In the Earth-fixed rotating frame a tether segment feels gravity mu/r^2 downward and centrifugal force w^2 r upward, so the net inward pull is g_eff(r) = mu/r^2 - w^2 r. It vanishes at the geostationary radius r_g = (mu/w^2)^(1/3) = 42,164 km. Below r_g the tether must carry the weight of everything under it; above r_g the net force is outward and the counterweight holds the cable taut.

For a cable that is equally stressed everywhere, tension T = sigma A obeys dA/dr = (rho g_eff / sigma) A, so A(r)/A(R) = exp[(Phi(r) - Phi(R)) / (sigma/rho)] with Phi = -mu/r - w^2 r^2/2. The taper ratio at GEO is exp(about 4.8e7 J/kg divided by the design specific strength) - about 5 for an ideal nanotube cable with a safety factor of 1.5, but more than a hundred million for Kevlar, which is why the idea needs a new material. The page uses sidereal rotation and the equatorial radius.

Release from height h gives an inertial speed w r; it exceeds escape speed sqrt(2 mu / r) above r = (2 mu / w^2)^(1/3), about 53,000 km radius. Simplifications: point-mass Earth (no J2), constant material properties, no atmosphere, wind, Moon or climber loading, and no oscillation modes of the cable.