In the rotating frame of a geostationary tether the effective gravity is g(r) = μ/r² − ω²r, zero at the synchronous radius r_s = (μ/ω²)^(1/3) ≈ 42,164 km (35,786 km altitude). Below r_s the tether hangs, above it the centrifugal term pulls up.
Uniform design stress σ requires a tapered cross-section: dA/A = (ρ/σ) g(r) dr, which integrates to A(r) = A_s · exp(−(ρ/σ)(Φ(r_s) − Φ(r))) with effective potential Φ = −μ/r − ω²r²/2. The taper ratio from surface to GEO depends only on the specific strength σ/ρ.
The counterweight balances tension at the top: M_cw = σ A(r_top) / (ω² r_top − μ/r_top²). Climber power at constant speed is P = m g(r) v; lifting energy to GEO is m(Φ(r_s) − Φ(R)). A payload released at r moves at ωr and escapes Earth if ωr ≥ √(2μ/r), i.e. beyond (2μ/ω²)^(1/3) ≈ 53,100 km.
Simplifications: 2D equatorial slice, rigid tether in a static effective field (no oscillations, Coriolis lateral loads, atmosphere, debris, or Moon/Sun perturbations), constant design stress, no safety factor beyond the chosen σ, climber mass ignored in the tether load. Materials values are illustrative design strengths, not product data.