Question

The remaining problems in this section deal with free damped motion, a mass $m$ is attached to both a spring (with given spring constant $k$ ) and a dashpot (with given damping constant c). The mass is set in motion with initial position $x_{0}$ and initial velocity $v_{0}$. Find the position function $x(t)$ and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form $x(t)=$ $C_{1} e^{-p t} \cos \left(\omega_{1} t-\alpha_{1}\right) .$ Also, find the undamped position function $u(t)=C_{0} \cos \left(\omega_{0} t-\alpha_{0}\right)$ that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so $c=0) .$ Finally, construct a figure that illustrates the effect of damping by comparing the graphs of $x(t)$ and $u(t)$. $m=\frac{1}{2}, c=3, k=4 ; x_{0}=2, v_{0}=0$

          The remaining problems in this section deal with free damped motion, a mass $m$ is attached to both a spring (with given spring constant $k$ ) and a dashpot (with given damping constant c). The mass is set in motion with initial position $x_{0}$ and initial velocity $v_{0}$. Find the position function $x(t)$ and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form $x(t)=$ $C_{1} e^{-p t} \cos \left(\omega_{1} t-\alpha_{1}\right) .$ Also, find the undamped position function $u(t)=C_{0} \cos \left(\omega_{0} t-\alpha_{0}\right)$ that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so $c=0) .$ Finally, construct a figure that illustrates the effect of damping by comparing the graphs of $x(t)$ and $u(t)$.  
$m=\frac{1}{2}, c=3, k=4 ; x_{0}=2, v_{0}=0$

Added by Julie D.

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