$($JUST LETTERS D AND E PLEASE$)$ 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 as shown in the figure. The power is transmitted through

the shaft and delivered to the belt on pulley B$.$ Assume the belt tension on the loose side at B is $15$

percent of the tension on the tight side.

Note that all dimensions are in mm $.$

Ignore the transverse shear stresses for your analysis.

$($a$)$ Determine the tensions in the belt on pulley $B$ assuming the shaft is running at a constant

speed. In other words, the torques generated at both pulleys should balance each other.

$($b$)$ Find the magnitudes of the bearing reaction forces at O and C assuming the bearings act as

simple supports i$.$e$.$ no reaction moments are generated at the ends.

$($c$)$ Draw shear$-$force and bending$-$moment diagrams for the shaft. Following the example

covered in Lecture $4,$ make one set for the horizontal plane $(x-z)$ and another set for the

vertical plane $(x-y).$

$($d$)$ Determine the location of the cross$-$section along the beam that experiences the highest

bending moment, $M=\sqrt[\phantom{2}]{M_y^2+M_Z^2}.$

$($e$)$ Determine the location of the critical point on the cross$-$section identified above. Show the

stress element at this location with the stress arrows.

$($f$)$ Calculate the bending stress and torsional shear stress at this critical point? Ignore the

transverse shear stresses for your analysis.

$($g$)$ For the state of stress at the critical point, determine the principal stresses $({}_1,{}_2,{}_3)$ and

the maximum shear stress $({}_{\text{m}\text{a}\text{x}}).$ Ignore the transverse shear stresses for your analysis.