# Spring constant in an underdamped system

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:$$m\ddot{x} + c\dot{x} = -k x$$...?

• What is $c$? When asking questions here, airways define all symbols. Commented Mar 15, 2017 at 6:18
• 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. Commented Mar 15, 2017 at 9:13

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.