In simple harmonic motion, two quantities describe the timing of oscillations: the period $T$ and the frequency $f$.

Period $(T)$

The period $T$ is the time required for one complete cycle. After time $T$, the system returns to the same position with the same velocity and direction of motion. The SI unit of period is the second ($\text{s}$).

Frequency $(f)$

The frequency $f$ is the number of cycles that occur per unit time. The SI unit of frequency is the hertz ($\text{Hz}$), where

$$1 \text{ Hz} = 1 \text{ s}^{-1}$$

Since period and frequency describe the same oscillation from different perspectives, they are reciprocals of each other:

$$f = \frac{1}{T}$$ $$T = \frac{1}{f}$$

If a system completes one cycle in time $T$, then $1/T$ cycles occur each second. Conversely, if the frequency is $f$, each cycle takes time $1/f$.

A higher frequency corresponds to a shorter period. If the period doubles from $T$ to $2T$, the frequency becomes $f/2$.

Angular Frequency $(\omega)$

The angular frequency $\omega$ describes how rapidly the oscillation progresses through its cycle in terms of radians rather than complete cycles.

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