A closed organ pipe of length 99.4 cm is vibrating in its first overtone and in always resonanc

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A closed organ pipe of length 99.4 cm is vibrating in its...

A closed organ pipe of length $99.4 \mathrm{~cm}$ is vibrating in its first overtone and in always resonance with a tuning fork having frequency $f=(300-2 t) \mathrm{Hz}$, where $t$ is time in second. The rate by which radius of organ pipe changes when its radius is $1 \mathrm{~cm}$ is (speed of sound in organ pipe $=320 \mathrm{~m} / \mathrm{s}$ ) (A) $\frac{1}{72} \mathrm{~m} / \mathrm{s}$ (B) $\frac{1}{36} \mathrm{~m} / \mathrm{s}$ (C) $\frac{1}{18} \mathrm{~m} / \mathrm{s}$ (D) $\frac{1}{9} \mathrm{~m} / \mathrm{s}$

A closed organ pipe of length $99.4 \mathrm{~cm}$ is vibrating in its first overtone and in always resonance with a tuning fork having frequency $f=(300-2 t) \mathrm{Hz}$, where $t$ is time in second. The rate by which radius of organ pipe changes when its radius is $1 \mathrm{~cm}$ is (speed of sound in organ pipe $=320 \mathrm{~m} / \mathrm{s}$ )
(A) $\frac{1}{72} \mathrm{~m} / \mathrm{s}$
(B) $\frac{1}{36} \mathrm{~m} / \mathrm{s}$
(C) $\frac{1}{18} \mathrm{~m} / \mathrm{s}$
(D) $\frac{1}{9} \mathrm{~m} / \mathrm{s}$

Key Concepts

Implicit Differentiation

Implicit differentiation is a mathematical tool used when variables are interrelated by an equation and not expressed explicitly in terms of one another. By differentiating both sides of the equation with respect to time, one can solve for the rate of change of one variable when another variable is also changing with time.

Chain Rule in Differentiation

The chain rule is a fundamental principle in calculus that allows differentiation of composite functions. In situations where one variable depends on another through an intermediate function, the chain rule provides a method to differentiate with respect to time, capturing the rates of change of all composite layers within the relationship.

Relationship between Frequency, Speed of Sound, and Effective Length

The frequency of the sound produced in a pipe is directly related to the speed of sound and inversely related to the effective length of the pipe. For a closed organ pipe vibrating in one of its allowed modes, the frequency can be determined using formulas that incorporate the effective length (physical length plus end correction) and the specific harmonic under consideration.

End Correction in Acoustic Pipes

The concept of end correction accounts for the fact that the effective length of an open pipe is slightly longer than its physical length due to the migration of the antinode beyond the open end. This extra length is typically a fixed fraction of the pipe’s radius, which must be considered when calculating the standing wave conditions in the pipe.

Resonance in Closed Organ Pipes

Closed organ pipes support standing waves that only allow certain harmonics, namely the odd harmonics. This means that the resulting resonant frequencies are determined by the relationship between the speed of sound and the effective length of the pipe. For the fundamental mode, the wavelength is four times the effective length, and for overtones (such as the first overtone), the relationship scales accordingly, ensuring resonance only at these specific frequencies.

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