Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / Section B / B1 / Solution

A normal mode is a small motion in which all coordinates oscillate at one common angular frequency with fixed relative amplitudes. For conservative linearized motion, these are eigenvectors of the mass-weighted stiffness matrix.

All three springs are initially stretched from $l$ to $2l$, giving equal tension $T_0=kl$ and zero resultant force on each mass. This prestress is essential for the transverse modes found below.

For the final mixed-axis trajectory of a prestressed spring pair, the initial displacements excite only the symmetric mode in each direction. Therefore, to first order in $ϵ/l$,The second mass has the same displacements about $4l$. Over the first transverse period $T_y=2\pi \sqrt{2m/k}$, write $s=t/T_y$:The trajectory is not closed after $T_y$, because the frequency ratio is $\sqrt{2}$. It ends at approximately $(-0.858,1)$ in these scaled coordinates rather than $(1,1)$.