Problem 16 - Physics

Question

Two out-of-tune flutes play the same note. One produces a tone that has a frequency of \(262 \mathrm{~Hz},\) while the other produces \(266 \mathrm{~Hz}\). When a tuning fork is sounded together with the \(262-\mathrm{Hz}\) tone, a beat frequency of \(1 \mathrm{~Hz}\) is produced. When the same tuning fork is sounded together with the 266 -Hz tone, a beat frequency of 3 Hz is produced. What is the frequency of the tuning fork?

Step-by-Step Solution

Answer

The frequency of the tuning fork is 263 Hz.

Step 1: Understand the Beat Frequency

Beat frequency is the difference between two frequencies. When the tuning fork is sounded with the 262 Hz tone and produces a beat frequency of 1 Hz, the possible frequencies of the tuning fork are either 261 Hz or 263 Hz.

Step 2: Determine Tuning Fork Frequency with 262 Hz

Using the information from the beat frequency with the 262 Hz flute: \[ |f_{tuning} - 262| = 1 \] This gives us two possibilities for the tuning fork frequency: 1. \( f_{tuning} = 261 \) Hz2. \( f_{tuning} = 263 \) Hz

Step 3: Determine Tuning Fork Frequency with 266 Hz

Using the information from the beat frequency with the 266 Hz flute: \[ |f_{tuning} - 266| = 3 \] This gives us two possibilities for the tuning fork frequency: 1. \( f_{tuning} = 263 \) Hz2. \( f_{tuning} = 269 \) Hz

Step 4: Find Common Frequency

Compare the two sets of possible frequencies. From the calculations: - From 262 Hz: possible tuning fork frequencies are 261 Hz and 263 Hz. - From 266 Hz: possible tuning fork frequencies are 263 Hz and 269 Hz. The common frequency in both scenarios is 263 Hz.

Key Concepts

Tuning Fork

A tuning fork is a simple yet essential tool in the world of music and acoustics. It consists of a U-shaped metal bar with a handle, designed to vibrate at a specific frequency when struck. This specific frequency is a standard reference for tuning musical instruments.
When a tuning fork is activated, its tines vibrate and produce sound waves at a consistent frequency. This natural frequency is known as the fork's "fundamental frequency." It serves as a reliable and precise reference point.
Tuning forks are used to ensure that musical instruments produce the correct pitch. When played alongside an instrument, they help musicians achieve the desired tone.

Frequency Difference

Frequency difference is a fundamental concept in understanding sound wave interactions.
It represents the numerical difference between two frequencies. When two similar frequencies interact, frequency difference becomes critical in determining the sound's characteristics.
In the context of the exercise, frequency difference forms the basis of the beat phenomenon. If two sound waves with slightly different frequencies are played together, the result is a fluctuating intensity or "beat."

• This beat frequency is calculated by subtracting one frequency from another.
• For example, with 262 Hz and 266 Hz, the difference is 266 - 262 = 4 Hz.
• In this exercise, knowing the frequency difference allowed us to calculate the tuning fork's exact frequency, accurately identifying it as 263 Hz.

Sound Waves

Sound waves are vibrations that travel through the air or other mediums, allowing us to hear different pitches and tones.
These waves are typically generated by vibrating objects, like vocal cords, musical instruments, or tuning forks.
The frequency of a sound wave determines its pitch; higher frequencies correspond to higher pitches, while lower frequencies correspond to lower pitches.
Sound waves can interact in various ways, resulting in phenomena such as beats, harmonics, and resonance.

• In this exercise, two sound waves from flutes interacted with a third wave from the tuning fork.
• Their interaction generated beats, enabling us to calculate the tuning fork's frequency by observing the beat frequency.

Acoustics

Acoustics is the scientific study of sound. It encompasses the production, transmission, and perception of sound waves, and applies to everything from musical instruments to architectural design.
Understanding acoustics helps in designing spaces like concert halls, where sound quality and clarity are of utmost importance.
In music, acoustics allows us to understand how instruments produce sound and how those sounds interact. This knowledge leads to better instrument design and musician skills.

• In this exercise, the acoustic principle of beat frequency helped determine the precise frequency of the tuning fork when played with the flutes.
• Through acoustics, waveform interactions predictably form recognizable patterns, such as beats.