The illustrated system has the following system parameters: $m=10kg,J_0=1kg-m^2,r=5cm,k_1=1000\frac{N}{m},k_2=500\frac{N}{m},c=500N-\frac{s}{m},F_0=50N,$ and $=20ra\frac{d}{s}.$ Determine the following:

$($a$)$ Draw the FBD and use kinetics to determine the equation of motion. Hint: draw a FBD of the mass, $m,$ and get the equation of motion for the block in terms of the tension, T$.$ Then draw the FBD of the pulley and solve for $T.$ Substitute and get the final equation of motion in terms of $.$

$($b$)$ Solve the differential equation obtaining the total solution for small oscillations, in terms of $,$ given initial conditions: ${}_0=5^0,{}_0^{}=4ra\frac{d}{s}.$ Plot the complementary, particular, and total solutions on one plot $($position vs$.$ time$)$ for a time duration of $4\mathrm{.}0$ seconds.

Pulley, mass moment of inertia $J_0$