What is the relationship between frequency and period?

Oscillations are repeating motions that occur at regular time intervals. The time for one complete cycle of the motion before it repeats is called the period. For example, the period of the Earth orbiting the Sun is approximately 365 days, and the period of the Earth’s rotation around its axis is one day. If a person is playing jump rope, the period of the rope’s turns could be approximately two seconds, or for a spinning ceiling fan, it might take just half a second for a complete cycle. The SI symbol for period is $\textit{T}$. The examples here show that period is a time measurement per cycle. The concept of a period can also be written as an equation:

$\hspace{3em}$ $\ce{period}=\dfrac{\ce{time}}{\ce{cycle}}$

Frequency is a value that tells us the number of cycles of a repeating motion that occur in a given amount of time. The symbol for frequency is $f$. If we say that a person has a heart rate of 55 beats per minute, we are stating the frequency of their heartbeat. Vinyl records used for playing music typically rotate at 33 rpm or revolutions per minute, which is also a measure of frequency. We can say that the frequency of Earth’s rotation is one rotation per day. Notice that all of the examples above involve a number of cycles per unit time. The definition of frequency can also be written as an equation:

$\hspace{3em}$ $\ce{frequency}=\dfrac{\ce{cycles}}{\ce{time}}$

The relationship between frequency and period is that they are inverses of one another. This relationship can be expressed through the equations:

$\hspace{3em}$ $f=\dfrac{1}{T} \hspace{3em}$and $\hspace{3em}T=\dfrac{1}{f}$

We can show how this equation connects the definitions of period and frequency from the first two paragraphs.

$\hspace{3em}$ $\ce{period}=\dfrac{\ce{time}}{\ce{cycle}}$

$\hspace{3em}$ $\ce{frequency}=\dfrac{1}{\ce{period}}=\dfrac{1}{\frac{\ce{time}}{\ce{cycle}}}=\dfrac{\ce{cycle}}{\ce{time}}$

Note that in IB Standard Level Physics and IB Higher Level Physics, period is normally given in the units of seconds (s), and frequency has an SI unit of hertz (Hz). Through the relationship between frequency and period, we can explore the relationship between these two units.

$\hspace{3em}$ $f=\dfrac{1}{T} \hspace{3em}$therefore$\hspace{3em}\ce{Hz}=\dfrac{1}{\ce{s}} =\ce{s}^{-1}$

As an example, consider an object that oscillates with a frequency of 5 Hz. If we want to find the period, we can apply the equation

$\hspace{3em}$ $T=\dfrac{1}{f}=\dfrac{1}{5\ \ce{Hz}}=$ 0.2 s

Therefore, it takes 0.2 s for the object to complete one oscillation.

Waves are generated by an oscillating source. The source causes a disturbance in the medium, resulting in the particles in that medium undergoing oscillations. The frequency of a wave is equal to the frequency of these individual oscillations of its particles. The period of the wave can be found from the same inverse relationship as before,

$\hspace{3em}$ $T=\dfrac{1}{f}$