Problem $5.$ In the system below, a solid thin disk rolls without slipping on a horizontal surface. The disk at attached to two springs with equal stiffnesses $k$ at point $s,$ which is at an offset a from the center of the disk. The disk has a radius $r$ and a mass $m.$ The two springs in the system are both unstretched in the equilibrium position shown below. Assuming the disk is only able to undergo small angle oscillations $($ is small$),$ answer the following questions:

a$.$ Write the equation of motion for the disk $($symbolically$)$

b$.$ Find the natural frequency ${}_n$ of the disk $($symbolically$)$

c$.$ Find the natural frequency ${}_n$ of the disk if $m=1kg,r=0\mathrm{.}4m,k=10\frac{N}{m}$ and $a=0\mathrm{.}1$ m $.$

Hint: You need to be cognizant of the relationship between linear translation $($position$,$ acceleration$)$ and angular motion $($in terms of $,{}^{}).$ The displacement of point $s$ can be thought of as: Linear translation of center of gravity $r+$ angular motion due to rotation about the center of the disk.