# I SHM and the sign of k

#### Mr Davis 97

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 $\omega$.

#### Math_QED

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k is positive by definition, at least in this physics context.

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