$3\mathrm{.}9$ Consider the mechanical system in Figure P$3\mathrm{.}9,$ where $f$ and $p$ are external forces; viscous friction $(B)$ exists between $m_1$ and $m_2$; and a massless pulley changes the direction of the tension force of an unstretchable rope, which suspends a weight $(M).$

Figure P$3\mathrm{.}9$

$($a$)$ Draw free$-$body diagrams for the mass elements and auxiliary plots for the spring and damping elements.

$($b$)$ Derive the governing equations of motion for the system.