Problem 78 - Precalculus Mathematics for Calculus

Question

Sound Vibrations A tuning fork is struck, producing a pure tone as its tines vibrate. The vibrations are modeled by the function $$ v(t)=0.7 \sin (880 \pi t) $$ where \(v(t)\) is the displacement of the tines in millimeters at time \(t\) seconds. (a) Find the period of the vibration. (b) Find the frequency of the vibration, that is, the number of times the fork vibrates per second. (c) Graph the function \(v\)

Step-by-Step Solution

Answer

(a) \( T = \frac{1}{440} \) seconds. (b) \( f = 440 \) Hz. (c) The graph is a sine wave with amplitude 0.7 and period \( \frac{1}{440} \) seconds.

Step 1: Identify the General Form of the Sine Function

The given function is \( v(t) = 0.7 \sin(880\pi t) \). The general form of a sine function is \( y = A \sin(Bx) \), where \( A \) is the amplitude, \( B \) relates to the period of the function, and \( x \) is the variable.

Step 2: Find the Period of the Vibration

The period \( T \) of a sine wave is given by the formula \( T = \frac{2\pi}{B} \). Here, \( B = 880\pi \). Substituting in the formula, we get \( T = \frac{2\pi}{880\pi} = \frac{1}{440} \). Hence, the period of the vibration is \( \frac{1}{440} \) seconds.

Step 3: Find the Frequency of the Vibration

The frequency \( f \) is the number of cycles per second and is given by the inverse of the period, \( f = \frac{1}{T} \). From Step 2, we know the period \( T = \frac{1}{440} \), so the frequency \( f = 440 \) Hz. Therefore, the tuning fork vibrates 440 times per second.

Step 4: Graph the Function

To graph \( v(t) = 0.7 \sin(880\pi t) \), plot the sine wave with an amplitude of 0.7, a period of \( \frac{1}{440} \), and no vertical shift. The x-axis represents time \( t \) in seconds, and y-axis represents the displacement \( v(t) \) in millimeters. The wave will complete one cycle from its peak, through zero, to its trough, and back to zero within \( \frac{1}{440} \) seconds.

Key Concepts

Period of a Function

In mathematics, the period of a function refers to the length of the interval after which the function repeats itself. For trigonometric functions like sine functions, this concept is vital as it describes the repeating nature of these waves.
The function given in the exercise is modeled as \( v(t) = 0.7 \sin(880\pi t) \). This is a standard sine wave where you can identify the period using the coefficient \( B \) from the general sine function form \( y = A \sin(Bx) \).

• The formula to find the period \( T \) is \( T = \frac{2\pi}{B} \), where \( B \) in this case is \( 880\pi \).
• Upon substituting, we calculate \( T = \frac{2\pi}{880\pi} = \frac{1}{440} \).

This means that every \( \frac{1}{440} \) seconds, the tuning fork's vibration repeats. Understanding this helps to analyze how often the sound repeats, giving insight into its harmony and frequency.

Frequency of Vibrations

Frequency is a measure of how often a repeating event occurs per unit of time. In the context of sound waves, it measures how many vibrations occur in one second, defining the pitch of the sound produced.
To find the frequency of the vibration from the given function \( v(t) = 0.7 \sin(880\pi t) \), we use the relationship between period and frequency.

• The frequency \( f \) is the reciprocal of the period, expressed as \( f = \frac{1}{T} \).
• Given that the period \( T \) is \( \frac{1}{440} \), the frequency \( f = 440 \) Hz.

This means the tuning fork vibrates 440 times each second, producing an "A" note in music, which is commonly used as a tuning standard.

Sine Wave Graphing

Graphing a sine wave like the one described by the function \( v(t) = 0.7 \sin(880\pi t) \) reveals the oscillating nature of the wave.
To graph this function, follow these steps:

• Recognize the function has an amplitude of 0.7, meaning the wave reaches 0.7 millimeters above and below the x-axis.
• Identify the period as \( \frac{1}{440} \) seconds, showing how quickly the wave repeats within a second.
• The starting point is zero, and the wave moves to its maximum, back through zero to the minimum, and returns to zero, completing one full cycle.

The graph will demonstrate how rapidly the tuning fork vibrates, creating the auditory experience. Understanding how to graph these sine waves is crucial for visualizing sound waves and their properties.