Period and Frequency

The period is the time for one complete revolution (or rotation).

The frequency is the number of revolutions/rotations per given time period.

The normal SI unit for time is s or seconds.

Period, T, and frequency, f, have the simple inverse proportionality:

\[ \rm \mathbf{f = \frac{1}{T}} \]

The unit of frequency is then s^-1. This unit has a special name, Hertz (Hz):

1 Hz = 1 s^-1

On the other hand, T is also the time to travel once around the circle, covering a total distance of 2$\pi$r. The average speed is then given by

\[ \rm \mathbf{ \frac{2\pi r}{T}} \]

Combining this result with v = r$\omega$ gives:

\[ \rm \mathbf{r \omega = \frac{2 \pi r}{T} = 2 \pi r f} \] or

$\omega$ = 2$\pi$ f

Sometimes $\omega$, the absolute value of the angular velocity vector, is called the "angular frequency". It is given in radians per second as opposed to rotations or cycles per second for frequency. (Omission of the factor 2$\pi$ between the frequency and angular frequency is a very common error!)

Examples:

Period of rotation of the Earth: T = 1 day = 1 d $\cdot$ 24h/1d $\cdot$ 60 min/1h $\cdot$ 60 s/1min = 86,400 s
The Earth's rotation frequency is then: f = 1/day = 1.157$\cdot$ 10^-5 s^-1
The Earth's angular frequency is thus: $\omega$ = 2$\pi$/day = 7.272$\cdot$ 10^-5 rad$\cdot$ s^-1