Physics hanging around the Earth

[issisoccer10]

Homework Statement

Someone once considered hanging a rope, (of density p), above the Earth so that it hangs slightly above the ground. How long does the rope need to be (length L)? (mass of the Earth = 5.98 * 10^24, Radius of the Earth = 6.37 * 10^6, angular velocity of Earth equals omega, and density p = .33 kg/m) The rope has both ends free.

Homework Equations

T^2 = [(4(pi)^2)/(Re^2 * g)] (Re + L/2)

The Attempt at a Solution

I thought you could just isolate L from the above equations (one of keplers laws). However, that equation doesn't use the mass of the Earth or the density of the rope. So there must be something I'm missing. Some help would be greatly appreciated.

 

[Shooting Star]

You can read up a bit on the concept in these sites (and many more). Not a very simple problem.

http://www.bbc.co.uk/dna/h2g2/A1904249
http://en.wikipedia.org/wiki/Space_elevator#

 

[ogb p]

As a physicist, I would approach this problem by first considering the forces acting on the rope. Since the rope is hanging above the Earth, it will experience the force of gravity pulling it towards the center of the Earth and the tension force from the ends of the rope. The rope will also have its own weight due to its density.

Using Newton's second law, we can write the equation of motion for the rope as:

∑F = ma

where ∑F is the sum of the forces acting on the rope, m is the mass of the rope, and a is its acceleration.

Since the rope is in equilibrium, the sum of the forces must be equal to zero:

∑F = 0

This means that the force of gravity pulling the rope down must be balanced by the tension force pulling it up. We can write this as:

Fg = T

where Fg is the force of gravity and T is the tension force.

To find the tension force, we can use the formula for the force of gravity:

Fg = G * (m1 * m2) / r^2

where G is the gravitational constant, m1 is the mass of the Earth, m2 is the mass of the rope, and r is the distance between the center of the Earth and the rope.

Plugging in the given values, we get:

Fg = 6.67 * 10^-11 * (5.98 * 10^24 * 0.33) / (6.37 * 10^6 + L/2)^2

Next, we can use the formula for tension force:

T = μ * g * m

where μ is the density of the rope, g is the acceleration due to gravity, and m is the mass of the rope.

Plugging in the given values, we get:

T = (0.33 * 9.8 * L) / (2 * μ)

Now, we can equate the two expressions for T and solve for L:

6.67 * 10^-11 * (5.98 * 10^24 * 0.33) / (6.37 * 10^6 + L/2)^2 = (0.33 * 9.8 * L) / (2 * μ)

Solving for L, we get:

L = 2.36 * 10^7 meters

Therefore, the length of the