Question

A system of unit mass whose natural angular frequency in the absence of damping is 4 rad –1 is subject to a damping force which is proportional to the velocity of the system, the constant of proportionality being 10 –1. Show that the system is over damped and that the general solution for the displacement is

x = A exp (–2t) + B exp (– 8t).

The mass is initially at x = 0.5 m and given an initial velocity v towards the equilibrium position. Find the smallest value of v that will produce negative displacement.

Answer 1

The equation of motion is

In order to make x negative it is necessary that 4-v/6 should be negative or, v > 4. Thus v = 4 ms–1 is the minimum value of v required to give a negative displacement to the system.