A massmis supported by a Hookian spring with a spring constantk. The damper provides a damping force that is proportional to the speed with a damping constant?. All constants are positive valued. At equilibrium, the length of the spring is shrunk bys, i.e.ks=mg. This equilibrium position is chosen to bey=0. The differential equation describing the motion of the mass is

my??+?y?+ky=0.

(a). [4] Identify the conditions for the three damping cases: 1) Over damped; 2) Critically damped; 3)Under damped.

(b). [3] Find the general solutiony(t) for the critically damped case. Furthermore, whent??

t??, what does solutiony(t)look like?

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