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Chapter 10: Problem 1493

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

Short Answer

Expert verified
The new fundamental frequency of the air column when the tube is half-submerged in water is twice the initial frequency. Therefore, the correct answer is (D) \(2f\).

Step by step solution

01

Formula for the fundamental frequency

For an open cylindrical tube, the fundamental frequency (\(f\)) is given by the formula: \[f = \frac{v}{2L}\] where: \(v\) is the speed of sound in air, \(L\) is the length of the air column.
02

Calculate the length of the air column when half-submerged in water

When the tube is half-submerged in water, the length of the air column is halved. Let \(L'\) represent the new length of the air column: \[L' = \frac{L}{2}\]
03

Calculate the new fundamental frequency

Using the formula for the fundamental frequency and the new length of the air column (\(L'\)), we can calculate the new fundamental frequency (\(f'\)): \[f' = \frac{v}{2L'}\]
04

Substitute the new length in the frequency equation

Replace \(L'\) with \(\frac{L}{2}\) in the equation for the new fundamental frequency: \[f' = \frac{v}{2\left(\frac{L}{2}\right)}\]
05

Simplify the equation and compare the initial and final frequencies

Now we simplify the equation and compare the new frequency (\(f'\)) to the initial frequency (\(f\)): \[f' = \frac{v}{L}\] Comparing this to the initial frequency equation: \[f = \frac{v}{2L}\] We can clearly see that the new fundamental frequency of the air column is twice the initial frequency. Therefore, the correct answer is (D) \(2f\).

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