If > 2√/km, the system mx" + px' + kx = 0 is over- damped. The system is allowed to come to equilibrium. Then the mass is given a sharp tap, imparting an instanta- neous downward velocity vo (a) Show that the position of the mass is given by where Vo x(t) = e-ur(2m) sinh yt. u²-4mk Y = 2m (b) Show that the mass reaches its lowest point at 2my н t= tanh1 Y 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/u.

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