Appendix A:    Angular Frequency vs Angular Velocity

The angular frequency is a rather abstract concept which deserves some discussion.

\displaystyle \omega =\frac{{2\pi }}{T}=2\pi f

At the most basic level, from \displaystyle \omega =2\pi f, you should appreciate that the angular frequency ω is simply the frequency of the oscillation multiplied by \displaystyle 2\pi .

At a more abstract level, from \displaystyle x={{x}_{0}}\sin \omega t, we can think of \displaystyle \omega t=\theta as the phase angle of the oscillation. So ω is the “velocity” at which the phase of the oscillation progresses.

The angular frequency ω in SHM is actually very similar in concept to the angular velocity ω in circular motion. (This explains why they are both given the symbol ω.)

For oscillations, \displaystyle \omega =\frac{{d\theta }}{{dt}}=\frac{{2\pi }}{T}  is the rate of change of phase angle.

For circular motion, \displaystyle \omega =\frac{{d\theta }}{{dt}}=\frac{{2\pi }}{T} is the rate of change of angular displacement.

Lastly, a circular motion collapsed into one dimension is actually an SHM. For example, the displacement in the y-direction of the circular motion illustrated below is actually an SHM. The circular motion’s radius R, is the SHM’s amplitude R. The angular velocity ω of the circular motion is also the angular frequency ω of the SHM.

Animation

Circular Motion vs Oscillation

Pendulum Wave

Demonstration

Pendulum Wave (Harvard Natural Sciences)

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