Why is $\omega $ the natural/angular frequency?

Question

Pardon me cause I'm a little confused. If we have something like: $y=A\sin \left( \omega t-\delta \right)$ why would $\omega$ be considered the natural frequency? I always thought the frequency of a function like that would be $\frac{2\pi }{\omega }$. Why is it that $\omega$ is the frequency and has the unit: rad/s?

Answer 1

$sin(\omega t)$ will repeat with period $ T = \frac{2 \pi}{\omega}$ as you point out, but the frequency is defined as $f = \frac{1}{T}$ $= \frac{\omega}{2\pi}$. Then the angular frequency is defined as $2 \pi f = 2\pi \frac{\omega}{2\pi} = \omega$.

The reason we use the angular frequency, $\omega$, is because the $2\pi$ is always present and so to know how quickly the function repeats, i.e. to have an intuitive idea of it's frequency we don't need the $2 \pi$ bit to be mentioned every time. It is enough to know $\omega$ since large $\omega$ means the function repeats quickly and small $\omega$ mean the function takes a long time to repeat.