Space Elevator Rope: Homework Solution & Equations

[LonePhysicist]

Homework Statement

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.

Homework 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)

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.)

 

[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?

 

[haruspex]

w^2*r*m = m*me*G/r^2 + T

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?

me = (4*pi*R^3*p)/(3*r^2)

If the left hand side stands for M_e, the mass of the Earth, why the division by r²?