Problem #$2$: $(35$ points$)$ A $10$ lb block $B$ remains in contact with a bar $OA,$ which rotates in a

vertical plane $($with gravity acting downward$),$ as shown in Figure $2.$ At any time $t,$ the radial

position of the block and the orientation of the bar are given by position $R(t)$ and angle $(t),$

respectively. The coefficients of static and kinetic friction between block $B$ and bar $OA$ are ${}_s$ and

${}_k,$ respectively.

At the instant of time shown in the figure, the radial position of block $B$ is $R=1\mathrm{.}2ft,$ and the bar

$OA$ rotates with an angular velocity ${}^{}=5ra\frac{d}{sec}$ and an angular acceleration of ${}^{}=3ra\frac{d}{se{c}^2}.$

a$)$ Draw a free body diagram of block $B$ at any time $t.$

b$)$ At the instant shown in the figure, find the normal force exerted by bar $OA$ on block $B$ if the

block does not slide along bar, and find the magnitude $($and the direction$)$ of the friction force

needed to prevent block $B$ from sliding along bar $OA$ at that instant.

c$)$ If the coefficients of static and kinetic friction between block $B$ and bar $OA$ are ${}_s=0\mathrm{.}25$ and

${}_k=0\mathrm{.}20,$ respectively, then determine whether or not the block will slide along the bar at

the instant shown in the figure.

d$)$ Assuming that, at the instant shown, the block slides away from point $O$ at a radial speed

$R^{}=2f\frac{t}{s}($and the kinetic coefficient of friction is ${}_k=0\mathrm{.}20,$ as given in part c $)),$ find the

normal force $N$ exerted by bar $OA$ on block $B,$ and find $R^{}$ of the block along the bar at that

instant.

e$)$ For any instant of time $t$ at which the block does slide along bar $OA,$ find the equation that

relates $R(t),R^{}(t),$ and $R^{}(t).($You don't need to solve the equation!$)$