Problem $4(30$ points$)$

A $1$DOF spring$-$mass$-$damper system with $m=2kg,$ damping coefficient $c=5k\frac{g}{s},$ and stiffness

$k=40\frac{N}{m}$

$(1)$ Find the equation of motion

$(2)$ Determine whether the system is overdamped, underdamped, or critically damped

$(3)$ Find natural frequency ${}_n,$ damped natural frequency ${}_d,$ Period $T_d$

Critical damping $c_{cr},$ Damping ratio $$

$(4)$ Assume initial velocity is $0\mathrm{.}6\frac{m}{s},$ initial displacement is $0$ mm

Determine displacement solution $x(t),$ velocity $x^{}(t),$ and acceleration $x^{}(t)$