solve x(t) please This problem is an example of critically damped harmonic motio

Question

solve x(t) please

This problem is an example of critically damped harmonic motion. A mass m = 3 kg is attached to both a spring with spring constant k = 147 N/m and a dash-pot with damping constant c = 42 N middot s/m. The ball is started in motion with initial position x_0 = 6 m and initial velocity v_0 = - 44 m/s. Determine the position function x (t) in meters. x (t) = Graph the function x (t). Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Solve the resulting differential equation to find the position function u (t). In this case the position function u (t) can be written as u (t) = C_0 cos (omega _0 t - alpha _0). Determine C_0, omega _0 and alpha _0. C_0 = omega _0 = alpha _0 = (assume 0 lessthanorequalto alpha _0

Explanation / Answer

Given that mass m = 3 kg; spring constatnt k = 147 N/m; damping constatnt c = 42N. s/m;

Initial conditions x0 = 6m and initial velocity v0 = -44m/s.

The position function under critically damped harmonic motion is,

      x(t) = e-pt (C1+C2t)

     p = c/ 2m = 42/(2*3) = 7

       C1 = x0 = 6

      C2 = v0+x0p = -44 + 6*7 = -2

Threre fore,

x(t) = e-7t (6 + -2t)

     

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