Problem 10 - Technical Mathematics with Calculus

Question

Find the period and the frequency of a sine wave that has an angular velocity of $$2.58 \mathrm{rad} / \mathrm{s}$$

Step-by-Step Solution

Answer

The frequency is approximately 0.411 Hz, and the period is approximately 2.435 seconds.

Step 1: Understanding the Relationship Between Angular Velocity and Frequency

The angular velocity (often denoted as \( \omega \)) of a sine wave is related to its frequency (\( f \)) by the formula \( \omega = 2\pi f \). To find the frequency, we can rearrange the formula: \( f = \frac{\omega}{2\pi} \).

Step 2: Calculating the Frequency

Substitute the given angular velocity (\(2.58 \mathrm{rad}/\mathrm{s}\)) into the rearranged formula to find the frequency: \( f = \frac{2.58}{2\pi} \). Calculate the value of \( f \) to get the frequency in hertz (Hz).

Step 3: Calculating the Period

The period (\( T \)) of a sine wave is the reciprocal of the frequency: \( T = \frac{1}{f} \). Use the frequency calculated in Step 2 to find the period.

Key Concepts

Frequency of a Sine Wave

The frequency of a sine wave is a measure of how often the wave repeats itself over a set time interval, typically measured in cycles per second or Hertz (Hz). In short, it tells us how 'fast' the wave oscillates. Understanding frequency is crucial for various fields including physics, engineering, and music.

When describing the frequency of a sine wave, we are often given its angular velocity, denoted as \( \omega \). To find the frequency (\( f \) in Hertz) from angular velocity, we use the relationship \( f = \frac{\omega}{2\pi} \). This relationship emerges because a sine wave completes one full cycle when its angle has swept 2\(\pi\) radians; hence dividing the angular velocity by 2\(\pi\) gives us the number of full cycles per second.

Calculating Period

The period (\( T \) in seconds) of a sine wave is the amount of time it takes to complete one full cycle. It’s the 'duration' of one cycle of the wave, a concept particularly important in oscillatory systems like pendulums, electrical circuits, and sound waves. To calculate the period, we take the reciprocal of the frequency: \( T = \frac{1}{f} \).

For example, if we were to find the period of a sine wave with a given frequency, we simply invert the frequency (provided the frequency is nonzero). This straightforward calculation is a fundamental step in understanding wave behavior and is often a starting point for analyzing oscillatory systems.

Relationship Between Angular Velocity and Frequency

Angular velocity and frequency are intimately connected. Angular velocity (\( \omega \) in radians per second) refers to the rotational speed, and it indicates how fast the angle changes in a rotating or oscillating object. For sine waves, which oscillate back and forth in a smooth, periodic manner, the angular velocity corresponds to the rate at which the phase of the wave (its angle) changes.

The relationship is encapsulated by the equation \( \omega = 2\pi f \), linking angular velocity (\( \omega \) ) directly with frequency (\( f \) ). This equation is pivotal as it unites the concepts of time and angle, both of which describe fundamental aspects of a wave's motion. This connection is used not only in theoretical physics but also practically in designing electronic circuits and understanding mechanical vibrations.