Question

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(1 point) This problem is an example of over-damped harmonic motion. A mass m = 3 kg is attached to both a spring with spring constant k = 36 N/m and a dash-pot with damping constant c= 24 N · s/m. The ball is started in motion with initial position xo = -4 m and initial velocity vo = 2 m/s. Determine the position function x(t) in meters. X(t) = Graph the function x(t).

Answer 1

The differential equation is mx"+cx' + x = 0 3x" + 24 x' +36 x = 0 auxilary equation is 3r^2+24r +36 = 0 root r=-2,-6 Hence solution is x(t) = (c1)^(-2 t) + (c2)^(-6t) x'(t) =-2(c1)^(-2 t) +-6(c2)^(-6t)

Put x(0) = -4, x'(0) = 2 c1 + c2=-4 -2 cl +-6 c2 = 2 on solving cl=-5.5 c2 = 1.5 Hence solution is x(t) = -5.5 e^(-2 t) +1.5 e^(-6t) x(t) = - 11 e^(-2 t)/2+3 e^(-6 t)/2