Problem $1.$ For the mechanical system shown below:

a$)$ Write the equations of motion

b$)$ Plot the free response when $x_3(0)=0\mathrm{.}5m$

c$)$ Determine its natural frequencies ${}_{mi}$ where $i=1$dots$3$

Where the input force $f(t)=0$ and the mass $m_3$ of the state $x_3$ is negligible. Our Example $6\mathrm{.}18$ developed in the class, L$9,$ will be very useful for this problem.