$(35$ points$)$ A solid circular bar ABCD with fixed supports is acted upon by torques $T_0$ and $2T_0$ at the location shown in the figure. $T_0=80N*m.$ The length of the bar $L=4m.$ The polar inertial of moment of the bar $I_P=1{10}^{-9}{m}^4.$ The shear modulus of the bar $G=100$Gpa.

$($a$)$ Calculate the reactions $R_A$ at points $A,$ and $R_B$ at point $B.$

$($b$)$ Calculate the angle of twist ${}_C$ at point $C,$ and ${}_D$ at point $D.$