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Chapter 29: Problem 2831

An air column in a pipe, which is closed at one end will be in resonance with a vibrating tuning fork of frequency \(264 \mathrm{~Hz}\), What is the length of the column if it is in \(\mathrm{cm} ?\) (speed of sound in \(\operatorname{air}=330 \mathrm{~m} / \mathrm{s}\) ) (A) \(62.50\) (B) \(15.62\) (C) 125 (D) \(93.75\)

Short Answer

Expert verified
The length of the air column needed for resonance with the tuning fork is approximately 31.25 cm. The correct answer is not given in the options provided.

Step by step solution

01

Recall the fundamental resonance mode for a closed pipe

: The fundamental mode of resonance (n=1) for a closed pipe is given by the formula: \[f = \frac{2n - 1}{4L}v\] Where \(f\) is the frequency, \(L\) is the length of the air column, and \(v\) is the speed of sound in air.
02

Substitute the given values into the formula

: We are given the frequency \(f = 264 \mathrm{~Hz}\) and the speed of sound in air \(v = 330 \mathrm{~m} / \mathrm{s}\). To solve for the length of the air column (L), we will substitute these values into the formula and solve for L. The given resonance mode for the closed pipe is the fundamental mode, n = 1: \[\frac{2(1) - 1}{4L} \times 330 = 264\]
03

Solve for the air column length L

: Now, we need to solve the equation for L: \[\frac{1}{4L} \times 330 = 264\] To get L, we first multiply each side of the equation by 4L: \[330 = 264 \times 4L\] Now, divide each side by 264: \[L = \frac{330}{264 \times 4}\]
04

Calculate the air column length and convert to cm

: Calculate the value of L: \[L = \frac{330}{1056} \approx 0.3125 \mathrm{~m}\] To convert this length to centimeters, multiply by 100: \[L = 0.3125 \times 100 \approx 31.25 \mathrm{~cm}\] The length of the air column needed for resonance with the tuning fork is approximately 31.25 cm. The closest answer in the options is (B) \(15.62\). However, this answer is not correct and thus the correct answer is not given in the options provided. The correct answer should be 31.25 cm.

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Most popular questions from this chapter

A closed organ pipe of length \(\mathrm{L}\) and an open organ pipe contain gases of densities \(\rho_{1}\) and \(\rho_{2}\) respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. What is the length of the open organ pipe? (A) \(\left.(4 \mathrm{~L} / 3) \sqrt{(} \rho_{1} / \rho_{2}\right)\) (B) \((\mathrm{L} / 3)\) (C) \((4 \mathrm{~L} / 3) \sqrt{\left(\rho_{2} / \rho_{1}\right)}\) (D) (4L / 3)

In a resonance tube experiment, the first resonance is obtained for $10 \mathrm{~cm}\( of air column and the second for 32 \)\mathrm{cm}$. The end correction for this apparatus is equal to \(\ldots \ldots \ldots\) (A) \(1.9 \mathrm{~cm}\) (B) \(0.5 \mathrm{~cm}\) (C) \(2 \mathrm{~cm}\) (D) \(1.0 \mathrm{~cm}\)

An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by $100 \mathrm{~Hz}$ than the fundamental frequency of the open pipe. What is the fundamental frequency of the open pipe? (A) 240 (B) 200 (C) 300 (D) 480

Two vibrating strings of the same material but of lengths Land \(2 \mathrm{~L}\) have radii \(2 \mathrm{r}\) and \(\mathrm{r}\) respectively. They are stretched under the same tension. Both the string vibrate in their fundamental modes. The one of length \(\mathrm{L}\) with frequency \(\mathrm{U}_{1}\) and the other with frequency \(\mathrm{U}_{2}\). What is the ratio $\left(\mathrm{U}_{1} / \mathrm{U}_{2}\right)$ ? (A) 8 (B) 2 (C) 1 (D) 4

A tuning fork of \(512 \mathrm{~Hz}\) is used to produce resonance in a resonance tube experiment. The level of water at first resonance is $30.7 \mathrm{~cm}\( and at second resonance is \)63.2 \mathrm{~cm}$. What is the error in calculating velocity of sound? Assume the speed of sound $330 \mathrm{~m} / \mathrm{s}$. (A) \(58(\mathrm{~cm} / \mathrm{s})\) (B) \(204.1(\mathrm{~cm} / \mathrm{s})\) (C) \(280(\mathrm{~cm} / \mathrm{s})\) (D) \(110(\mathrm{~cm} / \mathrm{s})\)
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