The tuning fork and sonometer wire were sounded together and...

The tuning fork and sonometer wire were sounded together and produce 4 beats/second when the length of sonometer wire is $95 \mathrm{~cm}$ or $100 \mathrm{~cm} .$ The frequency of tuning fork is [MP PMT 1990] (a) $156 \mathrm{~Hz}$ (b) $152 \mathrm{~Hz}$ (c) $148 \mathrm{~Hz}$ (d) $160 \mathrm{~Hz}$

The tuning fork and sonometer wire were sounded together and produce 4 beats/second when the length of sonometer wire is $95 \mathrm{~cm}$ or $100 \mathrm{~cm} .$ The frequency of tuning fork is [MP PMT 1990]
(a) $156 \mathrm{~Hz}$
(b) $152 \mathrm{~Hz}$
(c) $148 \mathrm{~Hz}$
(d) $160 \mathrm{~Hz}$

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Key Concepts

Fundamental Frequency of a Vibrating String

The fundamental frequency of a vibrating string is determined by the equation f = (1/2L)?(T/?), where L is the length of the string, T is the tension in the string, and ? is its mass per unit length. This relationship explains how the frequency of the string varies inversely with its length and directly with the square root of its tension.

Beat Frequency

Beat frequency is the difference between two frequencies when they are sounded together, resulting in periodic variations in amplitude. This phenomenon is crucial in determining small differences between frequencies, and it is observable as beats per second.

Standing Waves and Harmonics

When a string is fixed at both ends, it supports standing waves at discrete frequencies known as harmonics or modes. The fundamental mode, which is the lowest frequency mode, has a simple wave pattern, and its frequency depends directly on the string's physical properties such as length, tension, and mass density. This concept is foundational for understanding how adjustments in these properties, like changing the length of the string, affect the frequency produced.

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