A thin rod has length $L=5\mathrm{.}00m$ and mass $M=90\mathrm{.}0kg.$ The rod is pivoted about one end and held at an angle of $41\mathrm{.}0$ degrees above the horizontal by a light rope. The rope is attached at a distance of $3\mathrm{.}50$ m along the rod from the frictionless pivot.

$($a$)$ What is the tension $T$ in the rope?

$($b$)$ What are the $x-$ and $y-$components of the force provided by the pivot?

$($c$)$ Now the rope is cut! Immediately afterwards, what is the angular acceleration of the rod about the frictionless pivot?

$($d$)$ After the rod has fallen through $41\mathrm{.}0$ degrees and is parallel to the $x-$axis, what is the angular velocity of the rod at this instant?