Consider the mass$-$spring$-$damper system shown in Figure $??.$ The mass is $1$ kg and the damping is $6N.\frac{s}{m}.$

$($a$)$ We first use a spring with stiffness of $10\frac{N}{m}.$ Obtain the poles of the system, determine if the system

is stable, and the time constant of the system. Does the system oscillate? If it does what is the

frequency of oscillation?

$($b$)$ Now we replace the spring with a softer spring with stiffness of $5\frac{N}{m}.$ Repeat part $($a$).$

$($c$)$ Put your pencil down for a minute and think a little about the physics of the system. Is it possible for

a mass$-$spring$-$damper system with arbitrary mass, stiffness and damping to be unstable $($growing free

response$)?$ Try and see if you can support your answer by some math.

Figure $1$: The mass$-$spring$-$damper system.