= Uniform unloaded space-elevator cable
{title2=$\eta'(\eta'+1)=2\eta^3$}
A radial co-rotating cable of constant <linear mass density> $\lambda$ extends from planetary radius $R$ to $b=\eta'R$, with zero applied endpoint <tension>. If the <geostationary orbit> has radius $\eta R$, integration of the effective <force> gives $\eta'(\eta'+1)=2\eta^3$ for a nonzero cable. At position $r$ its <tension> is $\lambda[GM(1/R-1/r)-\omega^2(r^2-R^2)/2]$. Differentiation shows the maximum at the <geostationary orbit>. An anchor load or counterweight alters the integrated condition.
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