13.2.1. Period and frequency

We have seen that any motion that repeats itself at regular intervals of time is called periodic motion. The smallest interval of time after which the motion is repeated is called its period. Let us denote the period by the symbol T. Its SI unit is second.

The concept of 'period' refers to the duration for one complete cycle of a periodic motion. For example, if you were to swing back and forth, the time from the start of the swing to when you return to that starting point is one complete cycle, which is your period T. It's important to note that the SI unit of this period is seconds, meaning we can measure how long it takes using a clock.

Think of a Ferris wheel: the period is the time it takes for a cabin to go all the way around and return to its starting position. If the cabin takes two minutes to complete this cycle, then the period T is 2 minutes.

For periodic motions which are either too fast or too slow on the scale of seconds, other convenient units of time are used. The period of vibrations of a quartz crystal is expressed in units of microseconds (10–6 s) abbreviated as µs. On the other hand, the orbital period of the planet Mercury is 88 earth days. The Halley’s comet appears after every 76 years.

Depending on the speed of the periodic motion, the unit of measurement for the period can change. For extremely fast motions, such as vibrations in crystals, we might use microseconds, which helps us measure very short periods. Conversely, for astronomical events like the orbits of planets or comets, we need much larger units like days or years.

Imagine you have a very noisy clock that 'ticks' at a high frequency. The period might be in milliseconds (ms) for a busy clock, but for measuring something like the return of a comet, you might use years, highlighting the vast difference in time scales.

The reciprocal of T gives the number of repetitions that occur per unit time. This quantity is called the frequency of the periodic motion. It is represented by the symbol ν. The relation between ν and T is ν = 1/T.

Frequency ν refers to how many cycles occur in a second, essentially measuring the rate of repetition of the periodic motion. If we know the period T, we can find frequency by taking the inverse. For instance, if a pendulum takes 2 seconds to complete one swing, its frequency is half a cycle per second, or 0.5 Hz.

Think of a drummer. If a drummer beats a drum once every second, their frequency is 1 Hz, which means one full motion each second. If the drummer speeds up and hits the drum twice in that same second, the frequency is now 2 Hz.

The unit of ν is thus s–1. After the discoverer of radio waves, Heinrich Rudolph Hertz (1857 –1894), a special name has been given to the unit of frequency. It is called hertz (abbreviated as Hz). Thus, 1 hertz = 1 Hz = 1 oscillation per second = 1 s–1.

Frequency is measured in hertz (Hz), where 1 Hz indicates one complete cycle per second. The unit hertz honors Heinrich Hertz, who contributed significantly to the understanding of waves. When we say something vibrates at 60 Hz, we mean it completes 60 cycles every second.

Imagine a strobe light at a concert flashing 60 times a second. If the strobe light is at 60 Hz, it means that light is turning on and off 60 times each second, creating a rhythm with the music that you can visually perceive.

Note, that the frequency, ν, is not necessarily an integer.

Frequency can take on any positive value, allowing for both whole numbers and fractions. This means that a system can oscillate at a frequency that isn’t a whole number—like 1.5 Hz, which would mean it completes one and a half cycles in a second. This flexibility in measurement is important for understanding more complex motions.

Think about a spinning record. If it spins fast, it might complete 33 and a third rotations every minute, indicating a frequency that isn’t a neat whole number—this is quite common in real-world applications.

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