Space elevator

NASA is planning to stretch a long cable from the equator into space, so that persons and loads can be transported to the orbiting height of the space station, and that the cable would always remain above the same point on Earth. Now, a short rope would fall under its own weight to Earth, but once its length exceeds a certain limit, the rope will start to support itself due to the centrifugal force.

What minimum length must such a rope have, approximately, so that it is just carrying itself?

Details and Assumptions:

  • The cable is not elastic, has a uniform density, and is tightly stretched without any curvature.
  • Set up the force balance for incremental rope pieces and integrate them over the length of the rope.
  • The circumference of Earth is about \(2 \pi R \approx 40,000 \,\text{km}.\)
  • Acceleration due to gravity g = \(10 \text{ m/s}^2.\)

Bonus question: What is the maximum tensile stress along the rope? Is there any known material that can resist these forces?

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