Problem 68 - University Physics with Modern Physics

Question

The frequency of the note \(\mathrm{F}_{4}\) is 349 \(\mathrm{Hz} .\) (a) If an organ pipe is open at one end and closed at the other, what length must it have for its fundamental mode to produce this note at \(20.0^{\circ} \mathrm{C}\) (b) At what air temperature will the frequency be 370 \(\mathrm{Hz}\) , corresponding to a rise in pitch from F to F-sharp? (Ignore the change in length of the pipe due to the temperature change.)

Step-by-Step Solution

Answer

(a) The length must be 0.246 meters. (b) The frequency will be 370 Hz at approximately 54.47°C.

Step 1: Equation for Frequency of Pipe

An open-closed pipe produces a fundamental frequency according to the formula: \( f = \frac{v}{4L} \), where \( f \) is the frequency, \( v \) is the speed of sound in air, and \( L \) is the length of the pipe.

Step 2: Calculate Speed of Sound at 20°C

The speed of sound in air as a function of temperature can be calculated using: \( v = 331.4 + 0.6 \times T \) m/s, where \( T \) is the temperature in Celsius. At \( 20.0^{\circ} \mathrm{C} \), \( v = 331.4 + 0.6 \times 20 = 343.4 \) m/s.

Step 3: Find Length for Frequency 349 Hz

Rearrange the formula to solve for \( L \): \( L = \frac{v}{4f} \). Substitute \( v = 343.4 \) m/s and \( f = 349 \) Hz: \( L = \frac{343.4}{4 \times 349} = 0.246 \) meters.

Step 4: Calculate New Speed of Sound for 370 Hz

Rearrange the formula to solve for the new speed of sound \( v' \) when \( f = 370 \) Hz: \( v' = 4Lf = 4 \times 0.246 \times 370 = 364.08 \) m/s.

Step 5: Find New Temperature for 370 Hz

The new speed of sound equation is \( v' = 331.4 + 0.6 \times T' \). Set \( v' = 364.08 \) m/s and solve for \( T' \): \( 364.08 = 331.4 + 0.6 \times T' \). Rearranging gives: \( T' = \frac{364.08 - 331.4}{0.6} \approx 54.47 \)°C.

Key Concepts

Frequency of sound

The frequency of sound refers to the number of vibrations or cycles per second, measured in Hertz (Hz). It's like the beat in a song; more beats mean a higher frequency and pitch. Different musical notes have specific frequencies. For example, the note F4 has a frequency of 349 Hz.
This means if you hear the note F4, the air is vibrating at 349 cycles per second. Understanding frequency helps in various applications, from music to acoustics. In our exercise, we need to find an organ pipe's length that produces this note. Frequency is crucial because it defines the sound you hear.

Speed of sound

The speed of sound is how fast sound waves travel through a medium, like air. It's affected by several factors, including temperature. At room temperature, which is around 20°C, the speed of sound in air is about 343.4 meters per second.
The formula to calculate the speed of sound in air is: \[ v = 331.4 + 0.6 imes T \] where \( T \) is the temperature in Celsius.

• As temperature increases, sound travels faster.
• At 20°C, we calculate it to be 343.4 m/s.

This concept helps when considering how sound will behave in different environments, such as calculating how a pipe length needs to change with temperature.

Fundamental frequency

The fundamental frequency is the lowest frequency produced by any vibrating object, including musical instruments. For pipes, this is the simplest form of vibration. Pipes can be open at both ends or one end closed. Our exercise uses a pipe closed at one end to produce a fundamental frequency of 349 Hz.
In acoustics for such a pipe, the formula for the fundamental frequency is: \[ f = \frac{v}{4L} \]

• \( f \) is the frequency.
• \( v \) is the speed of sound.
• \( L \) is the length of the pipe.

Using this equation, we can find the appropriate length of the pipe to achieve the desired frequency, as done in our exercise.

Organ pipe acoustics

Organ pipes work on the principle of acoustics where sound waves resonate within a cylindrical shape, like the pipe. There are two types of pipes: open-open and open-closed. In open-closed pipes, the closed end reflects sound waves, allowing a standing wave to form, which gives rise to the fundamental frequency.
In our exercise, the pipe is designed to produce a note at 349 Hz. To calculate its length when open at one end, we use the equation \[ L = \frac{v}{4f} \]Here, you need the speed of sound and your desired frequency. We used a calculated speed of sound at 20°C to determine the correct pipe length for this note to resonate perfectly.
This principle is crucial for tuning musical instruments, ensuring they produce the correct pitches when played.