$(15$ points$)$ A solid, circular bar ABC consists of two segments, as shown in the figure. One segment has a diameter of $d_1=20mm$ and length of $L_1=4m$; the other segment has a diameter of $d_2=10mm$ and length of $L_2=2m,$ Suppose $T=200N*m,$ shear modulus $G=10$Gpa.

$($a$)$ Calculate the polar moment of inertia of segment $AB$ and $BC.$

$($b$)$ Calculate the angle of twist between the ends of the bar.