Problem $3$: $(18$ points$)$

The slotted arm $OA$ rotates about a fixed axis through $O.$ The slider $B$ has a mass of $0\mathrm{.}3$ kg $.$ The path of the center of the pin is described by the equation of $r$ in terms of $,$ where $r$ is measured in meters and $$ in radians. At this instant, $=30,{}^{}=45\frac{\text{d}\text{e}\text{g}}{s},$ and ${}^{}=20\frac{\text{d}\text{e}\text{g}}{s^2}.$ Assume all frictions are negligible and the motion occurs in a vertical plane. Assume gravity $g=9\mathrm{.}81\frac{m}{s^2}.$ Show all work.

a$)$ Compute the expression of $r,$ in terms of $*(2$ points$)$

b$)$ Determine the expression of $r^{}$ and $r^{}.(4$ points$)$

c$)$ Determine the radial acceleration of the pin, $a_r.(2$ points$)$

d$)$ Determine the transverse acceleration of the pin, $a_{}.(2$ points$)$

e$)$ Determine the total acceleration of the pin, a$.(2$ points$)$

f$)$ Determine the forces applied by the arm $OA$ to slider $B,F_{\text{a}\text{r}\text{m}}.(3$ points$)$

g$)$ Determine the forces applied by the sides of the slot to slider $B,F_{\text{s}\text{i}\text{d}\text{e}}.(3$ points