# SHM and the sign of k

I have the equation for simple harmonic motion ##\displaystyle \frac{d^2x}{dx^2} + k^2 x = 0##. I have a simple question. Do we need to make an assumption about the sign of ##k## before we solve this? We have that the roots satisfy ##r^2=-k^2##. So ##r=\pm i \sqrt{k^2}##. Do I need to assume ##k## is either positive or negative before I can proceed?

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Simple harmonic motion is what happens with a system that has a mass attached to a spring. The equation is ## m \frac{d^2 x}{dt^2}=-kx ##. The mass ## m ## and spring constant ## k ## are always both positive. The mass times acceleration is equal to the force which is opposite the displacement. ## \\ ## In the problem above ## k^2>0 ##. The sign of ## k ## doesn't matter. Most often, your ## k ## is actually written as ## \omega ##.

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