As shown in Figure $4,$ there is a circular disk made of homogeneous material, with an area density of $=20k\frac{g}{m^2}.$ The disk lies on the $x-y$ plane and is connected to a pin at A via a rigid rodOA, rotating with a constant angular speed of ${}_2=2ra\frac{d}{s}($clockwise$).$ The radius of the circular disk is $r=1m.$ At this moment, the $x^{\text{'}}-$axis and $y^{\text{'}}-$axis are parallel to the $x$ axis and $y-$axis, respectively. Note that the rodOBA lies on the $x-y$ plane, and the angle between rodBA and the $x-$axis is $45.$RodOB and rodBA both have lengths of $2$ m $.$ Rod $OB$ is rotating around the $y-$axis with an angular speed of ${}_1=2ra\frac{d}{s}($counterclockwise$)$ and an angular acceleration of ${}_1=4ra\frac{d}{s^2}($counterclockwise$).$ The mass of the other components in the system can be neglected compared to the disk. The gravitational acceleration is $g=9\mathrm{.}8\frac{m}{s^2}.$

Determine the absolute angular velocity and absolute angular acceleration of the disk; Use the parallel axis theorem to calculate $I_x,I_y,I_z,I_{xy}$;

Based on Euler's equation, determine the moment $C$ acting on $O$

Figure $4$

$(18$ Marks$)$