Question

You are given four tuning forks. The fork with the lowest frequency vibrates at \(500 \mathrm{~Hz}\). By using two tuning forks at a time, the following beat frequencies are heard: \(1,2,3,5,7\), and \(8 \mathrm{~Hz}\). What are the possible frequencies of the other three tuning forks?

Short answer

The frequencies of the other three tuning forks are 502 Hz, 503 Hz, and 508 Hz.

Step-by-step solution

Understand Beat Frequency

A beat frequency is the absolute difference between the frequencies of two interfering waves, which are creating the beats. It's important to first understand this concept, that if two tuning forks are striking together and creating beats, the difference of their frequencies is the frequency of the beats.

Calculate Possible Frequencies

Next, calculate the possible frequencies of the tuning forks. This can be done by both adding and subtracting each of the beat frequencies (1 Hz, 2 Hz, 3 Hz, 5 Hz, 7 Hz, and 8 Hz) from the given frequency of the first tuning fork (500 Hz).

Identify the Frequencies

Finally, collect all the frequencies obtained. However, take into account that the frequency of a tuning fork can be neither negative nor exactly the same as the one given. Therefore, the only valid frequencies are the ones greater than 500 Hz. Hence, the frequencies of three additional tuning forks are the three unique frequencies obtained after excluding all the invalid options. Always remember that real physical quantities such as frequency, can never be negative in these types of problems.

Key concepts

Tuning Forks

Imagine the classroom setting, where the sound of a single tuning fork fills the room with a clear musical note. It's this tool, the tuning fork, which is a simple yet fascinating device used to produce a specific pitch. It consists of a handle attached to a U-shaped metal prong, designed to vibrate at a particular fixed frequency when struck.

When tuning forks are used, they are typically struck against a surface or a rubber pad to start their vibration. The vibration of the prongs produces sound waves, which we perceive as a musical note. The frequency of this note is determined by the material, length, and mass of the prongs. Musicians often use tuning forks to tune instruments, ensuring that each note played matches the standard musical pitch.

But what makes tuning forks truly intriguing is their ability to interact with each other. When two tuning forks of slightly different frequencies are struck together, they produce a phenomenon known as 'beats,' which leads us to the concept of interference of sound waves.

Interference of Waves

Have you ever watched ripples on a pond collide and create patterns? This visual can help us understand the interference of waves, which is the interaction that occurs when two waves meet while traveling along the same medium. In the world of acoustics, when two sound waves of similar frequencies come into contact, they combine to form a new wave pattern.

The principle at play here is superposition, where the amplitudes of the individual waves add together at each point of intersection. If the crests of two waves align, they constructively interfere, resulting in a wave of greater amplitude. Conversely, if a crest meets a trough, they destructively interfere, leading to a cancellation effect or a wave of reduced amplitude.

With tuning forks, this concept manifests as the beats we hear, which are essentially the fluctuations in sound intensity caused by the interference of their sound waves. The rate at which these beats occur, known as the beat frequency, is a direct result of the frequency difference between the two forks. If the forks have frequencies that are close together, the resulting beat frequency will be slow, and if they are further apart, the beat will be faster.

Frequency Calculation

Understanding how to calculate frequency is essential when dealing with waves, whether they be of sound, light, or another form of energy. Frequency refers to the number of waves that pass a point per second and is measured in hertz (Hz).

In the context of our tuning fork exercise, frequency calculation enables us to deduce the frequencies of unknown forks based on the beat frequencies produced. If we know one tuning fork vibrates at 500 Hz and we hear a beat frequency of 1 Hz when paired with a second fork, it means the second fork could vibrate at either 499 Hz or 501 Hz — since the beat frequency is the absolute difference between the two.

We can apply this logic to a list of beat frequencies to determine possible frequencies for the other forks. However, remember that real physical frequencies must be positive values. Thus, using mathematical calculations while keeping in mind the principles of wave interference, we can solve puzzles like our textbook problem, building foundational knowledge for both theoretical and practical applications in fields like music, physics, and engineering.