$3.(40$pts$)$ REVISED A solid Titanium Ti$-6$Al$-4$V shaft with a diameter of $75$ mm is subjected to a combination of distributed and discrete torque loads as shown. End C is firmly anchored. Looking from the front side, a CCW $800$N$\backslash (\backslash$cdot $\backslash )$m discrete torque is applied at the end. Between $0\mathrm{.}5$ m and $1\mathrm{.}0$ m from the end is a CW distributed torque load of $\backslash (\backslash$mathrm$\{$t$\}=4\backslash$mathrm$\{$kN$\}\backslash$cdot $\backslash$mathrm$\{$m$\}/\backslash$mathrm$\{$m$\}\backslash ).$ A distance $1\mathrm{.}2$ m from the end a CCW discrete $\backslash (400\backslash$mathrm$\{$~N$\}\backslash$cdot $\backslash$mathrm$\{$~m$\}\backslash )$ torque load is applied.

a$.$ Show that the support at C exerts an $\backslash (800\backslash$mathrm$\{$~N$\}\backslash$cdot $\backslash$mathrm$\{$~m$\}\backslash )$ CCW torque on the shaft, when viewed from the free end. b$.$ Find the internal torques $($in $\backslash (\backslash$mathrm$\{$N$\}\backslash$cdot $\backslash$mathrm$\{$m$\}\backslash ))$ at distances $\backslash (0\mathrm{.}2\backslash$mathrm$\{$~m$\},0\mathrm{.}4\backslash$mathrm$\{$~m$\},0\mathrm{.}9\backslash$mathrm$\{$~m$\}\backslash ),$ and $1\mathrm{.}4$ m from the support at C $.$ Indicate direction $($CW or CCW$)$ as viewed from the free end.

c$.$ Find the angles of twist $($in both radian and degrees$)$ at distances $\backslash (0\mathrm{.}2\backslash$mathrm$\{$~m$\},0\mathrm{.}4\backslash$mathrm$\{$~m$\},0\mathrm{.}9\backslash$mathrm$\{$~m$\}\backslash ),$ and $1\mathrm{.}4$ m from the support at C$..$ Indicate direction $($CW or CCW$)$ as viewed from the free end.

d$.$ Find the maximum shear stress $\backslash (\backslash$tau $\backslash )($in MPa$)$ at distances $\backslash (0\mathrm{.}2\backslash$mathrm$\{$~m$\},0\mathrm{.}4\backslash$mathrm$\{$~m$\},0\mathrm{.}9\backslash$mathrm$\{$~m$\}\backslash ),$ and $1\mathrm{.}4$ m from the support at $\backslash ($ C $\backslash ).$

e$.$ Sketch the torque and the angle of twist as a function of $\backslash ($ x $\backslash ),$ with $\backslash ($ x$=0\backslash )$ at end $"$C$."$