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A slender uniform rigid bar of mass $m$ is hinged at $O$ and supported by two springs with stiffnesses $3k$ and $k$, and a damper with damping coefficient $c$, as shown in the figure. For the system to be critically damped, the ratio $c/\sqrt{km}$ should be

1. 2
2. 4
3. 2√7
4. 4√7

Correct answer: 4√7

Solution

The springs provide an equivalent restoring moment proportional to angular displacement, and the damper provides a resisting moment proportional to angular velocity. Using the geometry from the figure, the equivalent rotational parameters lead to a critical damping condition that gives $c/\sqrt{km}=4\sqrt{7}$.

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