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{"ops":[{"insert":"A block of mass m is attached to a spring, with spring constant k, and a dashpot. The dashpot provides a retarding force with damping rate b and proportional to the velocity \u02d9x. When the block is a distance d0 from the left wall the spring is relaxed. When extended beyond d0 and released the block exhibits damped oscillatory motion. You may assume that gravitational effects are negligible. (a) Show that the equation of motion of this sytem has the form x\u00a8 + 2\u03b3x\u02d9 + \u03c9 2 0x = 0 and identify the variables, \u03b3 and \u03c90. (b) Consider the case when \u03b3 < \u03c9 (under-damping). The block is now pulled to the right a distance x = a, from its equilibrium position, and at time t = 0 it is released from rest. Show that the subsequent motion is given by, x(t) = ae\u2212\u03b3t \u0012 cos \u03c91t + \u03b3 \u03c91 sin \u03c91t \u0013 where \u03c91 = p \u03c9 2 0 \u2212 \u03b3 2. (c) At what time does the block first change its direction of motion after being released?\n"}]}

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