Question $9$ The system shown in the static equilibrium position is a thin, uniform cross$-$section, homogeneous, rigid L$-$shaped rod hinged at the corner that carries a singular mass M on it$.$ The rod is connected to the ground by a linear spring, k $,$ and a viscous damper, $c.m_1$ and $m_2$ are masses of vertical and horizontal rods, respectively.

c$)$ Obtain the equation of motion of the small amplitude vibrations of the system using Euler's equation.

d$)$ Ignore damping and calculate the natural frequency of the system using numerical values $m_1=4kg,m_2=4kg,M=3kg,k=800\frac{N}{m},l=1m.$ In this undamped system, calculate the vibration amplitude, $,$ the phase angle, $,$ and particular solution that corresponds to initial conditions ${}_0=0\mathrm{.}2ra{d}_r{}_0^{}=2\mathrm{.}2ra\frac{d}{s}.$ Take into account $g=10\frac{m}{s^2}.$