Problem $3$: $(50$ points$)$

Consider a very thin $\backslash (45^\{\backslash$circ$\}\backslash )$ right triangle of side $\backslash ($ a $\backslash )$ as shown in the figure. The triangle lies in the $\backslash ($ x y $\backslash )$ plane as shown and has a mass $\backslash ($ M $\backslash ).$

$($a$)$ What is the inertia tensor for this object, for the origin as shown in the figure? Enter your final answer in the box below. Hint: all of the elements inside the parenthesis should be integers, and I have given you one of the elements so you can check that you are on the right track.

$($b$)$ Suppose the triangle has an angular velocity $\backslash (\backslash$omega $\backslash )$ that is directed along the $\backslash ($ y $\backslash )-$axis. What is the angular momentum in this case? Work out the three components of the angular momentum.

$($c$)$ What is the angle between the angular velocity vector and the angular momentum vector? Express your answer numerically in degrees.

$($e$)$ Use the parallel axis theorem to find the inertia tensor for the triangle if we move the origin to the center of mass. Write your answer in the box below. Hint $-$ all $9$ elements inside the parenthesis should be integers.