Problem 76 - Algebra and Trigonometry

Question

Sound Vibrations \(\quad\) 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) \( \frac{1}{440} \) seconds; (b) 440 Hz; (c) Graph is a sine wave with amplitude 0.7 and period \( \frac{1}{440} \).

Step 1: Identify the formula for period of a sine function

The period of the sine function can be determined using the formula \( T = \frac{2\pi}{b} \) where \( b \) is the coefficient of \( t \) inside the sine function. In the given function, \( v(t) = 0.7 \sin (880 \pi t) \), \( b = 880\pi \).

Step 2: Calculate the period

Substitute \( b = 880\pi \) into the period formula: \[ T = \frac{2\pi}{880\pi} = \frac{1}{440} \] So, the period of the vibration is \( \frac{1}{440} \) seconds.

Step 3: Define frequency in terms of period

The frequency is the reciprocal of the period. It tells us how many cycles occur in one second. Frequency \( f \) is given by \( f = \frac{1}{T} \).

Step 4: Calculate the frequency

Using the period \( T = \frac{1}{440} \), compute the frequency: \[ f = \frac{1}{\frac{1}{440}} = 440 \, \text{Hz} \] Thus, the frequency of the vibration is 440 Hz.

Step 5: Graph the function

To graph \( v(t) = 0.7 \sin (880\pi t) \), note the amplitude is 0.7 and the period is \( \frac{1}{440} \). Plot the sine wave starting at \( t = 0 \), with peaks at \( v(t) = 0.7 \), troughs at \( v(t) = -0.7 \), and zero crossings at each period interval.

Key Concepts

sine function

The sine function is a fundamental trigonometric function often used to model periodic phenomena such as sound waves, light waves, and many natural phenomena. In the given exercise, the vibration of the tuning fork is expressed using a sine function. The formula for a basic sine function is:\[ v(t) = A \sin(Bt + C) + D \]Here, \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical shift. For the tuning fork problem:

• Amplitude \(A = 0.7\), which determines the maximum displacement of the tines.
• Phase shift \(C\) and vertical shift \(D\) are 0, as these are not present in the formula.
• The coefficient \(B = 880\pi\) directly influences the period of the function.

The sine function is periodic, meaning it repeats its values in regular intervals. This property is key in modeling waves and vibrations since they consistently cycle through their motion.

period of a function

The period of a function indicates the duration of one complete cycle of the wave before it repeats. Understanding the period is essential for analyzing any periodic signal, such as sound waves from a tuning fork. To find the period of a sine function in the form \( \sin(Bt) \), we use the formula:\[ T = \frac{2\pi}{B} \]In the tuning fork example, the sine function is \( \sin(880\pi t) \). Substituting \(880\pi\) for \(B\) in the formula gives:\[ T = \frac{2\pi}{880\pi} = \frac{1}{440} \]This result shows that the period of the tuning fork's vibration is \( \frac{1}{440} \) seconds. This means that the complete vibration cycle repeats 440 times every second, hinting at the connection between period and frequency.

frequency calculation

Frequency is the measure of how often an event repeats in a given time period. In the context of a sine wave, it represents the number of complete cycles per second, measured in Hertz (Hz).Frequency is calculated as the reciprocal of the period:\[ f = \frac{1}{T} \]By determining the period of the tuning fork's vibration as \( \frac{1}{440} \) seconds, we can find its frequency:\[ f = \frac{1}{\frac{1}{440}} = 440 \, \text{Hz} \]This means the tuning fork vibrates at a frequency of 440 Hz, often recognized as the musical note A above middle C, a standard pitch used for tuning musical instruments. Understanding frequency helps in various fields, such as acoustics and audio engineering, as it directly affects the pitch of sounds we hear.