A swing completes a back-and-forth cycle every 2 seconds. What is the frequency of the swing? A. \(0.5 \mathrm{~Hz}\) B. \(1 \mathrm{~Hz}\) C. \(0.2 \mathrm{~Hz}\) D. \(2 \mathrm{~Hz}\)

When we talk about the period and frequency of a swing, we are discussing how often something happens in a specific amount of time. The period is the time it takes for one complete cycle of an event. For example, the period of a swing is the time it takes to go from one side, back to the other side, and return to the starting point.

Frequency, on the other hand, measures how many cycles happen in one second. The relationship between period (T) and frequency (u) is quite simple: they are inversely related.
This means that if you know the period, you can find the frequency using the formula: \[ u = \frac{1}{T} \]
Here, \[ u \] is the frequency in Hertz (Hz), and \[ T \] is the period in seconds. So, if something has a longer period, it has a lower frequency, and vice versa.

Understanding this relationship helps us solve problems related to cycles, like the swinging motion of playground swings.

Hertz (Hz) is the unit used to measure frequency. It tells us how many cycles occur in one second. For instance, if a swing has a frequency of 1 Hz, it means the swing completes one full cycle every second.

The term Hertz is named after Heinrich Hertz, a German physicist who made significant contributions to the study of electromagnetism. Using Hertz as a unit allows us to easily compare the frequencies of various cyclical events.
To illustrate, let's consider the swing example where the period is 2 seconds. Using the formula for frequency:
\[ u = \frac{1}{2} = 0.5 \text{ Hz} \]
So, the swing's frequency is 0.5 Hz, meaning it completes half a cycle every second. This measure helps us understand how frequently the swing is moving back and forth.

Let's take a detailed look at our swing example. The swing completes one full back-and-forth motion every 2 seconds. This 2-second duration is called the period (T).

To find out how often the swing completes a back-and-forth motion per second, we need to calculate its frequency (u). As we already know, frequency is the inverse of the period:
\[ u = \frac{1}{T} \]
Given that \[ T \] is 2 seconds, we substitute it into the formula:
\[ u = \frac{1}{2} = 0.5 \text{ Hz} \]
So, the swing has a frequency of 0.5 Hz, meaning it completes half a cycle in one second.

Analysing the swing's period and frequency helps us understand its movement. In real-world applications, this analysis is crucial in areas like engineering and physics, where understanding cyclical motions can inform everything from designing safe playground equipment to creating effective mechanical systems.