A mechanical system is presented in Fig. $1.$ A cylinder of radius $R,$ mass $m,$ and moment of inertia $I_O(I_O=m\frac{R^2}{2})$ about its centers of mass $\text{'}O\text{'}$ is rolling without slip on a horizontal surface. The cylinder's center point $O$ is connected through a spring with stiffness coefficient $k$ to a wall, and through another spring with stiffness coefficient $k$ to a thin$-$rod$-$pendulum pivoting about a fixed point $C,$ which is also its center of mass. The rod is of mass $m,$ of length $2L,$ and of mass moment of inertia $I_C$ about $C$