Homework H$\mathrm{.}6.$E

Given: A homogeneous wheel of mass $m$ and outer radius $R$ rolls without slipping on a horizontal surface. Block A $($also having a mass of $m)$ is pinned to the center $O$ of the wheel and is able to slide without friction on the same horizontal surface. A spring $($of stiffness $k)$ and a dashpot $($of damping constant $c)$ are connected between block A and ground. Let $x$ represent the motion of block A measured positively to the right. When $x=0m,$ the spring is unstretched.

Find: For this problem:

$($a$)$ Draw individual free body diagrams for block A and the wheel;

$($b$)$ Derive the single differential equation of motion for the system in terms of the coordinate $x,$ its time derivatives, and the following parameters: $m,R,c,$ and $k$;

$($c$)$ Determine numerical values for: the undamped natural frequency ${}_n,$ the damping ratio $,$ and the damped natural frequency ${}_d$; and

$($d$)$ Determine the response of the system $x(t)$ for $t>0,$ assuming the system is released when the springs are unstretched with $x^{}(0)=v_0.$

Use the following parameters in your analysis: $m=4kg,k=2250\frac{N}{m},R=0\mathrm{.}1m,c=60k\frac{g}{s},$ and $v_0=8\frac{m}{s}.$