Question

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

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

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.

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: \[\frac{2(1) - 1}{4L} \times 330 = 264\]

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: \[L = \frac{330}{264 \times 4}\]

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.