For a "1 mile length" rope, I think it depends exactly how you are imagining the system - i.e. what is the initial linear and angular velocity of the rope? An angular velocity of 1 rev/year for the rope would change the situation - think about it...

Also, if the Earth is in orbit round the sun, I think there are two sets of approximations you can make. In a short timescale, the orbiting could be ignored. In a longer timescale where the orbiting is significant, you need to include the gravitational effects of both the sun and the Earth on the rope, to get a consistent model of the physics.

 

[KingNothing]

if the rope say a 1 mile length, is aligned in a straight line with the center of the earth

What do you mean by this? The center of the Earth is a single point. Do you mean Earth's axis?

Could you draw a diagram?

 

[wolram]

What do you mean by this? The center of the Earth is a single point. Do you mean Earth's axis?

Could you draw a diagram?

I meant that if a line was taken along the length of the rope and projected towards earth, it would point to physical center.

 

[DaveC426913]

I do believe the forces acting on an object in orbit tend to stretch it along the radial axis and compress it along the transverse axis. Thus, the rope will end up stretched out pointing toward Earth.

 

[Voltage]

I'm not an expert on this, but a body in a low orbit circumnavigates the Earth in maybe 90 minutes, whilst a body in a high orbit takes longer. Your rope is rather like two objects, one in a low orbit and one in a high orbit, so it sounds like the lower portion is going to slew forward of the upper portion.

↑
— O


\→
O

But I don't know to be honest.

 

[D H]

For a rigid rod, it is easy to show that the vertical (radial) orientation is a stable equilibrium while the horizontally orientation is an unstable equilibrium. For an orientation slightly off from the vertical or the horizontal, the perturbation forces that force the rigid rod to a vertical orientation are tensile. Thus the radially-aligned rope is in a stable equilibrium and the horizontally aligned rope is in an unstable equilibrium.

My intuition (caveat: the behavior of objects on orbit is often counter-intuitive) tells me the "rope in a ball configuration" is also an unstable equilibrium. Consider a ball of rope with one end protruding. The gradient in the gravitational field will pull that protuding end vertically away from the ball. This is tensile, so the rope will unravel into a vertical orientation.

 

[wolram]

Quite different from what i thought, thanks.