Consider a mass$-$spring$-$damper system with mass m$,$ damping b$,$ and spring constant k$,$ hanging from the ceiling. Define the displacement x$($t$)$ as positive when moving downward, and assign x $=0$ when the system is in an equilibrium due to its own weight $($so that you don$$t need to further consider the weight as a force acting on the mass$).$

a$)$ Assuming a spring constant of k $=100$ N$/$m$,$ find the mass that will lead to an oscillation period of $1$ second if the mass is released from some nonzero initial position, if we assume no damping.

b$)$ Using the results of part $($a$),$ figure out the amount of damping that would be required to make the response critically damped.

c$)$ Include simulations showing both of your results, for x$(0)=1$ cm$.$ Make sure to label the axes, and provide descriptive titles. Also turn in a printout of whatever code that you used to generate the plots.