A shaft with a length $L=2\mathrm{.}0m$ is subjected to torques $T=5\mathrm{.}10kNm$ at either end. $($See

Fig. Q$4).$ Segment $AB(L_1=1\mathrm{.}0(m))$ is made of brass $(G_b=41\mathrm{.}3$GPa$)$ and has a square cross

section $(a=75mm).$ Segment $BC(L_2=1\mathrm{.}0(m))$ is made of steel $(G_s=74\mathrm{.}3$GPa$)$ and has a

circular cross section $(d=75mm).$ Ignore stress concentrations near $B.$

$($a$)$ Find the maximum shear stress and angle of twist for each segment of the shaft.

$($b$)$ Find a new value for the dimension $a$ of bar $AB,$ if the maximum shear stress in

$AB$ and $BC$ are to be equal.

$($c$)$ Repeat part $($b$)$ if the angles of twist of segments $AB$ and $BC$ are to be equal.

$($d$)$ If dimension $a$ is reset to $a=75mm$ and bar $BC$ is now a hollow pipe with an

outer diameter $d_2=a,$ find the inner diameter $d_1$ such that the angles of twist of

segments $AB$ and $BC$ are equal.

$($Ex$.3-13$ of Goodno & Gere, $9$e$,$ units changed to SI$)$