$8.$ A countershaft carrying two V$-$belt pulleys is shown in the figure. Pulley A receives power from a motor through a belt with the belt tensions shown. The power is transmitted through the shaft and delivered to the belt on pulley B$.$

$($a$)$ Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports.

$($b$)$ Draw shear$-$force and bending$-$moment diagrams for the shaft. If needed, make one set for the horizontal plane and another set for the vertical plane.

$($c$)$ At the point of maximum bending moment, determine the bending stress and the torsional shear stress.

$($d$)$ At the point of maximum bending moment, determine the principal stresses and the maximum shear stress.