Problem #$3$: $(25$ points$)$ A nonuniform arm OA rotating in a vertical plane with clockwise angular

velocity $$ about point $O$ is attached to a spring of stiffness $k=3l\frac{b}{i}n$ and is subjected to a constant

clockwise couple of magnitude $M,$ as shown in Figure $3.$ The arm $OA$ has a weight $W=16\mathrm{.}1lb$

$($with a corresponding mass of $m=\frac{W}{g}=0\mathrm{.}50$ slugs$)$ and a moment of inertia about its center of

mass $G$ equal to $I_G=18$ slug$-$in ${?}^2.$ At any instant, arm $OA$ makes an angle $$ with the horizonal

direction, as shown.

The arm is initially at rest in the vertical position $($i$.$e$.=90)$ when the spring is unstretched and

when the couple $M$ is first applied.

a$)$ Calculate the moment of inertia $I_O$ of arm $OA$ about point $O,$ and write down the kinetic

energy $T$ in terms of its angular velocity $$ at any instant.

b$)$ Calculate the work $U$ done by all the external loads acting on arm $OA$ when it reaches its

horizontal $($i$.$e$.=0)$ position.

c$)$ Calculate the angular velocity $$ of arm $OA$ when it reaches its horizontal $($i$.$e$.=0)$

position. Express the result in terms of couple $M.$

d$)$ Find the couple $M$ that will give arm $OA$ an angular velocity $=5ra\frac{d}{sec}$ when it reaches

its horizontal $($i$.$e$.=0)$ position.