(Solved): A mass m=4kg is attached to both a spring with spring constant k=401N/m and a dash-pot with dam ...

A mass $m=4\mathrm{kg}$ is attached to both a spring with spring constant $k=401\mathrm{N}/\mathrm{m}$ and a dash-pot with damping constant $c=4\mathrm{N}?\mathrm{s}/\mathrm{m}$. The mass is started in motion with initial position $x_0=3\mathrm{m}$ and initial velocity $v_0=7\mathrm{m}/\mathrm{s}$. Determine the position function $x(t)$ in meters. $x(t)=$ Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form $x(t)=C_1{e}^{?pt}\cos \left({?}_{1}t?{?}_1\right)$. Determine $C_1,{?}_1,{?}_1$ and $p$. $\begin{array}{ll}{C}_1=& \\ {?}_1=& \text{ (assume }0?{?}_1<2?\text{ ) }\\ {?}_1=& \text{ (ass. }\end{array}$ $p=$ Graph the function $x(t)$ together with the "amplitude envelope" curves $x=?C_1{e}^{?pt}$ and $x=C_1{e}^{?pt}$. Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected ( so $c=0$ ). Solve the resulting differential equation to find the position function $u(t)$. In this case the position function $u(t)$ can be written as $u(t)=C_0\cos \left({?}_{0}t?{?}_0\right)$. Determine $C_0,{?}_0$ and ${?}_0$. $\begin{array}{l}{C}_0=\\ {?}_0=\\ {?}_0=\end{array}$ $\text{ (assume }0?{?}_0<2?\text{ ) }$ Finally, graph both function $x(t)$ and $u(t)$ in the same window to illustrate the effect of damping.