Problem $4(30$ Points$)$

A countershaft of uniform cross section carrying two $V-$belt pulleys is shown in the figure. Pulley A receives power from a motor through a belt with the belt tension as shown. The power is transmitted through the shaft and delivered to the belt on pulley $B.$ The shaft is made from AISI $1020$ CD steel with a yield strength $S_y=390$MPa. Assume that the belt tension on the loose side at pulley $B$ is $12\%$ of the tension on the tight side, and supports $O$ and $C$ are treated as ball bearings.

$(1)$ Determine the support reactions at the ball bearings $O$ and $C.$

$(2)$ Draw the bending moment diagram and internal torque diagram of the shaft. If needed, made one set for the horizontal plane and another set for the vertical plane.

$(3)$ Determine the required diameter $d$ of the shaft if its initiation of yielding obeys the maximum distortional strain energy $($von Mises$)$ and maximum shear stress failure criteria, respectively. The factor of safety of the design is specified as F$.$S$.=2\mathrm{.}65.$ Stress concentration due to fillets and notches is ignored.