(Solved): The mass-spring-dashpot system shown in Figure \( \mathrm{Q} 2 \) is displaced downwards from a pos ...

The mass-spring-dashpot system shown in Figure \( \mathrm{Q} 2 \) is displaced downwards from a position of static equilibrium. The spring has a natural length \( L_{0} \) and stiffness \( k \), whereas the dashpot has a damping coefficient, c. Figure Q2. Mass-spring-dashpot system Answer the following questions: (a) Draw the free body diagram of the system, clearly label and define the symbols and forces acting on the system. [6] (b) If \( x \) is measured from the support, determine the position of static equillbrium \( x_{\text {e. }} \). [6] (c) Show that the governing equation of motion is given by: \[ x+\frac{k}{m}\left(x-x_{e}\right)+\frac{c}{m} \dot{x}=0 \] [6] (d) If \( c=50[\mathrm{Ns} / \mathrm{m}], \mathrm{m}=500+10 \mathrm{Y}[\mathrm{kg}] \), and \( \mathrm{k}=3400+20 \mathrm{Y}[\mathrm{N} / \mathrm{m}] \) (where \( \mathrm{Y} \) is the last digit of your university ID) determine the damping ratio of the system [6] (c) For what value of damping coefficient will the system return to equilibrium as fast as possible without oscillation?