11.5: Potential energy of a torsion spring
A special type of couple is a torsion spring which generates a couple proportional to the angle by which it is rotated. For rotation of the torsion spring by an angle \(\phi\) with respect to its zero angle at \(\phi=0\) around the \(z\)-axis, its angle-dependent moment is:
\[\overrightarrow{\boldsymbol{M}}_{\text {couple }}=-\kappa \phi \hat{\boldsymbol{k}} \tag{11.28} \label{11.28}\]
where \(\kappa\) is the torsion coefficient, having units of \(\mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\). Substituting this into Equation 11.2 we obtain:
\[\begin{align} \Delta V_{\text {couple }} & =\int_{\phi_{F, 1}}^{\phi_{F, 2}} \kappa \phi \hat{\boldsymbol{k}} \cdot \mathrm{d} \phi \hat{\boldsymbol{k}} \tag{11.29} \label{11.29}\\[4pt] V_{\text {couple }} & =\frac{1}{2} \kappa \phi^{2} \tag{11.30} \label{11.30}\end{align}\]