Question

Find the frequency (in hertz) and the angular velocity of a repeating waveform whose period is If a periodic waveform has a frequency of \(60.0 \mathrm{Hz}\), how many seconds will it take to complete 200 cycles?

Short answer

Total time for 200 cycles at 60.0 Hz is 3.33 seconds.

Step-by-step solution

Understand Frequency

Frequency (f) is the number of cycles per second of a periodic waveform. It is measured in hertz (Hz). If a waveform has a frequency of 60.0 Hz, this means it completes 60 cycles in one second.

Calculate time for one cycle

To find the time for one cycle, we take the reciprocal of the frequency. The time (T) for one cycle, also known as the period, is given by: \( T = \frac{1}{f} \).

Calculate the total time for 200 cycles

Since the time for one cycle is the period T, to find the total time for 200 cycles, we multiply the period by the number of cycles: \( \text{Total time} = T \times 200 \). Use the period from step 2.

Execute the calculation

Plug in the frequency value into the formula from step 2 to calculate the period: \( T = \frac{1}{60.0 \text{ Hz}} = \frac{1}{60.0} \text{ s} \). Then multiply the period by 200 cycles to get the total time.

Key concepts

Angular Velocity

When delving into the world of waveforms and their behaviors, it's essential to understand angular velocity. This concept refers to the rate at which an object rotates or revolves around a central point – in waveforms, this central point is commonly a cycle's beginning.

Mathematically, angular velocity, often denoted by the Greek letter omega (\r), is calculated by multiplying 2\raid\rf JLaTeX, the unit for angular velocity is radians per second (\rfrac{radians}{s}\r). To convert frequency, which is in hertz (Hz), to angular velocity we use the formula:
\[\r = 2\raid \r Imposses'f\]
For a waveform with a frequency of 60 Hz, the angular velocity would be:\[\r = 2\raid \raid(60 \r Hz)\].
We find that the waveform completes a full 360-degree rotation 60 times per second, which shows how frequently it repeats the rotational cycle.

Waveform Period Calculation

Understanding the period of a waveform is another fundamental aspect in analyzing periodic phenomena. The period is the duration of time it takes for a complete cycle to occur. Essentially, it's the time between two consecutive points that are in the same phase within the waveform.

In our problem, we determined the period (T) by inverting the frequency:
\[T = \rfrac{1}{f}\].
For a waveform at 60.0 Hz, its period is:\[T = \rfrac{1}{60.0 \r;\rHz} = \rfrac{1}{60.0} \r;\rs\]
which corresponds to roughly 0.0167 seconds. Calculating the period is crucial because it helps us predict when the waveform will complete its cycles and how it will interact with other waveforms or systems.

Hertz (Hz)

The hertz (Hz) is the unit of frequency in the International System of Units (SI) and is vital to understanding waveforms. One hertz represents one cycle per second. For example, a waveform with a frequency of 60 Hz completes 60 cycles every second.

Frequencies in hertz can be of any value and typically range from the very low (like the slow undulation of a geological movement) to the extremely high (like the oscillation of light waves).
When working with electronics, sound waves, and other periodic signals, you'll often find hertz used to describe their oscillation rates. It's a simple yet powerful way to quantify the repeat rate of any periodic event.

Cycles per Second

The term 'cycles per second' is a straightforward way of expressing the frequency of a repeating event. Each cycle represents a complete set of occurrences within the waveform, returning to its starting point and ready to begin anew.

Frequency and cycles per second are synonymous, with the former usually expressed in terms of hertz. As seen in our problem, a 60 Hz frequency is equivalent to saying there are 60 cycles every second.
When we talk about calculating the time for multiple cycles, like the 200 cycles in our exercise, we're simply multiplying the period of one cycle by the number of cycles:
\[\rtext{Total time} = T \times 200\].
This provides us with the total duration needed for the waveform to complete 200 full cycles, enabling precise timing necessary for understanding and manipulating periodic phenomena in practical applications.