Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 11 / b / Solution

To find the normal modes of two arc-spring-coupled beads on a hoop, take $x_i=a_i{e}^{i\Omega t}$ for a normal mode. The resulting generalized eigenvalue problem isIts determinant isThus the angular frequencies areThe ordinary nonzero frequency is $\omega /(2\pi )$. The zero mode has mode shape $(1,1)$ and represents a common rotation with no change in spring length. The oscillatory mode shape is $(1,-1/q)$.