$(6$ pts $)($Pendulum problem$)$ The motion of an ideal pendulum $($no damping$)$ of unit

length $)=(1m$ is given by

${}^{}+$gsin$=0$

where $g$ is a gravitational acceleration constant.

$($a$)(1pt)$ Linearize the dynamics around the equilibrium point $(0,0).$

$($b$)(2$ pts$)$ Suppose the initial conditions are given by $(0)={}_0$ and ${}^{}(0)={}_0.$ Write

the solution to the linearized system.

$($Hint: assume the form of solution in terms of sine and cosine functions, as done in

class. In addition to their coefficients, here you will also need a frequency parameter.$)$

$($c$)(2pts)$ Simulate the nonlinear dynamics $(1)$ in Matlab and plot the solution $(t)$

with the initial conditions ${}_0=0\mathrm{.}05$rad and ${}_0=0ra\frac{d}{s}$ for the following cases :

$($i$)$ on Earth $)=(9\mathrm{.}8\frac{m}{s^2}$ and $($ii$)$ on the moon $)=(1\mathrm{.}6\frac{m}{s^2}.$

$($d$)(1pt)$ For the pendulum on Earth, compare the numerical solution $(t)$ obtained in

$($c$)$ against the explicit analytical solution of the linearized system found in $($b$).$