Problem $3$: $(25$ points$)$

URGENT

A Mass $m$ is supported on a spring of stiffness $K($see Figure $1)$ and is connected to a ram $5$ of the crank mechanism via a viscous damper $C.$ The crank is rigid but massless and it's radius is $r.$ The ramS is massless and slides between the two walls without any friction. The crank starts from rest $($at the position shown$)$ and reaches a constant angular velocity of or$.$

$(1)$ Draw the Free Body Diagram $($FBD$)(5$ points$).$

$(2)$ Derive the equation of the motion using any method you like. $(5$ points$)$

$(3)$ Find the homogeneous solution $(m=1kg,K=144\frac{N}{m},C=6N-\frac{s}{m}).(5$ points$)$

$(4)$ Find the particular solution, if $r=0\mathrm{.}15m,{}_5=10ra\frac{d}{s}.(10$ points$)$