Problem 39 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

The beat frequency is 1.30 Hz.

Step 1: 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 \( f = \frac{343}{4.56} \approx 75.22 \space Hz \).

Step 2: 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 \( f' = \frac{343}{4.64} \approx 73.92 \space Hz \).

Step 3: Find Beat Frequency

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

Key Concepts

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.