Question $4(20$ points$)$: A spring$-$mass system is placed on a titled surface with friction. The key

parameters include friction coefficient $\backslash$mu $=0\mathrm{.}05,$m$=250$kg$,\backslash$theta $=20\backslash$deg $,$ and k$=3000($N$)/($m$).$

$(1)$ Derive the equation of motion of this system.

$(2)$ Use Matlab to simulate the displacement and velocity responses of this system, when the

initial displacement x$\_(0)$ is $0\mathrm{.}1$ m and the initial velocity v$\_(0)$ is $0\mathrm{.}1($m$)/($s$)$

$(3)$ How long does it take for the vibration to die out?