# Angular Frequency interpretation?

I just began a topic on waves and oscillations and came across a term angular frequency ($\omega$) which was stated as $2\pi f$. However, I have seen the same symbol $\omega$, used for angular velocity. I know how to interpret angular velocity as the rate of change of angular displacement and how it relates to rotational motion, however I'm really confused with angular frequency as it relates to Simple Harmonic Motion and how I'm meant to interpret it.

You can take it as simply a definition, but the best way to motivate the definition is to observe that simple harmonic motion is the motion of the projection onto a straight line of the motion of a particle undergoing uniform circular motion (i.e. with constant angular velocity about the circle's center).

So, imagine a particle undergoing uniform circular motion on a unit radius circular path centered at the origin of a Cartesian co-ordinate system and with the motion confined to the $x-y$ plane. If the particle's constant angular speed is $\omega$, then the $x$-component of the particle's motion is described by $x=\cos(\omega\,t+\delta)$, for some constant phase $\delta$.

• So angular frequency is simply the projection of the angular velocity on a straight line? ? – MathLearner May 21 '17 at 14:59
• Might be helpful for some users to note that the projection scheme is sometimes called the "reference circle" and that from a slightly more technical perspective it is formally a "phasor". – dmckee --- ex-moderator kitten May 21 '17 at 18:18

The function describing a simple harmonic oscilator is:

$$x(t) = A \cos(\omega t+\phi)$$

which satisfies

$$m\frac{d^2x(t)}{dt^2}=-k\,x(t).$$

You can also verify that

$$\omega=\sqrt{\frac{k}{m}}.$$

Now imagine having a vector with length A and imagine that this vector starts rotating with a constant angular velocity $\omega$ and phase $\phi$. It's clear that the projection of this vector on the x-axis is exactly equal to the first equation.

We conclude that these two motions can be described with equivalent equations and therefore we call the parameter $\omega$ the angular velocity of the HO.