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Chapter 17: Problem 51

The A string of a violin is a little too tightly stretched. Beats at 4.00 per second are heard when the string is sounded together with a tuning fork that is oscillating accurately at concert A (440 Hz). What is the period of the violin string oscillation?

Short Answer

Expert verified
The period of the violin string's oscillation is approximately 0.00225 seconds.

Step by step solution

01

Understanding Beats

The phenomenon of beats occurs when two sound waves of slightly different frequencies interfere with each other. The beat frequency is the absolute difference between the two frequencies. In this case, the beat frequency is given as 4 Hz.
02

Determine Possible Frequencies of Violin String

Given that the beats frequency is 4 Hz, the frequency of the violin string could be either 4 Hz higher or 4 Hz lower than the frequency of the tuning fork. Thus, the possible frequencies are: \(f_{v} = 440 \, \text{Hz} + 4 \, \text{Hz} = 444 \, \text{Hz}\) or \(f_{v} = 440 \, \text{Hz} - 4 \, \text{Hz} = 436 \, \text{Hz}\).
03

Find the Period of Each Frequency

The period \(T\) of an oscillation is the reciprocal of the frequency \(f\). Use the formula \(T = \frac{1}{f}\).- For \(444 \, \text{Hz}\), the period is \(T = \frac{1}{444} \, \text{seconds}\).- For \(436 \, \text{Hz}\), the period is \(T = \frac{1}{436} \, \text{seconds}\).
04

Compute Values

Calculate the period of each frequency:\[T_{444} = \frac{1}{444} \approx 0.00225 \, \text{seconds}\]\[T_{436} = \frac{1}{436} \approx 0.00229 \, \text{seconds}\]
05

Conclusion on True Frequency

Since the question mentions that the string is too tightly stretched, the frequency of the violin string is higher than the tuning fork. Therefore, the frequency for the violin string is 444 Hz, and its period is approximately 0.00225 seconds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Beats in Sound Waves
Beats in sound waves occur when two waves of slightly different frequencies interfere with each other. It's a fascinating phenomenon where the waves alternate between constructive and destructive interference.
This causes the sound to oscillate in volume, which is experienced as a "beating" sound.
  • The beat frequency is determined by the difference between the two frequencies.
  • For example, if two frequencies are 440 Hz and 444 Hz, the beat frequency would be 4 Hz.
The listener hears the sound getting louder and then softer at the rate of the beat frequency.
In the exercise, the beat frequency is 4 Hz, indicating that the frequencies of the violin string and the tuning fork differ by 4 Hz.
Sound Frequency and Its Importance
Frequency in sound waves is the number of oscillations or cycles per second, measured in hertz (Hz). It's a key concept when dealing with sound and music.
High frequencies correspond to a high pitch, and low frequencies correspond to a low pitch.
  • The experiments with the violin string demonstrate frequency changes when the string is too tight or too loose.
In our exercise, the violin's A string produces sound by vibrating slightly off the correct frequency (440 Hz).
The beat frequency helps us figure out how much the actual frequency deviates. By understanding and calculating, the frequency of the string can be identified as either 436 Hz or 444 Hz.
Given that the string is too tightly stretched, it implies a higher frequency, leading to 444 Hz being the accurate frequency for this scenario.
Exploring the Oscillation Period
The oscillation period is the time taken for one complete cycle of vibration to occur, effectively depicted as the inverse of the frequency: \( T = \frac{1}{f} \). This aspect is critical when determining the behavior of sound waves.
The period of a higher frequency sound wave is shorter, conveying that it completes more cycles in a given time frame.
  • If you calculate the period for our two frequencies:
  • For 444 Hz: \( T = \frac{1}{444} \approx 0.00225 \) seconds
  • For 436 Hz: \( T = \frac{1}{436} \approx 0.00229 \) seconds
These calculations show very small differences, but they are pivotal in defining the correct frequency.
With a slightly tighter string, resulting in a higher frequency, we conclude that the violin's string period is approximately 0.00225 seconds.

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Most popular questions from this chapter

Suppose a spherical loudspeaker emits sound isotropically at \(10 \mathrm{~W}\) into a room with completely absorbent walls, floor, and ceiling (an anechoic chamber). (a) What is the intensity of the sound at distance \(d=3.0 \mathrm{~m}\) from the center of the source? (b) What is the ratio of the wave amplitude at \(d=4.0 \mathrm{~m}\) to that at \(d=3.0 \mathrm{~m} ?\)

At a certain point, two waves produce pressure variations given by \(\Delta p_{1}=\Delta p_{m} \sin \omega t\) and \(\Delta p_{2}=\Delta p_{m} \sin (\omega t-\phi) .\) At this point, what is the ratio \(\Delta p_{r} / \Delta p_{m},\) where \(\Delta p_{r}\) is the pressure amplitude of the resultant wave, if \(\phi\) is (a) \(0,\) (b) \(\pi / 2,\) (c) \(\pi / 3\), and (d) \(\pi / 4 ?\)

A whistle of frequency \(540 \mathrm{~Hz}\) moves in a circle of radius \(60.0 \mathrm{~cm}\) at an angular speed of \(15.0 \mathrm{rad} / \mathrm{s}\). What are the (a) lowest and (b) highest frequencies heard by a listener a long distance away, at rest with respect to the center of the circle?

A sound source sends a sinusoidal sound wave of angular frequency \(3000 \mathrm{rad} / \mathrm{s}\) and amplitude \(12.0 \mathrm{nm}\) through a tube of air. The internal radius of the tube is \(2.00 \mathrm{~cm} .\) (a) What is the average rate at which energy (the sum of the kinetic and potential energies) is transported to the opposite end of the tube? (b) If, simultaneously, an identical wave travels along an adjacent, identical tube, what is the total average rate at which energy is transported to the opposite ends of the two tubes by the waves? If, instead, those two waves are sent along the same tube simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) \(0,\) (d) \(0.40 \pi\) rad, and (e) \(\pi\) rad?

A certain sound source is increased in sound level by \(30.0 \mathrm{~dB}\). By what multiple is (a) its intensity increased and (b) its pressure amplitude increased?
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