Two forks A and B when sounded together produce 4 beats / s. The fork A is in unison with 30 cm

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Two forks A and B when sounded together produce 4 beats / s....

Two forks $\mathrm{A}$ and $\mathrm{B}$ when sounded together produce 4 beats $/ \mathrm{s}$. The fork A is in unison with $30 \mathrm{~cm}$ length of a sonometer wire and $B$ is in unison with $25 \mathrm{~cm}$ length of the same wire at the same tension. The frequencies of the fork are (A) $24 \mathrm{~Hz}, 28 \mathrm{~Hz}$ (B) $20 \mathrm{~Hz}, 24 \mathrm{~Hz}$ (C) $16 \mathrm{~Hz}, 20 \mathrm{~Hz}$ (D) $26 \mathrm{~Hz}, 30 \mathrm{~Hz}$

Two forks $\mathrm{A}$ and $\mathrm{B}$ when sounded together produce 4 beats $/ \mathrm{s}$. The fork A is in unison with $30 \mathrm{~cm}$ length of a sonometer wire and $B$ is in unison with $25 \mathrm{~cm}$ length of the same wire at the same tension. The frequencies of the fork are
(A) $24 \mathrm{~Hz}, 28 \mathrm{~Hz}$
(B) $20 \mathrm{~Hz}, 24 \mathrm{~Hz}$
(C) $16 \mathrm{~Hz}, 20 \mathrm{~Hz}$
(D) $26 \mathrm{~Hz}, 30 \mathrm{~Hz}$

Key Concepts

Beat Frequency

Beat frequency is the absolute difference between two sound frequencies that are played together. When two sound waves of slightly different frequencies interfere, they produce periodic variations in amplitude called beats, and the number of beats per second is equal to the absolute difference between the two frequencies.

Fundamental Frequency of a Vibrating String

The fundamental frequency of a vibrating string under tension is determined by its length, tension, and linear density as given by the formula f = (1/2L)?(T/?). When the tension and mass per unit length are constant, the frequency is inversely proportional to the length of the string, meaning shorter strings produce higher pitches.

Inverse Proportionality Between Frequency and Length

In cases where the tension and mass density are fixed, the frequency of a vibrating string is inversely proportional to its length. This relationship allows one to determine the frequency ratio of two strings by simply comparing their lengths, with the shorter string having a higher frequency.

Tuning with Sonometer Wires

A sonometer is a device used to study the vibrations of a stretched wire. By changing the length of the wire (using marked points) and adjusting to the fundamental mode of vibration, one can produce a specific frequency that matches a known sound source such as a tuning fork, providing a practical method to compare and measure frequencies.

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