Problem #$1$: $(40$ points$)$ In the two$-$bar linkage shown in Figure $1,$ bar $AB$ is pinned to the ground

at $A,$ and is pinned to bar $BD$ at $B.$ Bar $BD$ is pinned to a collar at $D$ that slides along a horizontal

bar, as shown in Figure $1.$ Bar $AB$ has a counterclockwise angular velocity ${}_{AB}=16ra\frac{d}{sec}$ and a

counterclockwise angular acceleration ${}_{AB}=8ra\frac{d}{se{c}^2}.$

Answer the following questions for the instant when link $AB$ is inclined at an angle $=30$ with

respect to the horizontal direction.

Unit vectors i and $j$ are in the horizontal and vertical directions, respectively. Express all vector

results in terms of i and $j.$

a$)$ Calculate the absolute velocity $v_B$ and the absolute acceleration $a_B$ of point $B.$

b$)$ Calculate the angular velocity ${}_{BD}$ of link $BD$ and the absolute velocity $v_D$ of point $D.$

c$)$ Calculate the angular acceleration ${}_{BD}$ of link $BD$ and the absolute acceleration $a_D$ of point

D$.$

Figure $1$