For all problems, you can assume gravity is $9\mathrm{.}81\frac{m}{?}$

This problem uses MATLAB. Turn in your program and plots, as well as answers to the following parts. Please print your MATLAB code as a pdf to include in your submission. Be sure to include axis labels with units and legends with your plots.

Consider the spring$-$damper system shown, where and the spring is initially at rest with $x(0)=0m.$

$3-1.(10$ pts$)$ Derive a first$-$order linear differential equation to describe the position of the spring with applied force $F(t).$

$3-2.(5$ pts$)$ Assuming that $F(t)$ is a constant value of F $,$ determine the particular solution to the spring position, $x(t).$

$3-3.(5$ pts$)$ Provide equations which describe: $(1)$ the energy stored in the spring as a function of time $U(t)$ and $(2)$ the power dissipated by the damper as a function of time, $P(t)$

$3-4.(10$ pts $)$ Simulate System A for $8$ seconds with a timestep $dt=0\mathrm{.}01$ seconds using the forward Euler method in MATLAB given that $k=2000\frac{N}{m},b=1200N-\frac{s}{m},$ and $F=150N$ :

a$)$ Plot the position $x(t)$ calculated by Euler's method and the position function found in part $3-2$ on the same graph $($Euler$\text{'}$s in red, Analytical Solution in blue$)$

b$)$ Plot the energy stored in the spring calculated through Euler's method and the energy stored in the spring with your analytical solution $($Euler$\text{'}$s in red, Analytical Solution in blue$)$

$3-5.(10$ pts $)$ Simulate System A for $8$ seconds with a timestep $dt=0\mathrm{.}01$ seconds using the forward Euler method in MATLAB given that $k=2000\frac{N}{m},b=1200N-\frac{s}{m},$ and a time varying force $F(t)=150cos(2t)N$ :

a$)$ Plot the position $x(t)$ calculated by Euler's method

b$)$ Plot the energy stored in the spring, $U(t),$ calculated through Euler's method