Question

Find the period and the frequency of a sine wave that has an angular velocity of $$455 \mathrm{rad} / \mathrm{s}$$

Short answer

The period of the sine wave is \( \frac{2\pi}{455} \) seconds, and the frequency is \( \frac{455}{2\pi} \) Hz.

Step-by-step solution

Understanding the Relationship Between Angular Velocity and Period

The period of a wave, denoted as T, is the time in seconds for one complete cycle. The angular velocity, denoted as \( \omega \), is the rate of change of the angle with respect to time. The relationship between the period and the angular velocity is given by the equation \( T = \frac{2\pi}{\omega} \).

Calculating the Period

Given the angular velocity \( \omega = 455 \, \text{rad/s} \), we can calculate the period T by using the formula \( T = \frac{2\pi}{\omega} \). Substitute the value of \( \omega \) into the equation to find the period.

Finding the Frequency

The frequency, denoted as f, is the number of cycles per second. It is the inverse of the period, which can be represented by the equation \( f = \frac{1}{T} \). Once we have the period, we can calculate the frequency by taking its reciprocal.

Key concepts

Period of a Wave

When studying waves, one of the fundamental properties to understand is the period of a wave. Simply put, the period (T) refers to the duration it takes for one complete oscillation or cycle of the wave. In practical terms, if you were to watch a wave in the ocean, the period would be the time between two consecutive crests (the highest points) passing by a fixed point.

Using this concept, students can understand how to determine the length of time associated with each cycle of a wave, which is crucial in various fields such as physics, engineering, and even music. For a sine wave, which is a smooth and repetitive oscillation, the period is especially meaningful because all sine waves are defined by their regularity and predictability.

Cyclical Nature of Waves

The period represents the cyclical nature of a wave and is a measure of how 'fast' or 'slow' the wave oscillates. It is important to note that the period is always a positive value and is measured in seconds (s).

In our given exercise, by using the relationship between angular velocity and the period, students can calculate the precise timing of each cycle in a mechanical or electromagnetic wave, enhancing their understanding of wave phenomena.

Sine Wave Frequency

Closely related to the period is the concept of sine wave frequency. The frequency (f) of a wave indicates how many cycles occur in one second and is measured in hertz (Hz). Low-frequency waves have long periods and thus fewer cycles per second, whereas high-frequency waves have short periods with more cycles per second.

The frequency of a sine wave is particularly important because sine waves form the basis of many types of signals, such as those found in audio and radio communications. By understanding frequency, students can analyze and appreciate the vast range of sine wave applications, from the pitches of musical notes to the transmission of data over airwaves.

Inverse Relationship Between Period and Frequency

The relationship between period and frequency is inverse; as the period increases, the frequency decreases, and vice versa. Hence, calculating one allows you to easily find the other, which is a beneficial skill when analyzing oscillatory systems or designing electronic circuits where signal timing is paramount.

Angular Velocity

In the realm of rotational motion, angular velocity plays a key role. Angular velocity (omega) represents the rate of change of the angle as an object rotates, typically measured in radians per second (rad/s). It is a vector quantity, which means it has both a magnitude and a direction.

For students dealing with physical scenarios involving rotation such as wheels, turbines, or celestial bodies, comprehending angular velocity is essential. It allows not only for the determination of how quickly something is spinning but also for deeper insights into the dynamics of the rotational system.

Connecting Angular Velocity to Wave Motion

When applied to wave motion, angular velocity helps to make the bridge between linear and circular motion concepts. Every point on a sine wave can be thought of as projecting the circular motion of a point rotating with angular velocity. This concept is visualized in applications like signal processing and the generation of alternating current (AC), emphasizing the practical importance and the cross-disciplinary nature of angular velocity.