Transverse stiffness of a stretched spring ($k_{\perp }=T_0/L$)

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 $-T_{0}y/L$. 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.