a$)$ What is the order of the equation: mx$\text{'}\text{'}+$ bx$\text{'}+$ cx $=$ f$.$

b$)$ What is the order of the equation: mv$\text{'}+$ bv $=$ f$.$

c$)$ Is the following equation linear or nonlinear: mv$\text{'}=$ f $+$ cos$($v$).$

d$)$ Is the following equation linear or nonlinear: mv$\text{'}+$ a$($t$)$v $=$ f$.$

e$)$ The equation mv$\text{'}+$ bv $=$ f$($t$)$ is classified as time$-$invariant, but clearly f is varying with time and v will vary with time as a result. So$,$ why is it called time$-$invariant?

f$)$ Why are linear systems great to work with $($beyond the fact that the math is generally easier$)?$

g$)$ Assume a linear system has an input of the general form Asin$($wt$).$ What is the general form of the output? Be complete and precise.

h$)$ State the superposition principle. Be concise and precise... don't draw a diagram... just state the principle!

i$)$ What is the general approach to writing an equation of motion $($when using a free$-$body diagram or a circuit diagram $-$ we are NOT considering energy methods here$)?$ In other words, what "process" works for both mechanical and electrical systems?

j$)$ What is Newton's $2$nd Law $($do not assume constant mass$)?$

k$)$ What are the two forms of Kirkoff's Laws?

l$)$ Give an example of two linear constitutive relationships for typical mechanical systems.

m$)$ Given an example of two linear constitutive relationships for typical electrical systems.

n$)$ What is the characteristic shape of the time response for a first order system?

o$)$ For a first order system, where does the ~$63\%$ estimate come from when we say that this is the extent of motion from initial to final value after $1$ time constant? Explain the computation that results in the $0\mathrm{.}63$ factor.

p$)$ As the time constant gets larger, is a system more or less "responsive"?

q$)$Give an example of a zero$-$order system.

r$)$ What are the two primary motivating factors for using feedback to control a system?

s$)$ What are the three categories of second order system responses?

t$)$ Can a natural system be unstable?

u$)$ Can a controlled system be unstable?

v$)$ True or False: A system in force equilibrium must be stationary.

w$)$ What are the fundamental SI units of Force $($write the solution in terms of kg$,$ m$,$ sec$)?$

x$)$ Imagine a toy top is standing straight up on a table and ready to be spun $-$ but not currently spinning. You are looking down at the top from above, looking down the axis of rotation $($although the top isn't yet spinning$).$ You adjust the top slightly so that it is leaning just a little bit $-$ so that it is no longer perfectly upright. What happens and why? Be specific in describing the behavior, stating directions, and using physical principles to explain the behavior.

y$)$Now$,$ imagine the same situation as in $\text{'}$x$\text{'}$ except that the top is spinning rapidly $-$ the direction of spin is $"$up$"$ in the Right Hand Rule sense. Again, you adjust the top slightly so that it is leaning just a little bit $-$ so that it is no longer perfectly upright. What happens and why? Be specific in describing the behavior, stating directions, and using physical principles to explain the behavior.

z$)$ Imagine a hanging pendulum. You use a proportional controller in order to control the angular position of the pendulum so that it is at a position $45$ degrees off vertical. You can't seem to get the controller to work such that there is zero steady state error. What is the problem?