Problem 67 - Pearson Physics

Question

To tune middle \(\mathrm{C}\) on a piano, a tuner hits the key and at the same time sounds a \(261 \mathrm{~Hz}\) tuning fork. If the tuner hears three beats per second, what are the possible frequencies of the piano key?

Step-by-Step Solution

Answer

The piano key could be at 258 Hz or 264 Hz.

Step 1: Understanding Beats

Beats occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency is equal to the absolute difference between the two sound frequencies.Given that the beat frequency is 3 beats per second (Hz), and the tuning fork frequency is 261 Hz, the possible frequencies of the piano key can be calculated as:\( f_{piano} = f_{fork} \pm f_{beat} \)

Step 2: Calculate Higher Frequency

Calculate the possibility where the frequency of the piano is higher than that of the tuning fork:\[ f_{piano} = 261 \, Hz + 3 \, Hz = 264 \, Hz \]

Step 3: Calculate Lower Frequency

Calculate the possibility where the frequency of the piano is lower than that of the tuning fork:\[ f_{piano} = 261 \, Hz - 3 \, Hz = 258 \, Hz \]

Step 4: Conclusion

The possible frequencies of the piano key are 258 Hz and 264 Hz. These are the two frequencies at which the beats heard would match the observed 3 beats per second.

Key Concepts

Sound Waves

Sound waves are vibrations that travel through the air or any other medium. These waves are generated by vibrating objects, such as a guitar string plucked or a piano key struck. When the object vibrates, it causes the air particles around it to vibrate as well, creating a wave of compressions and rarefactions that move through air to reach our ears.
Sound waves are characterized by their frequency, which is the number of vibrations or cycles per second. This frequency is measured in Hertz (Hz). For example, a tuning fork vibrating at 261 Hz produces a sound wave with 261 cycles per second.
The higher the frequency, the higher the pitch of the sound we perceive. Conversely, a lower frequency results in a lower pitch. Understanding these fundamental properties of sound waves is crucial when dealing with musical instruments and tuning.

Frequency Interference

When two sound waves of slightly different frequencies are played together, they interfere with each other to create a phenomenon known as beats. This is a type of frequency interference. The listener perceives a fluctuating volume of sound, which sounds like a periodic "wa-wa-wa" effect.
The rate at which these beats are heard is known as the beat frequency. It is calculated as the absolute difference between the frequencies of the two sound waves. Mathematically, this is expressed as:

• Beat Frequency = \(|f_1 - f_2|\)

Where \(f_1\) and \(f_2\) are the frequencies of the two sound waves.
In the context of the piano tuning exercise, the beat frequency of 3 Hz indicates that the piano's frequency differs by 3 Hz from the tuning fork's frequency. This is why two possible frequencies exist for the piano key, which leads us to the concept of adjusting the frequency to eliminate beats.

Tuning Fork

A tuning fork is a tool used to produce a specific pitch or frequency. It typically consists of a metal fork-shaped instrument that resonates at a specific frequency when struck.
In our exercise, the tuning fork resonates at 261 Hz, corresponding to the note "Middle C". The vibrations of the tuning fork produce sound waves that travel through the air. These sound waves are used as a reference to tune musical instruments.
To tune an instrument like a piano, the musician strikes the piano key and the tuning fork simultaneously. They listen for the presence or absence of beats. If beats are heard, the piano's frequency is slightly off and must be adjusted. The goal is to bring the piano frequency to match the tuning fork exactly, eliminating the beats, ensuring the piano is in tune with the standard pitch.