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A tuning fork of unknown frequency makes three beats per second with a standard fork of frequency \(384 \mathrm{~Hz}\). The beat frequency decreases when a small piece of wax is put on a prong of the first fork. What is the freauency of this fork?

Short Answer

Expert verified
The frequency of the first fork is 387 Hz.

Step by step solution

01

- Understand the Beat Frequency

The beat frequency is given as the difference between the frequencies of two tuning forks. Let’s denote the unknown frequency of the first fork as \( f \). The beat frequency is 3 beats per second with a standard fork of frequency 384 Hz. This means: \[ |f - 384| = 3 \]
02

- Determine Possible Frequencies

Since the absolute value is used, the unknown frequency can be either higher or lower than the standard fork's frequency. Therefore, we have two potential solutions: \[ f = 384 + 3 = 387 \text{ Hz} \] or \[ f = 384 - 3 = 381 \text{ Hz} \]
03

- Effect of Adding Wax

Adding a small piece of wax to the first fork decreases its frequency. Given that the beat frequency decreases when wax is added, this implies that the frequency of the first fork is higher than 384 Hz initially. If the frequency were lower, adding wax would have made the beat frequency increase.
04

- Conclusion

Given that the beat frequency decreases when wax is added, the first fork's initial frequency must be higher than 384 Hz. Therefore, the correct frequency of the first fork is: \[ f = 387 \text{ Hz} \]

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Key Concepts

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

Tuning Fork
A tuning fork is a simple tool used to generate a specific pitch. It usually consists of a handle and two prongs (tines) that vibrate when struck. The vibrations produce sound waves at a particular frequency, which is often used as a reference tone in musical settings. Tuning forks are commonly used to tune musical instruments and for scientific experiments involving sound and vibration. They are made from metal, typically steel, and are designed to vibrate at a set frequency, known as its natural frequency. The sound it produces is clear and precise, making it ideal for calibration purposes.
Frequency
Frequency refers to the number of vibrations or cycles per second of a sound wave, and it is measured in Hertz (Hz). In the context of a tuning fork, frequency is the rate at which the fork's prongs vibrate. Higher frequency means a higher pitch, while lower frequency means a lower pitch. For example, a fork vibrating at 440 Hz produces the note A above middle C in musical terms. Frequency determines the pitch of the sound we hear. It’s crucial in various applications, including music, acoustics, and telecommunications. Understanding frequency helps us comprehend how different sounds interact and influence one another.
Beats Per Second
Beats per second refer to the phenomenon of 'beats,' which 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 frequencies. For instance, if one tuning fork vibrates at 387 Hz and another at 384 Hz, the beat frequency is \(|387 - 384| = 3\text{ beats per second}\). These beats are heard as a rhythmic variation in volume, which is particularly useful when tuning instruments. Musicians adjust the instrument until the beats disappear, indicating that the two frequencies match.
Sound Waves
Sound waves are vibrations that travel through a medium, such as air, water, or solid materials. These waves are longitudinal, meaning that the medium particles vibrate parallel to the direction of wave propagation. When a tuning fork vibrates, it creates sound waves that move outwards in all directions. The frequency of these waves determines the pitch of the sound. Sound waves consist of compressions and rarefactions, which correspond to high-pressure and low-pressure phases. The human ear detects these pressure changes and sends signals to the brain, allowing us to perceive sound. Understanding sound waves is essential for manipulating and utilizing sound in various technologies.
Resonance
Resonance occurs when an object vibrates at its natural frequency due to the influence of another vibrating object. For a tuning fork, resonance happens when it causes another object to vibrate at the same frequency. This effect amplifies the sound produced. For example, placing a tuning fork near a set of strings can cause only the string tuned to the same frequency to vibrate. This principle is used in musical instruments and audio technology. Resonance is important because it can significantly increase the amplitude of the vibrations, resulting in louder sounds. It’s a key concept in acoustics, physics, and engineering.

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Most popular questions from this chapter

A sound source \(A\) and a reflecting surface \(B\) move directly toward each other. Relative to the air, the speed of source \(A\) is \(29.9 \mathrm{~m} / \mathrm{s}\), the speed of surface \(B\) is \(65.8 \mathrm{~m} / \mathrm{s}\), and the speed of sound is \(329 \mathrm{~m} / \mathrm{s}\). The source emits waves at frequency \(1200 \mathrm{~Hz}\) as measured in the source frame. In the reflector frame, what are (a) the frequency and (b) the wavelength of the arriving sound waves? In the source frame, what are (c) the frequency and (d) the wavelength of the sound waves reflected back to the source?

A girl is sitting near the open window of a train that is moving at a velocity of \(10.00 \mathrm{~m} / \mathrm{s}\) to the east. The girl's uncle stands near the tracks and watches the train move away. The locomotive whistle emits sound at frequency \(500.0 \mathrm{~Hz}\). The air is still. (a) What frequency does the uncle hear? (b) What frequency does the girl hear? A wind begins to blow from the east at \(10.00 \mathrm{~m} / \mathrm{s}\). (c) What frequency does the uncle now hear? (d) What frequency does the girl now hear?

A French submarine and a U.S. submarine move toward each other during maneuvers in motionless water in the North Atlantic (Fig. 18-33). The French sub moves at \(50.0 \mathrm{~km} / \mathrm{h}\), and the U.S. sub at \(70.0 \mathrm{~km} / \mathrm{h}\). The French sub sends out a sonar signal (sound wave in water) at \(1000 \mathrm{~Hz}\). Sonar waves travel at \(5470 \mathrm{~km} / \mathrm{h}\). (a) What is the signal's frequency as detected by the U.S. sub? (b) What frequency is detected by the French sub in the signal reflected back to it by the U.S. sub?

An ambulance with a siren emitting a whine at \(1600 \mathrm{~Hz}\) overtakes and passes a cyclist pedaling a bike at \(2.44 \mathrm{~m} / \mathrm{s}\). After being passed, the cyclist hears a frequency of \(1590 \mathrm{~Hz}\). How fast is the ambulance moving?

Two point sources of sound waves of identical wavelength \(\lambda\) and amplitude are separated by distance \(D=\) \(2.0 \lambda\). The sources are in phase. (a) How many points of maximum signal (that is, maximum constructive interference) lie along a large circle around the sources? (b) How many points of minimum signal (destructive interference) lie around the circle?

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