Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 14B / c

Seek $z=v{e}^{i\Omega t}$ and put $\mu =m/M$. The generalized characteristic equation isorAs a quadratic in $X={\Omega }^2$, this issoA corresponding displacement vector isThus the four normal modes are $v_{+}{e}^{\pm i{\Omega }_{+}t}$ and $v_{-}{e}^{\pm i{\Omega }_{-}t}$. Both squared frequencies are positive for every $\mu >0$, proving linear stability.

For $\mu \ll 1$,Therefore, labeling the larger and smaller positive frequencies by $\omega$ and ${\omega }^{\prime }$,so $\alpha ={\omega }_0$.