Using a carefully drawn FBD and Newton's$/$Euler$\text{'}$s laws of motion, derive the

equation of motion for $(t),$ where $$ is the counterclockwise rotational angle of the

pulley, measured from the equilibrium position. Assume small motions.

Determine expressions for the undamped natural frequency ${}_n$ and damping ratio $.$

Let $k=500\frac{N}{m},J=0\mathrm{.}4kg{m}^2,r_1=0\mathrm{.}1m,r_2=0\mathrm{.}2m,c=60N\frac{s}{m},(0)=0\mathrm{.}1$rad,

${}^{}(0)=0.$

i$)$ State whether the system is overdamped, underdamped, or critically damped.

ii$)$ Obtain the solution $($in written form$)$ for $(t).$

iii$)$ Sketch $(t)$ by hand, providing a few appropriate values

Set $c=0($i$.$e$.,$ remove the damper$)$ and use the energy method to obtain the undamped

equation of motion. This serves as a partial check to your work in part a$.$