$($b$)$ A cantilever circular solid bar is subjected to a clockwise torsional moment of $10$ kNm at point $\backslash (\backslash$boldsymbol$\{$B$\}\backslash )$ and an anti$-$clockwise torsional moment of $2$ kNm at its free end point $\backslash (\backslash$boldsymbol$\{$C$\}\backslash )$ as shown in Figure Q$\mathrm{.}2($b$).$ Given that the modulus of rigidity is $80$ GPa $.$ Calculate:

$($i$)$ the maximum shear stress induced in each portion of the bar, i$.$e$.$ portions

$\backslash (\backslash$boldsymbol$\{$A B$\}\backslash )$ and $\backslash (\backslash$boldsymbol$\{$B C$\}\backslash )$; and

$($ii$)$ the angle of twist at its free end $\backslash (\backslash$boldsymbol$\{$C$\}\backslash ).$

$(6$ marks$)$

$(6$ marks$)$