You have a mass$-$spring system that uses a $3$ kg mass with unknown spring constant $k$ and unknown dampening coefficient $.$ When a force of $F(t)=9cos(10t)N$ is applied to the system, it exhibits pure resonance.

$($a$)$ What does this information tell you about the values of $$ and $k?$

$($b$)$ In this scenario, what are some possible options$/$adjustments that could be made to have the system display the beats behavior instead?

$($c$)$ Assume that, with the same mass and applied force, we attach a spring with constant $\frac{14}{5}\frac{N}{m}$ and add a dampener with coefficient $3N\frac{s}{m}.$ Write a differential equation to model the behavior of this system over time.

$($d$)$ If you apply a constant force of magnitude $9$ N $($instead of the oscillating force$),$ how far do you expect the mass to move?

$($e$)$ If you were to solve out the entire problem, you would get a solution of the form

$y(t)=C_1{e}^{-t}cos(t)+C_2{e}^{-t}sin(t)+$Acos$(t)+$Bsin$(t).$

How would you determine if this system is exhibiting practical resonance based on the solution?