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How can Simple Harmonic Motion have angular frequency?

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It isn't making any intuitive sense. If it isn't moving in circular motion, how can it have angular frequency or speed?

Also,

[tex]v=\pm ω\sqrt { A^{ 2 }-x^{ 2 } }[/tex]

only applies to SHM with springs only, right?

Also, does anyone know how to derive this equation below?

[tex]x=\frac { \pm \sqrt { { { v }_{ max } }^{ 2 }+{ v }^{ 2 } } }{ ω }[/tex]

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