$3\mathrm{.}32($a$)$ Determine the angle of twist caused by a $40-$kip$-$in$.$ torque $T$ in

the $3-$in$.-$diameter solid aluminum shaft shown $(G=3\mathrm{.}7{10}^6).$

$($b$)$ Solve part $($a$),$ assuming that the solid shaft has been replaced

by a hollow shaft of the same outer diameter and a $1-$in$.$ inner

diameter.

Fig. P$3\mathrm{.}32$