I've seen in many books, internet and video lectures, that equation of SHM:
$$x(t)=A\cos(\omega t+\phi)$$
where they just say that (without telling anything about $\omega$): $$\omega=\frac{2\pi}{T}$$
where $\omega$ is Angular Frequency, $A$ is the amplitude, $\phi $ is the phase angle and $T$ is time-period.
But I want to know from where this formula ($\omega =\frac{2\pi}{T}$) came (or how to derive it)? And what's the intuition behind angular frequency?
Note: I don't want an explanation/answer based on/related to circular motion.
Best Answer
The cosine is a periodic function. So, for the function:$A\cos(\omega t + \phi)$, when $t = 0$ its value is $A\cos(\phi)$. After a time $T$ the function repeats itself when $T$ is such that: $A\cos(2\pi + \phi)$. It happens when, $$2\pi = \omega T \implies \omega = \frac{2\pi}{T} $$