#$1(40$ points$)$

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.

For guidance, look at the MATLAB program in the section "Generalizing computationally $-$ Euler's Method" of chapter "Step $3$: Solving differential equations of motion" in the webtext.

$1-1$ System A shows a spring$-$damper system. Parameters for the spring and damper are $k_1=1600\frac{N}{m}$ and $b_2=900N**\frac{s}{m}.$ The spring has an initial extension$/$displacement of $0\mathrm{.}04$ m $.$ Analyze System A with the following steps:

a$)(4$ pts$)$ Write force balance, geometric continuity, and constitutive equations.

b$),(4pts)$ Obtain a differential equation of motion for the extension of the spring.

c$)(8$ pts$)$ Derive the particular analytical solution for this system with the parameters and initial condition listed above. Hint: Refer to the section "Analytical Solutions to Differential Equations" of chapter "Step $3$: Solving differential equations of motion" in the webtext.

d$)(6$ pts$)$ Simulate this system for both timesteps $dt=0\mathrm{.}5$ and $0\mathrm{.}1$ seconds for a total of $5$ seconds using the forward Euler method in MATLAB. Plot the simulation for both timesteps as well as your analytical solution from Part c $($for $5$ seconds$)$ in one graph. Be sure to tnclude axis labels with units and a legend.

e$)(2$ pts$)$ Explain what happens to the MATLAB solution for the spring extension$/$displacement as dt decreases.