$3-6(10$ pts$)$ A mass is added to the system, as shown above; it's initial velocity is $v_m(t)=0\frac{(m)}{(s)}.$ Derive a second$-$order linear differential equation to describe the position of the spring with applied force $F(t).$

$3-7(5$ pts$)$ Given that $k=2000\frac{N}{m},b=1200N-\frac{s}{m},m=200kg,$ and $F=4000N,$ determine the particular solutions to the spring position, $x_s(t),$ and mass velocity $v_m(t).$

$3-8(5$ pts$)$ Provide equations which describe: $(1)$ the energy stored in the spring as a function of time $U(t)$ and $(2)$ the kinetic energy of the mass as a function of time $K(t).$

$3-9(30pts$ Simulate System A for $8$ seconds with a timestep $dt=0\mathrm{.}01$ seconds using the forward Euler method in MATLAB:

a$)$ Plot the position $x(t)$ calculated by Euler's method and the position function found in part $3-7$ on the same graph $($Euler$\text{'}$s in red, Analytical Solution in blue$)$

b$)$ Determine the total work done by the damper as a function of time, $W(t),$ and the work done by the external force $FW(t).$

c$)$ Produce a graph with $4$plots; $U(t),K(t),W(t),FW(t)$

d$)$ Which did more work: the damper or the input force? Does that result make sense? $($Answer in one to two sentences$).$