Relations Involving $\omega$

We remarked above that $\omega$ had to have units of $t^{-1}$. The following are some True Facts involving $\omega$ that You Should Know:

$$\omega = \frac{2\pi}{T}$$ (9.15)

$$= 2\pi f$$ (9.16)

where $T$ is the period of the oscillator the time required for it to return to an identical position and velocity) and $f$ is called the frequency of the oscillator. Know these relations instantly. They are easy to figure out but will cost you valuable time on a quiz or exam if you don't just take the time to completely embrace them now.

Note a very interesting thing. If we build a perfect simple harmonic oscillator, it oscillates at the same frequency independent of its amplitude. If we know the period and can count, we have just invented the clock. In fact, clocks are nearly always made out of various oscillators (why?); some of the earliest clocks were made using a pendulum as an oscillator and mechanical gears to count the oscillations, although now we use the much more precise oscillations of a bit of stressed crystalline quartz (for example) and electronic counters. The idea, however, remains the same.