Problem $\backslash$#$2$: $(30$ points$)$ An overhanging beam $\backslash ($ A C B $\backslash )$ of length $3\mathrm{.}7$ m is pinned at point $\backslash ($ A $\backslash ),$ and supported by a roller at point $\backslash ($ C $\backslash ),$ as shown in Figure $2.$ The beam is loaded by an $8$ kN downward force at the free end $\backslash ($ B $\backslash )$ and by a uniform downward load of $\backslash (3\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )$ over the overhanging part of the beam between the roller at $\backslash ($ C $\backslash )$ and the free end at $\backslash ($ B $\backslash ),$ as shown. In an $\backslash ($ x$-$y$-$z $\backslash )$ coordinate system $($not shown in the figure$)$ with origin at the centroid of the cross section, the positive $\backslash ($ x $\backslash )-$direction measures horizontal distance to the right of point $\backslash ($ A $\backslash ),$ and the positive $\backslash ($ y $\backslash )-$direction measures vertical distances above the centroid of the cross section.

a$)$ Find the reactions at pin $\backslash ($ A $\backslash )$ and roller $\backslash ($ C $\backslash ).$ Is the problem statically determinate or indeterminate? Explain.

b$)$ Find the equations for the shear force $\backslash ($ V$=$V$($x$)\backslash )$ and the bending moment $\backslash ($ M$=$M$($x$)\backslash )$ throughout the beam. For each cut required, draw a clear free body diagram.

c$)$ Neatly sketch the shear force and bending moment diagrams, and provide the important values and locations of each. In particular, find the magnitude of the maximum shear force $\backslash ($ V$\_\{\backslash$max $\}\backslash )$ in the beam, and find the value of $\backslash ($ x $\backslash )$ at which it occurs. Find the magnitude of the maximum bending $\backslash ($ M$\_\{\backslash$max $\}\backslash )$ in the beam, and find the value of $\backslash ($ x $\backslash )$ at which it occurs.