Problem B$-3-17$

Consider the mechanical system shown in Figure $3-46.$ Two pulleys small and large, are

bolted together and act as one piece. The total moment of inertia of the pulleys is $1.$ The

mass $m$ is connected to the spring $k_1$ by a wire wrapped around the large pulley. The

gravitational force $mg$ causes static deflection of the spring such that $k_{\text{u}\text{a}\text{r}\text{r}}=mm.$

Assume that the displacement $x$ of mass $m$ is measured from the equilibrium position.

Two springs $($denoted by $k_2)$ are connected by a wire that passes over the small pulley

as shown in the figure. Each of the two springs is prestretched by an amount $y.$

Obtain a mathematical model of the system. Also, obtain the natural frequency

Please answer using energy method!