Problem $2[20$ Points$]$: VW on a Four$-$Bar Mechanism

A four$-$bar mechanism is shown below at a specific position. A motor rotates the crank counterclockwise at $\backslash (12\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash )$ by providing a counterclockwise torque of $\backslash (30\backslash$mathrm$\{$~N$\}.\backslash$mathrm$\{$cm$\}\backslash ).$ Determine the angular velocity of the rocker $\backslash (\backslash$omega$\_\{4\}\backslash )$ and the external force $\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$C$\}\}\backslash )$ by following the below steps. Note: You may assume that the mass and moment of inertia of each link is $0.$

a$)(8$ points$)$ Calculate the angular velocity of the rocker, $\backslash (\backslash$omega$\_\{4\}\backslash ),$ and the velocity of point C$,$ by using an instant center or PVA analysis. C is located in the middle of link $4.$

b$)(12$ points$)$ Assuming frictionless joints and massless links, determine Fc using the virtual work method. Indicate the correct direction of $\backslash (\backslash$mathbf$\{$F$\}\_\{\backslash$mathbf$\{$C$\}\}\backslash )$ on the schematic after finding its value.