Space elevator problem: how long is the cable?

[raving_lunatic]

Homework Statement

"Consider a cable, of fixed mass per unit length, which extends radially from just above the surface to a distance ηR measured radially from the center of the Earth. By determining the total force on the cable, or otherwise, find an expression and numerical value for η. At what point in the cable does the tension reach the maximum value?"

The cable is to be used as part of a space elevator, so it must rotate with the Earth.
This is pretty much all the information that the question gives.

Homework Equations

g = GM/r²

Radius of geostationary orbit obeys r³ = GM/ω²

The Attempt at a Solution

I think I'm approaching this problem in a way that's far too simplistic because of how many marks it's worth on the exam paper it was part of. I thought that to satisfy the condition of rotating with the earth, the CM of the cable must be at a geostationary orbit height, and given that the cable's CM will be halfway along its length, we can simply double that geostationary orbit height and add on Earth's radius to get the total difference from the Earth's center. Formally I guess I just equated the centrifugal force to the gravitational force, solved for when they were equal, and said that this had to be the center of mass for the cable.

Then, to evaluate the tension, it seemed like we can find the maximum just by considering where the magnitude of the resultant of gravity and the centrifugal force is maximised, i.e differentiating rw^2 - GM/r^2 with respect to r -- and this again just gave the result that tension is maximised at the cable's CM, which again I'm not sure is correct. Any help would be greatly appreciated as I feel like I've missed the point of the problem

 

[Chestermiller]

Try doing a differential force balance on the section of rope between r and r + dr. Let T(r) be the tension at r, and let T(r+dr) be the tension are r + dr. If the linear density of the rope is ρ, how much mass is there over the length dr? Besides the difference in tension, the other force on the differential mass is gravitational. This analysis will lead you to a differential equation for dT/dr. (Don't forget to include the ma term for the differential section of rope). Integrate it from r = R to the location where the tension is again zero. This will give you the value of η. I guess this analysis is supposed to be for a rope that's over the equator.

Chet