A spring$-$mass$-$damper system is shown in Fig. $3.$ The mass $M=1kg,$ damping

constant $b=10N*\frac{s}{m}$ and spring constant $k=5\frac{N}{m}.$

Figure $3$ Spring$-$mass$-$damper system for problem $2$

a$.$ Obtain the differential equations describing the system assuming

and for $t0$

b$.$ Find $Y(s)$

c$.$ Find $y(t)($Use the partial fraction expansion$)$

d$.$ Determine the poles and zeros of $Y(s),$ plot them.

e$.$ Find the damping ratio $$ and frequency ${}_n$

f$.$ Can the final theorem be used for $Y(s)?$ If so$,$ use it to find the steady$-$state

value of $y(t)$