Problem #$1$: $(35$ points$)$ The collar $B$ shown in Figure $1$ slides along rod $OA$ of length $3$ mm in

such a way that its radial distance $r(t)$ from pivot point $O$ is given $($in mm $)$ by:

$r(t)=\frac{10}{(6+t)}$

$($where 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:

$(t)=\frac{4sint}{}$

$($Note: In the $x-y$ coordinate system not shown in Figure $1,$ the $x-$coordinate measures vertical

distances below point $O,$ and the $y-$coordinate measures horizontal distances to the right of point

O$.)$

a$)$ Find the velocity vector $v$ of collar $B$ at time $t=\mathrm{.}25s.$ Express the result in polar coordinates

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

the vector $v$ and find the angle $$ that $v$ makes with the positive $($vertical$)x-$direction.

b$)$ Find the acceleration vector $a$ of collar $B$ at time $t=\mathrm{.}25s.$ Express the result in polar

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

respectively. Sketch the vector a and find the angle $$ that a makes with the positive

$($vertical$)x-$direction.

c$)$ Determine the magnitude $v$ of the velocity and the magnitude $a$ of the acceleration of $B$ when

the $t=\mathrm{.}25s.$

d$)$ Find the velocity $v_{\frac{B}{A}}$ of collar $B$ relative end $A$ at $t=\mathrm{.}25s.$ Express the result in terms of

the unit vectors $u_r$ and $u$ in the radial and angular directions, respectively.

e$)$ Find the acceleration $a_{\frac{B}{A}}$ of collar $B$ relative end $A$ at $t=\mathrm{.}25s.$ Express the result in terms

of the unit vectors $u_r$ and $u_{}$ in the radial and angular directions, respectively.