/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Two organ pipes, open at one end... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 39

Two organ pipes, open at one end but closed at the other, are each 1.14 \(\mathrm{m}\) long. One is now lengthened by 2.00 \(\mathrm{cm}\) . Find the frequency of the beat they produce when playing together in theifundamcntal.

Short Answer

Expert verified
The beat frequency is 1.30 Hz.

Step by step solution

01

Calculate Initial Frequency of Original Pipe

The initial length of the original pipe is 1.14 m. Since the pipe is open at one end and closed at the other, it supports a fundamental wavelength given by four times its length. So, the wavelength \( \lambda \) is \( 4 \times 1.14 \space m = 4.56 \space m \). The frequency \( f \) is given by \( f = \frac{v}{\lambda} \), where \( v \) is the speed of sound in air (approximately 343 m/s). Substituting the values, we find \( f = \frac{343}{4.56} \approx 75.22 \space Hz \).
02

Calculate New Frequency of Lengthened Pipe

The second pipe is extended by 2.00 cm, making its new length \( 1.14 \space m + 0.02 \space m = 1.16 \space m \). The new wavelength it supports is \( 4 \times 1.16 \space m = 4.64 \space m \). The new frequency is \( f' = \frac{343}{4.64} \approx 73.92 \space Hz \).
03

Find Beat Frequency

The beat frequency is the absolute difference between the two frequencies. Using the calculated frequencies, the beat frequency \( f_{beat} \) is \( |75.22 - 73.92| = 1.30 \space Hz \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Organ pipes
Organ pipes are fascinating musical instruments found in organs, consisting of resonant air columns that produce sound when air is directed through them. Different types of pipes can have different sound characteristics based on their design. One common type is a pipe that is open at one end and closed at the other. This type is often referred to as a "closed pipe."

In this setup:
  • When air vibrates inside the pipe, a standing wave pattern is created.
  • Only odd harmonics are present, meaning you mainly hear the fundamental frequency and its odd multiples.
These pipes are a classic example of how acoustic resonance works, and they transform energy into sound by the vibration of air within. Distinguished by their lengths and whether both ends are open, organ pipes play a crucial role in creating different musical notes.
Fundamental frequency
The fundamental frequency is the lowest frequency at which a system naturally vibrates. Also known as the first harmonic, it determines the pitch of the note that an organ pipe produces. In a pipe that is open at one end and closed at the other, the fundamental frequency can be found using the formula:
  • For a closed pipe, the wavelength \( \lambda \) for the fundamental frequency is four times the length of the pipe.
  • The frequency \( f \) is determined by the equation \( f = \frac{v}{\lambda} \), where \( v \) represents the speed of sound in air.
In practical situations, as seen in the given exercise, if you have a pipe length, you can easily compute the frequency by substituting its wavelength into the formula. This helps us understand how changes in pipe length affect sound, allowing musicians and engineers to craft precise notes.
Beat frequency
Beat frequency is a captivating auditory phenomenon that happens when two sound waves of nearly equal frequencies are played together. This phenomenon results in a new sound that repeatedly increases and decreases in volume, which we hear as "beats."

Calculating the beat frequency is straightforward:
  • Find the absolute difference between the two individual frequencies of the sound waves.
  • The formula is given by \( f_{\text{beat}} = |f_1 - f_2| \).
In musical contexts, beats can create unique effects but may also indicate that two instruments are slightly out of tune. As in the exercise, using beat frequency shows the practical application of this phenomenon, helping differentiate between very close frequencies.
Speed of sound
The speed of sound is an important factor in acoustics, determining how fast sound waves travel through a medium. In air, at room temperature, the speed of sound is approximately 343 meters per second (m/s). This speed can vary slightly depending on environmental conditions such as temperature, humidity, and air pressure.

Understanding this speed is crucial in various calculations:
  • Knowing the speed allows scientists and musicians to link wavelengths to frequencies in pipes or other instruments.
  • It is used in the equation \( f = \frac{v}{\lambda} \) to determine the frequency of sound based on its wavelength.
In our example scenario, the speed of sound provides a way to find out how the length changes in organ pipes, translating into a different frequency. Consequently, this shows how theoretical concepts directly apply to practical acoustics and real-world instruments.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) What is the sound intensity level in a car when the sound intensity is 0.500\(\mu \mathrm{W} / \mathrm{m}^{2} 7\) (b) What is the sound intensity level in the air near a jackhammer when the pressure amplitude of the sound is 0.150 \(\mathrm{Pa}\) and the temperature is \(20.0^{\circ} \mathrm{C}\) ?

A New Musical Tnstrument. You have designed a new musical instrument of very simple construction. Your design consists of a metal tube with length \(L\) and diameter \(L / 10\) . You have stretched a string of mass per unit length \(\mu\) across the open end of the tube. The other end of the tube is closed. To produce the musical effect you're looking for, you want the frequency of the third-harmonic standing wave on the string to be the same as the fundamental frequency for sound waves in the air column in the tube. The speed of sound waves in this air column is \(v_{v}\) (a) What must be the tension of the string to produce the desired effect? (b) What happens to the sound produced by the instrument if the tension is changed to twice the value calculatod in part (a)?(c) For the tension calculated in part (a), what other harmonics of the string, if any, are in resonance with standing waves in the air column?

Doppler Radar. A giant thunderstorm is moving toward a weather station at \(45.0 \mathrm{mi} / \mathrm{h}(20.1 \mathrm{m} / \mathrm{s}) .\) If the station sends a radar beam of frequency 200.0 \(\mathrm{MHz}\) toward the storm, what is the difference in frequency between the emitted beam and the beam reflected back from the storm? Be careful to carry plenty of significant figures! (Hint The storm reflects the same frequency that it receives.)

A person is playing a small fute 10.75 \(\mathrm{cm}\) long, open at one end and closed at the other, near a taut string having a fundamental frequency of 600.0 \(\mathrm{Hz}\) . If the speed of sound is 344.0 \(\mathrm{m} / \mathrm{s}\) , for which harmonics of the flute will the string resonate? In each case, which harmonic of the string is in resonance?

Standing sound waves are produced in a pipe that is 1.20 \(\mathrm{m}\) long. For the fundamental and first two overtones, determine the the locations along the pipe (measured from the left end) of the dis- placement nodes and the pressure nodes if (a) the pipe is open at both ends and (b) the pipe is closed at the left end and open at the right end.
See all solutions

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Read Explanation

Waves Physics

Read Explanation

Mechanics and Materials

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Electromagnetism

Read Explanation

Electrostatics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.