Question

Find the period and the angular velocity of a repeating waveform that has a frequency of $$68.0 \mathrm{Hz}$$

Short answer

The period of the waveform is approximately 0.0147 seconds, and the angular velocity is approximately 427.256 rad/s.

Step-by-step solution

Understanding the Relationship between Frequency and Period

The period of a repeating waveform is the duration of time it takes to complete one full cycle of the wave. This is inversely related to the frequency, which is the number of cycles that occur in one second. The relation can be described by the formula: \( T = \frac{1}{f} \), where \( T \) is the period and \( f \) is the frequency.

Calculating the Period of the Waveform

Given the frequency \( f = 68.0 \text{ Hz} \), we can calculate the period by taking the reciprocal of the frequency. \( T = \frac{1}{68.0} \).

Finding the Angular Velocity

Angular velocity, \( \omega \), is related to the frequency by the formula \( \omega = 2\pi f \). So, after finding the period, we can use the given frequency to calculate the angular velocity.

Perform the Calculations

First we find the period: \( T = \frac{1}{68.0} \approx 0.0147 \text{ s} \). Then we calculate the angular velocity using the frequency: \( \omega = 2\pi \times 68.0 \approx 427.256 \text{ rad/s} \).

Key concepts

Waveform Period

When studying repeating waveforms, like those in acoustics or electronics, it's crucial to understand the concept of the waveform period. The period, often denoted as T, is the time it takes for one complete cycle of the waveform to occur. Imagine a sine wave, which starts at the origin, reaches its maximum, descends back to the origin, drops to its minimum, and then returns once more to the origin, completing one full oscillation. The duration of this process is the period of the waveform.

For our example, with a frequency of 68 Hz, the period is the inverse of this frequency. Grasping this relationship helps in visualizing waveforms in terms of time rather than just their cyclic nature. This understanding is fundamental in fields like signal processing, telecommunications, and physics, where the timing of wave events is as significant as their spatial characteristics.

Frequency

At the heart of analyzing waveforms is the concept of frequency, symbolized as f. It quantifies how often a particular event, like a wave cycle, occurs within a unit of time, usually one second. Frequency is measured in hertz (Hz), where one Hz represents one cycle per second.

High frequency means more cycles occur each second, which relates to higher pitched sounds in acoustics or tighter wave patterns in optics. In our exercise, with a frequency of 68 Hz, there are 68 complete cycles of the waveform happening every second. Understanding frequency allows us to comprehend and design systems that use waves to convey information, from radio broadcasts to Wi-Fi signals.

Relationship between Period and Frequency

The relationship between the period (T) and frequency (f) of a wave is a fundamental concept that links time and rate. They are inversely proportional to each other, which means when one increases, the other decreases. The formula that describes this relationship is T = 1 / f, or conversely, f = 1 / T.

From our calculation, the period of a waveform with a 68 Hz frequency is about 0.0147 seconds. This inverse relationship allows us to easily convert between the time it takes for one cycle and how many cycles occur per second. It is beneficial when dealing with various phenomena such as electromagnetic waves, sound waves, and alternating currents. This concept is also essential in tuning instruments or configuring electronic circuits to operate at specific intervals.