= Transverse stiffness of a stretched spring
{title2=$k_\perp=T_0/L$}
A taut <spring> of equilibrium <length> $L$ and initial <tension> $T_0$ whose ends acquire a small relative transverse displacement $y$ tilts through $y/L$. Its leading transverse <force> is therefore $-T_0y/L$. The <length> change is quadratic in $y$, so using only incremental axial extension misses this first-order <force>. A Hooke <spring> with natural <length> $l_0$ has $T_0=k(L-l_0)$ and $k_\perp=k(1-l_0/L)$. The unstretched limit has no linear transverse stiffness under this geometry.
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