Part I

$m{x}^{}+c{x}^{}+kx=0$

$x^{}+2w_n{x}^{}+w_n^{2}x=0$

$x(0)=x_0$ and $x^{}(0)=v_0$

Consider the differential equation of motion given above:

Determine solutions for four cases given below:

$=0$;;$,=1$;$,>1$

Illustrate the cases using numerical values:

All units in metric: m in $kg$;$k$ in $\frac{N}{m}$;$c$ in $N\frac{s}{m_0}=0\mathrm{.}1m$;$v_o=10\frac{m}{s}$;$m=2kg$;$k=8\frac{N}{m}$

Cases: $c=0$;$c=1N\frac{s}{m}$;$c=8N\frac{s}{m}$;$c=20N\frac{s}{m}$

plot the time response.