Problem #$2$: $(35$ points$)$ The collar $B$ shown in Figure $2$ slides along rod $OA$ in such a way that its

radial distance $r(t)$ from pivot point $O$ is given $($in m$)$ by:

$r(t)=0\mathrm{.}50+0\mathrm{.}30t^2$

where $r$ is in meters and time $t$ is in seconds. The angular position $=(t)$ of rodOA $($where $$ is in

radians as measured from the vertical direction$)$ is given $($in radians$)$ by:

$(t)=0\mathrm{.}20t^3$

where $$ is in radians and $t$ is in seconds.

a$)$ Find the time $t($in seconds$)$ at which the angle $$ is equal to $55.($Note: convert degrees to

radians.$)$

b$)$ Find the velocity vector $v$ of collar $B$ when $$ is equal to $55.$ Express the result in polar

coordinates in terms of the unit vectors $u_r$ and $u_{}$ in the radial and angular directions,

respectively.

c$)$ Sketch the vector $v$ of collar $B$ when $$ is equal to $55,$ determine its magnitude $v,$ and find

the angle $$ that $v$ makes with the radial $r-$direction.

d$)$ Find the acceleration vector $a$ of collar $B$ when $$ is equal to $55.$ Express the result in polar

coordinates in terms of the unit vectors $u_r$ and $u_{}$ in the radial and angular directions,

respectively.

e$)$ Sketch the vector $a$ of collar $B$ when $$ is equal to $55,$ determine its magnitude $a,$ and find

the angle $$ that a makes with the radial $r-$direction.