Past exam of the physics B course of the University of Cambridge / 2021 / Paper 1 / Section B / B7 / c

Set $q={\ell }_1/{\ell }_2$ and use normal mode amplitudes ${\varphi }_j=A_j{e}^{i\omega t}$, with real physical displacements obtained at the end. In units of ${\omega }_0^2$, the stiffness matrix isWriting $\lambda ={\omega }^2/{\omega }_0^2$, the eigenvalue condition isThereforeFor finite positive fibre lengths both eigenvalues are positive. The corresponding amplitude ratio isThe lower normal mode has the spheres twisting in the same sense, and the higher one has opposite signs of angular displacement.