If , the system mx″ + μx + kx = 0 is over-damped. The system is allowed to come to equilib...

If , the system mx″ + μx + kx = 0 is over-damped. The system is allowed to come to equilibrium. Then the mass is given a sharp tap, imparting an instantaneous downward velocity υ0.

(a) Show that the position of the mass is given by

where

(b) Show that the mass reaches its lowest point at

a time independent of the initial conditions.

(c) Show that, in the critically damped case, the time it takes the mass to reach its lowest point is given by t = 2m/μ.