Question $1$:

For the system shown below, the disk of mass $m$ rolls without slips and $x$ measures the displacement of the disk from the unstretched position of the spring.

a$)$ Find the equations of motion.

b$)$ With $c=100(\frac{m}{s}),m=6kg,r=0\mathrm{.}10m,$ for what value of the spring stiffness k

is the damping ratio of the system one half of the critically damped value, so that $=$cr$/2?$

c$)$ With these parameter values, find the displacement of the disk if it is rolled $20$ cm to the right $($from static equilibrium$)$ and released from rest.