Past exam of the mathematical methods II course of the University of Oxford / 2018 / 10 / b / Solution

Put $\kappa =k/m$, $q=g/l$. The positive stiffness-per-mass matrix isSubstitution of a normal mode $x=v\cos \omega t$ requires $Kv={\omega }^{2}v$. Its characteristic equation is ${\lambda }^2-(\kappa +2q)\lambda +\kappa q=0$. ThereforeFor $k,m,g,l>0$, their sum and product are positive, so both squared angular frequencies are positive and ${\omega }_{+}>{\omega }_{-}>0$.