Small oscillations of a hoop with a sliding bead
= Small oscillations of a hoop with a sliding bead
For equal hoop and bead masses, with the hoop suspended from its circumference, the quadratic mass and stiffness matrices are proportional to
$$
M=\begin{pmatrix}3&1\\1&1\end{pmatrix},
\qquad
K=\begin{pmatrix}2&0\\0&1\end{pmatrix}.
$$
The dimensionless squared frequencies are the roots of $2c^2-5c+2$, namely $c=1/2$ and $c=2$.