/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 304 A pipe open at both ends has a f... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 304

A pipe open at both ends has a fundamental frequency \(f\) in air. The pipe is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column is now (A) \(\frac{3 f}{4}\) (B) \(2 f\) (C) \(f\) (D) \(\frac{f}{2}\)

Short Answer

Expert verified
The new fundamental frequency of the air column when half of the pipe is submerged in water is equal to the original fundamental frequency, \(f\). Therefore, the correct answer is (C).

Step by step solution

01

Understanding the fundamental frequency of open and closed air columns

The fundamental frequency of an open air column, like our open pipe, can be found using the following formula: \( f = \frac{v}{2L} \), where \(f\) is the fundamental frequency, \(v\) is the speed of sound, and \(L\) is the length of the air column. The fundamental frequency of a closed air column, on the other hand, can be found using this formula: \( f = \frac{v}{4L} \), where \(f\) is the fundamental frequency, \(v\) is the speed of sound, and \(L\) is the length of the air column.
02

Getting the initial frequency in terms of speed of sound and length

Initially, the pipe is open at both ends and has a fundamental frequency \(f\): \( f = \frac{v}{2L} \) Now we'll find the relationship between the speed of sound, \(v\), and the length of the pipe, \(L\): \[ L = \frac{v}{2f} \]
03

Finding the new length when half of the pipe is submerged in water

When half of the pipe is submerged in water, the air column in the remaining half becomes closed. Therefore, the length of the closed air column, which we'll call \(L'\), is equal to half the length of the pipe, \(L\): \[ L' = \frac{1}{2} L \]
04

Finding the new fundamental frequency

Now, we'll find the new fundamental frequency, which we'll call \(f'\), using the formula for the fundamental frequency of closed air columns: \[ f' = \frac{v}{4L'} \] We already know the relation between speed of sound and initial frequency: \[ L = \frac{v}{2f} \] Substitute for \(L'\): \[ L' = \frac{1}{2} L = \frac{v}{4f} \] Now we'll substitute this value of \(L'\) back into the formula for \(f'\): \[ f' = \frac{v}{4\left(\frac{v}{4f}\right)} \]
05

Simplifying and comparing the new fundamental frequency to the original

Simplify the fraction: \[ f' = \frac{v}{\frac{v}{f}} \] \[ f' = f \] Therefore, the new fundamental frequency \(f'\) is equal to the original fundamental frequency \(f\). This corresponds to option (C).

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Most popular questions from this chapter

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