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A single degree of freedom mass-spring-viscous damper system with mass $m$, spring constant $k$, and viscous damping coefficient $q$ is critically damped. The correct relation among $m$, $k$, and $q$ is

1. $q=\sqrt{2km}$
2. $q=2\sqrt{km}$
3. $q=\sqrt{2k/m}$
4. $q=2\sqrt{k/m}$

Correct answer: $q=2\sqrt{km}$

Solution

For a single degree of freedom mass-spring-damper system, the critical damping coefficient is $c_c=2\sqrt{km}$. Since the system is critically damped, the given viscous damping coefficient must equal this value.

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