Going up!

A space tether is a futuristic idea used commonly in science fiction. It consists of a long cable that extends from the surface of the Earth on the equator to an object anchored gravitationally or via propulsion in orbit around the earth. The cable rotates with the same period as the Earth so it always goes straight up from the surface of the Earth. If such a cable was made from one material, such as carbon nanotubes, and always had circular cross sections, then how far in meters above the center of the earth would the cable need to be thickest?

Details and assumptions

  • The cable must at least be able to support its own mass.
  • There are \(24~\mbox{hours}\) in \(1~\mbox{day}\).
  • The Earth is a perfect sphere of radius \(6370~\mbox{km}\) and mass \(6 \times 10^{24}~\mbox{kg}\).

Image source: BBC


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