Below geostationary orbit gravity dominates and the tether hangs in tension; above it centrifugal force from Earth's rotation dominates and the cable is pulled outward by a fixed counterweight. For a uniform cable the tension per unit linear density (in J/kg, directly comparable to a material's specific strength) at radius r is the closed-form integral of that force balance:
T(r)/λ = GM(1/r − 1/r_max) + ½ω²(r² − r_max²)
where GM is Earth's gravitational parameter, ω its rotation rate, and r_max the counterweight's orbital radius (fixed here at 1.4× geostationary altitude). A climbing payload of mass m at radius r_c adds m(GM/r_c² − ω²r_c) to every point of cable below it.
The anchor's instantaneous tension does not track that quasi-static value exactly — the cable is elastic, so it behaves like a lightly damped oscillator settling toward it:
x'' = −2ζωₙx' − ωₙ²(x − T_target)
with a fixed 4-minute fundamental period (ωₙ) representing the cable's lowest longitudinal mode. The damping-ratio slider controls how fast ringing dies out after the climber's load changes or after an "orbital disturbance" impulse (a stand-in for a tidal/lunar perturbation or a passing debris tap). Climber transit is time-compressed so a multi-day real ascent plays out in about two minutes — the oscillator itself still runs on this page's own clock.