SORU $2.$ In the system shown in the static equilibrium position, a thin, homogeneous, rigid L$-$shaped rod with a uniform cross$-$section hinged at the corner carries a singular mass $M$ at one end. The rod is connected to the ground by a linear spring, $k,$ and a viscous damper, $c.$

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

b$)$ Calculate the natural frequency of the system $($by ignoring damping$)$ using numerical values $m_1=2kg,m_2=6kg,M=2kg,k=440\frac{N}{m},l=1$ m $.$ In this undamped system, calculate the vibration amplitude, $,$ the phase angle, $,$ and particular solution that correspond to initial conditions $=0\mathrm{.}1$rad,${}^{}=2\mathrm{.}0ra\frac{d}{s}.$ Take into account $g=10\frac{m}{s^2}.$