A $12$ kg object is attached to a spring with spring constant $60$ kg$/$s$2.$ It is also attached to a dashpot with damping constant c$=72$ N$-$sec$/$m$.$ The object is pushed upwards from equilibrium with velocity $2$ m$/$s$.$ Find its displacement and time$-$varying amplitude for t$>0.$

y$($t$)=$

The motion in this example is

underdamped

overdamped

critically damped

Consider the same setup above, but now suppose the object is under the influence of an outside force given by F$($t$)=14$cos$($t$).$

What value for will produce the maximum possible amplitude for the steady state component of the solution? $$

What is the maximum possible amplitude?