Problem $4.$ Consider the nonlinear equation of motion for a pendulum

${}^{}(t)+\frac{g}{l}sin(t)=0$

where $(t)$ is measured from the downward position. In class, we showed that the linearization of $(1)$ about the downward equilibrium $($i$.$e$.,{}_e=0)$ is given by

${}^{}(t)+\frac{g}{l}(t)=0,$

where $(t)$ is the linear approximation of $(t)$ about ${}_e=0.$

Let $g=9\mathrm{.}8\frac{m}{s^2}$ and $l=0\mathrm{.}2m.$ Use Matlab or Simulink to simulate the nonlinear equation of motion $(1)$ and the linearized equation of motion $(2)$ for $4$ seconds. Run the simulation three times, using the initial conditions: $($i$)(0)=(0)=0\mathrm{.}1$rad,${}^{}(0)={}^{}(0)=0ra\frac{d}{s}$;$(ii)(0)=(0)=0\mathrm{.}5$rad,${}^{}(0)={}^{}(0)=0ra\frac{d}{s}$; and $(0)=(0)=1$ rad,${}^{}(0)={}^{}(0)=0ra\frac{d}{s}.$

For each set of initial conditions, plot $(t)$ and $(t)$ on the same axis, and plot ${}^{}(t)$ and ${}^{}(t)$ on the same axis. What do you observe as the initial angle $(0)=(0)$ increases? Please turn in a hardcopy of the Matlab code and the plots with all axes correctly labeled and with units specified.