Problem $2.$

Consider the slider$-$crank mechanism shown below with the corresponding vector loop.

Link $AB=2cm,$ Link $BC=3cm$ in length. Slider $C$ moves horizontally.

At the instant when angle $=30,$ the angular velocity of the crank ${}_{AB}={}_2^{}=10ra\frac{d}{s}$

$($CW$)$ and its angular acceleration ${}_{AB}={}_2^{}=1ra\frac{d}{s^2}(CCW).$

Note that the above values for speed, angular velocity, and angular acceleration are

magnitudes. You must take care of the signs depending on the given information.

At the instant when angle $=30,$ use the complex number approach to:

$($a$)$ Write the loop closure equation for the acceleration vector expression.

$($b$)$ Specify the zero and nonzero terms in the velocity and acceleration expressions.

$($c$)$ Write the real and imaginary equations of the velocity and acceleration

expressions.

$($d$)$ Solve the real and imaginary equations of the velocity and acceleration

expressions to determine

the distance $r_1$ and the angle $[$Hint: use sine and$/$or cosine laws$]$

the velocity and acceleration of slider $C[$Magnitude and direction$]$

the angular velocity and angular acceleration of link $BC$

$[$Magnitude and direction$]$

Hint: $R^{}=((r^{})-r{}^{}{?}^2)e^j+j(2r^{}{}^{}+r{}^{})e^j$