Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 14E (Classical Dynamics)

Write $r_1=(x_1,y_1)$, $r_2=(x_2,y_2)$,
${\rho }_1=|r_1|$, and ${\rho }_2=|r_2-r_1|$. ThenWith $e_y=(0,1)$, the Euler--Lagrange equations are

At the downward vertical equilibrium, the lower spring supports $m_{2}g$ and the upper spring supports $(m_1+m_2)g$. Thus the two vertically suspended springs have lengthsand positions

For equal masses and springs,A small horizontal displacement $x_1$ changes each spring direction to first order. The equilibrium tensions $2mg$ and $mg$ give transverse restoring stiffnessThereforeas required.

The other motions are vertical. If ${\eta }_1,{\eta }_2$ are downward displacements from equilibrium, their linearized equations areThe vertical normal modes of two equal suspended masses areNeither frequency contains $g$: gravity only shifted the equilibrium lengths.