Q$.1(25$ points$)$ The torques shown in Fig.Q$1$ are exerted on pulleys $\backslash ($ A$,$ B $\backslash ),$ and $\backslash ($ C $\backslash ).$

$($a$).$ Knowing that both shafts are solid and made of brass $(\backslash ($ G$=39\backslash$mathrm$\{$GPa$\}\backslash )),$ determine the angle of twist between $\backslash ($ A $\backslash )$ and $\backslash ($ C $\backslash ).$

$($b$).$ The shaft $\backslash ($ B C $\backslash )$ of the pulley assembly is to be redesigned. Knowing that the allowable shearing stress in shaft $\backslash ($ B C $\backslash )$ is $60$ MPa $,$ determine the smallest allowable diameter of shaft $\backslash ($ B C $\backslash ).$