Question

Twenty four tuning forks are arranged in such a way that each fork produces 6 beats/s with the preceding fork. If the frequency of the last tuning fork is double than the first fork, then the frequency of the second tuning fork is \(\ldots \ldots\) (A) 132 (B) 138 (C) 276 (D) 144

Short answer

The frequency of the second tuning fork is 144 Hz.

Step-by-step solution

Understanding the concept of beats/s between two tuning forks

Beat frequency is the difference in frequencies between two waves meeting at the same time and place. In this case, each tuning fork produces 6 beats/s with the preceding fork, which means the difference in frequency between consecutive forks is 6 Hz.

Create an equation representing the relationship between the frequencies of the forks

Let f1 represent the frequency of the first fork. Then the frequency of the second fork can be expressed as f1 + 6, the third fork as f1 + 12, and so on. So, the frequency of the nth fork can be represented as f1 + 6(n - 1), where n is the position of the fork in the sequence. We are given that the frequency of the last fork, which is the 24th fork, is double the frequency of the first fork. So, we can write this as f1 + 6(24 - 1) = 2f1.

Solve for the frequency of the first fork

Now, we can solve the equation f1 + 6(24 - 1) = 2f1 to find the frequency of the first fork, f1: f1 + 6(23) = 2f1 f1 + 138 = 2f1 138 = f1 So, the frequency of the first fork is 138 Hz.

Find the frequency of the second fork

Now that we know the frequency of the first fork, we can determine the frequency of the second fork using the difference given at the start: Frequency of the second fork = f1 + 6 = 138 + 6 = 144 Hz Thus, the frequency of the second tuning fork is 144 Hz. The correct answer is (D) 144.

Key concepts

Tuning Forks

Tuning forks are fascinating tools used to produce specific frequencies of sound. These metal objects have a U-shape with a handle at one end. When a tuning fork is struck, it vibrates and produces a sound wave at a particular frequency. This frequency corresponds to a musical note. Tuning forks provide a very pure tone because they vibrate at a single frequency, making them widely used in musical tuning and scientific experiments.
In the exercise, tuning forks are organized in a sequence, each producing sound slightly different from its neighbor. This small change in frequency helps illustrate a concept called beat frequency, which is explained in the next sections.

Frequency Doubling

Frequency doubling is an interesting phenomenon where one frequency is exactly twice another. In the context of the exercise, frequency doubling suggests that the last tuning fork has a frequency that is twice that of the first tuning fork.
Understanding this idea is key when solving problems that involve progression and sequences of frequencies. It's often used in wave physics and sound studies to depict cases where harmonics or specific relationships between frequencies are important. Frequency doubling can also be found in optical physics when frequencies of light are doubled to create new waveforms.

Difference in Frequency

The difference in frequency is an important concept when considering sound waves and beat frequencies. It represents the gap between the frequencies of two sound waves. When two frequencies are close to each other, they can interfere, creating a beat frequency that we hear as a rhythmic pattern in sound.
This phenomenon is used extensively in tuning instruments, where musicians listen for the beats to disappear as they match their instrument's pitch to a reference pitch. In the exercise, each tuning fork produces 6 beats per second with the preceding fork, indicating a frequency difference of 6 Hz between successive forks.

Wave Interference

Wave interference is a principle that occurs when two or more waves meet at the same place at the same time. This meeting can result in the waves adding together, cancelling each other out, or something in between.
Interference is constructive when waves amplify each other, and destructive when they reduce the overall amplitude. The concept of wave interference is vital to understanding beats, as beats arise when two waves of slightly different frequencies interfere with each other. The fluctuating interference creates patterns of loud and soft sound, which we perceive as beats. This principle is fundamental in acoustics, especially in tuning and adjusting musical instruments.