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.
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