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In Simple Harmonic Motion in one dimension, if we assume $$\text{Displacement}=x=A \text{sin} (\omega t+\phi)\implies \text{velocity}=v=A \omega \text{cos} (\omega t+\phi)$$

From here by substitution and use of the Pythagorean identity we can show that: $$|v|=|\omega\sqrt{A^2-x^2}|$$

However, can we remove the magnitude/mod signs and say: $$v=\omega\sqrt{A^2-x^2}$$

I am not sure for two reasons:

  • The sign of the square root
  • If $\omega$ can be negative
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However, can we remove the magnitude/mod signs...

In general, no.

For example, supposing that $\omega$ is positive (as one may do), you can still have a positive displacement at the same time as a negative velocity (which would require the negative signed root rather than the positive). In fact, half the time when the displacement is positive the velocity is negative and half the time when the displacement is positive the velocity is positive.

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  • $\begingroup$ but $\cos(-\omega t + \phi) = \cos(+\omega t - \phi)$, so the negative frequency is just a redefinition of phase...I guess related to the sign of $v_{(x=0)}$ when $\phi\equiv 0$. $\endgroup$
    – JEB
    Commented Apr 24 at 5:27

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