Problem 12 - A Graphical Approach to Precalculus with Limits

Question

A note on the piano has frequency \(F\). Suppose the maximum displacement at the center of the piano wire is given by \(s(0)\). Find constants a and \(\omega\) so that the equation \(s(t)=a \cos \omega t\) models this displacement. Graph s in the viewing window \([0,0.05]\) by \([-0.3,0.3]\). $$F=110 ; s(0)=0.11$$

Step-by-Step Solution

Answer

\(a = 0.11\), \(\omega = 220\pi\). Equation: \(s(t) = 0.11 \cos(220\pi t)\). Grah features cosine oscillations.

Step 1: Relate frequency to angular frequency

The angular frequency \( \omega \) is related to the frequency \( F \) by the formula \( \omega = 2\pi F \). Given \( F = 110 \), we find that \( \omega = 2\pi \times 110 = 220\pi \).

Step 2: Determine amplitude from initial displacement

The displacement function is given by \( s(t) = a \cos \omega t \). The maximum displacement at \( t = 0 \) is \( s(0) = a \cos(0) = a \), and we know from the problem statement that \( s(0) = 0.11 \). Thus, the amplitude \( a = 0.11 \).

Step 3: Write the displacement equation

Using the values found for \( a \) and \( \omega \), we write the displacement equation as \( s(t) = 0.11 \cos(220\pi t) \).

Step 4: Graph the function in the given range

To graph the function \( s(t) = 0.11 \cos(220\pi t) \), set the viewing window from \( t = 0 \) to \( t = 0.05 \) on the x-axis, and from \( s = -0.3 \) to \( s = 0.3 \) on the y-axis. The graph will create a waveform oscillating between \( -0.11 \) and \( 0.11 \).

Key Concepts

Frequency

Frequency is a fundamental concept in the study of waveforms, especially in trigonometry and physics. When we talk about the frequency of a wave, we are referring to how often the wave repeats itself over a specific period of time. This is typically measured in Hertz (Hz), with one Hertz equating to one complete cycle per second.
In the context of the problem, the frequency given is 110 Hz, meaning the waveform completes 110 cycles every second.
Understanding this helps us relate frequency to another important concept, angular frequency. Essentially, frequency provides the necessary information to describe how rapidly the wave pattern oscillates, impacting how we perceive sounds such as a note from a piano.

Amplitude

Amplitude is the term used to describe the size of the wave's oscillations. It represents the peak value of the waveform, or how far the wave moves from its central position. Being a measure of displacement, it helps in defining the wave's energy or strength.
In the exercise, the amplitude is represented by the variable 'a' in the cosine wave equation, which is determined to be 0.11. This figure tells us that the maximum displacement from the central, or rest position of the piano wire is 0.11 units. Understanding amplitude allows us to visualize how loud or intense a sound is, with larger amplitudes resulting in louder sounds.

Angular Frequency

Angular frequency is crucial in analyzing periodic wave functions. It provides insight into the rate of oscillation in terms of radians per unit time. Derived from frequency, angular frequency is represented by the symbol \( \omega \).
In trigonometric functions, angular frequency acts as a scaling factor in the argument of sine or cosine terms, converting time into radians. It is calculated by the formula \( \omega = 2\pi F \), where \( F \) is the frequency.
For our problem, given \( F = 110 \) Hz, the angular frequency can be calculated as \( \omega = 220\pi \) radians per second. This value is significant because it directly affects the speed of oscillations in the function \( s(t) = 0.11 \cos(220\pi t) \).

Graphing

Graphing trigonometric functions like \( s(t) = 0.11 \cos(220\pi t) \) involves plotting the values of the function over a specified range to visualize its oscillatory behavior.
In this task, the graph is constructed in a window defined by the x-axis range from 0 to 0.05 seconds and the y-axis range from -0.3 to 0.3. This particular setup captures a snapshot of the wave's behavior in a small timeframe.
The graph demonstrates how the waveform oscillates with its maximum and minimum values corresponding to the calculated amplitude, \( a \). Observing the graph can help us understand the periodic nature of waves. It quickly informs us how sound waves, like those of a piano note, appear in mathematical representations.