rrs$\text{'}2j+|||=y^{\text{'}}|g+ky|=0$

For a mass$-$spring oscillator, Newton's second law implies that the position $y(t)$ of the mass is governed by the second$-$order differential equation

$m{y}^{\text{'}\text{'}}(t)+b{y}^{\text{'}}(t)+ky(t)=0.$

$($a$)$ Find the equation of motion for the vibrating spring with damping if $m=20kg,b=80k\frac{g}{sec,k}=260k\frac{g}{se{c}^2,y(0)}=0\mathrm{.}2m,$ and $y^{\text{'}}(0)=-0\mathrm{.}1\frac{m}{sec}.$

$($b$)$ After how many seconds will the mass in part $($a$)$ first cross the equilibrium point?

$($c$)$ Find the frequency of oscillation for the spring system of part $($a$).$

$($d$)$ The corresponding undamped system has a frequency of oscillation of approximately $0\mathrm{.}574$ cycles per second. What effect does the damping have on the

frequency of oscillation? What other effects does it have on the solution?

$(a)y(t)=$

help with parts A$-$D