Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 355 / 1 / b / Solution

Choose tangent and normal vectorsso ${\hat{\mathbf{t}}}_s=-\kappa \hat{\mathbf{n}}$, ${\hat{\mathbf{n}}}_s=\kappa \hat{\mathbf{t}}$, and $\kappa ={\theta }_s$. Thenwhile the drag components are $f_n=-{\zeta }_{\perp }U\cos \theta$ and $f_t=({\zeta }_{\perp }U/2)\sin \theta$.

LetSince $N=A{\kappa }_s$, one has $\mathcal{N}=K^{\prime }$. Force balance becomesAt the free end, $\mathcal{T}(1)=\mathcal{N}(1)=K(1)=0$; at the held midpoint, position and orientation impose $\theta (0)=0$ in this convention. Reversing the normal reverses the corresponding signs without changing the shape.