$($a$)=0\mathrm{.}541\sqrt[\phantom{2}]{\frac{g}{b}},$

$($b$)=\sqrt[\phantom{2}]{\frac{g}{b}},$ and $($c$)=1\mathrm{.}414\sqrt[\phantom{2}]{\frac{g}{b}}.$

The assembly is constructed of homogeneous slender rod which has a mass $$ per unit length, and it rotates freely about a horizontal axis through the pivot at $O.$ If the assembly is nudged from the starting position shown, determine its angular velocity after it has rotated $($a$)45,($b$)90,$ and $($c$)180.$