Physics hanging around the Earth

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To determine the length of a rope hanging above the Earth, the relevant equation is T^2 = [(4(pi)^2)/(Re^2 * g)] (Re + L/2), where T represents the period of the rope's oscillation. The challenge lies in incorporating the mass of the Earth and the rope's density into the calculations, as the provided equation does not account for these factors. The discussion highlights the complexity of the problem, suggesting that additional considerations or equations may be necessary to solve for the rope's length accurately. Participants express the need for further clarification on how to integrate these variables into the solution. This problem exemplifies the intricate relationship between gravitational forces and physical properties in physics.
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Homework Statement


Someone once considered hanging a rope, (of density p), above the Earth so that it hangs slightly above the ground. How long does the rope need to be (length L)? (mass of the Earth = 5.98 * 10^24, Radius of the Earth = 6.37 * 10^6, angular velocity of Earth equals omega, and density p = .33 kg/m) The rope has both ends free.


Homework Equations


T^2 = [(4(pi)^2)/(Re^2 * g)] (Re + L/2)


The Attempt at a Solution


I thought you could just isolate L from the above equations (one of keplers laws). However, that equation doesn't use the mass of the Earth or the density of the rope. So there must be something I'm missing. Some help would be greatly appreciated.
 
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