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I just want to verify, the spring constant is independent of whether there is damping or not right? i.e. If I were to determine the spring constant of an undamped spring through $F=-kx$, can it also be used for equations such as $$\omega = \sqrt{\frac{k}{m}-\left( \frac{c}{2m}\right) ^2}$$ where $c$ is the damping co-efficient of the damped harmonic oscillator:$$m\ddot{x} + c\dot{x} = -k x$$...?

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  • $\begingroup$ What is $c$? When asking questions here, airways define all symbols. $\endgroup$
    – DanielSank
    Mar 15 '17 at 6:18
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    $\begingroup$ Actually, in this analysis, it is considered that the damped string is a combination of the undamped spring and the damping force. The damping force acts on the body itself, so the spring constant doesn't get modified (in the appropriate usual limits), only the motion of the spring does, on which the spring constant doesn't depend. $\endgroup$ Mar 15 '17 at 9:13
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As Kalpak Gupta points out, damping is a force which always reduces the resultant force on the spring, and thus also the amplitude of oscillations. All elastic materials or devices obey Hooke's Law within a small enough range of displacements. If the damping force does not take the resultant force outside of the current linear range, the spring constant remains the same.

If the linear range extends to zero amplitude and the current amplitude is within the linear range, then damping will have no effect on the spring constant, because damping always reduces amplitude.

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  • $\begingroup$ sammy gerbil, can you give a reference for non-ideal springs? $\endgroup$ Mar 15 '17 at 23:06
  • $\begingroup$ Thank you for your comment @DavidWhite, as a result of which I revised my answer. $\endgroup$ Mar 16 '17 at 0:02
  • $\begingroup$ sammy gerbil, note that my question above wasn't a criticism or request for a revised answer. I teach AP physics at the high school level, and we do two spring labs during the year. A good reference that details when real springs act in a non-ideal fashion would be somewhat helpful for me and my future students. $\endgroup$ Mar 16 '17 at 15:41
  • $\begingroup$ Sorry I don't know of any suitable references. The dependence of the spring constant in the linear range on spring parameters might be useful - access restricted. I found a few other useful sites, but none explaining when the linear range starts and stops. $\endgroup$ Mar 16 '17 at 16:38
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Hook's law is defined just in the linear force case and the Hook's constant $k$ is an intrinsic property for a spring. As long as the spring stays linear, i.e. the amplitude of ossicilation is small, the Hokk's force is $F=-kx$ everywhere. So, the motion equation and the frequency are all right. You can use it to determine Hook's constant.

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