# Space Elevator Rope

1. Oct 20, 2015

### LonePhysicist

1. The problem statement, all variables and given/known data
Consider a long rope with uniform mass density extending radially from just above the surface of the earth to a radius of n'R. Show that if the rope is to remain above the same point on the equator at all times, then n' must be given by

n'2+n' = (8 pi G p)/(3 w^2) G is the gravitational constant, p is the density of earth.

2. Relevant equations
w^2*r*m = m*me*G/r^2 + T (what I assume to be true)
me = (4*pi*R^3*p)/(3*r^2)

3. The attempt at a solution
Tried finding dr/dm but having the issue of not knowing T because that varies with r. Also tried dT/dr / dT/dm but the algebra became too messy after eliminating T from the problem and the integrals that resulted make no sense, such as integrating mass without initial conditions and using density = m/l, this substitution did not yield promising results after integrating. Also taken consideration of finding the point of Maximum T by using dT/dr and setting that equal to 0, but the answer does not give insight into of n' (I assumed this was half the height originally, but this is not true from what I have found.)

2. Oct 20, 2015

### RUber

In your equation in part 2 of the template, what are you using as r and R?
Is that R= radius of earth and r=n'R= distance from center of earth to top of rope? Or vice versa?

3. Oct 21, 2015

### haruspex

Shouldn't that be a differential equation? I.e. consider an element of the rope height dr. What is the change in tension along it?
If the left hand side stands for Me, the mass of the Earth, why the division by r2?