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The reason I am stating this is because on calculating the units of w(omega) I found is equal to s^-1 not regular the rad/s. Proof:

    F=-kx =>  k= -F/x
           Newton/metre = (Kg-m)/s^2m = Kg/s^2 (Units of K)

If we take the book definition of kx=mw^2x the we get

                            k=mw^2 then w^2= K/m

                Units of this (Kg/s^2Kg)^1/2 = 1/s or s^-1 meaning frequency. 

Which makes more sense to me when considering a spring where applying w(angular velocity) seems less effective than f(frequency).

But I'd like to know if I made any mistakes if yes then an explanation would be very appreciated.

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rad/s and s^-1 are the same unit. Radians are dimensionless.

Also in this case $\omega$ is an angular frequency, not an angular velocity. So you can use either $\omega$ or $f$. It doesn't matter. They are essentially the same thing. $\omega=2\pi f$

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