Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 14E / c / Solution

For equal masses $m$ and spring constants $k$, vertical displacements from equilibrium have stiffness and mass matrices
$$
K=k\begin{pmatrix}2&-1\\-1&1\end{pmatrix},
\qquad M=mI.
$$
The two squared frequencies and displacement ratios are
$$
\omega_\pm^2=\frac{k}{2m}(3\pm\sqrt5),
\qquad
\frac{\eta_2}{\eta_1}=\frac{1\mp\sqrt5}{2}.
$$
Gravity shifts the equilibrium but does not enter these vertical frequencies.


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 Incoming links: Vertical normal modes of two equal suspended masses