$1.[$By hand $-10$ points$]$ For the system shown below in Figure $1$:

a$.$ Draw the free body diagrams for mass $\backslash ($ M $\backslash )$ and for massless point $\backslash ($ A $\backslash ).$ Write the differential equations describing the system

b$.$ Write the state variable equations for the system $($in Figure $1),$ where the input is the displacement $\backslash ($ x$\_\{3\}($t$)\backslash ).$ The output is the force exerted by the dashpot $\backslash ($ B$\_\{2\}\backslash )$ on point $\backslash ($ A $\backslash ),$ with the positive sense to the left