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# Classify the motion of the mass on a spring as either underdamped, critically damped, or overdamped, or nd the range of the parameter value that will give the desired result. Assume x = x(t), that is, x is a function of t. $x^{\prime \prime}+b x^{\prime}+4 x=0$, underdamped

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Recall that to classify the motion of the mass on a spring whose movement is modeled by the equation

$mx^{\prime\prime} + b x^\prime +k x =0$

it is sufficient to determine the sign of

$\pmb{ b^2-4mk}$

If $b^2-4mk$ is positive, the motion is overdamped, if it equal to zero to motion is critically damped, and if $b^2-4mk$ is negative the motion is underdamped.

In our exercise we have that the equation is

$\textcolor{#4257b2}{ x^{\prime\prime} + bx^\prime +4x=0 }$

We have

$\pmb{ m=1, \,\, k=4}$

and we want the movement to be $\textbf{underdamped}$ which leads us to the inequality

\begin{aligned} & \pmb{ b^2-4mk < 0} \Leftrightarrow \\ & b^2 -16 < 0 \Leftrightarrow \\ & b^2 <16 \Leftrightarrow \\ & \pmb{ b < 4} \end{aligned}

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