Question

A tube, closed at one end and containing air, produces, when excited, the fundamental note of frequency \(512 \mathrm{~Hz}\). If the tube is opened at both ends what is the fundamental frequency that can be excited (in \(\mathrm{Hz}\) )? (A) 256 (B) 1024 (C) 128 (D) 512

Short answer

The fundamental frequency of the tube when opened at both ends is (B) 1024 Hz.

Step-by-step solution

Understand the fundamental frequency of a tube

The fundamental frequency of a tube depends on its length and whether it is open or closed at one or both ends. For a tube closed at one end, the fundamental frequency is determined by the following formula: \(f_1 = \frac{c}{4L}\), where \(f_1\) is the fundamental frequency, \(L\) is the length of the tube, and \(c\) is the speed of sound in air, which is approximately \(343 \frac{m}{s}\). For a tube opened at both ends, the fundamental frequency is given by: \(f_2 = \frac{c}{2L}\), where \(f_2\) is the new fundamental frequency.

Find the length of the tube using the closed tube formula

We are given the fundamental frequency of the tube when closed at one end (\(f_1\)) as \(512 Hz\). Therefore, we can find the length of the tube using the closed tube formula: \(L = \frac{c}{4f_1}\), Plugging in the given frequency and the speed of sound in air, we get: \(L = \frac{343}{4 \times 512} = \frac{343}{2048} m\)

Calculate the new fundamental frequency using the open tube formula

Now that we have the length of the tube, we can use the formula for the fundamental frequency of an open tube: \(f_2 = \frac{c}{2L}\), Plugging in the values of the speed of sound in air and the length of the tube, we get: \(f_2 = \frac{343}{2 \times \frac{343}{2048}} = 1024 Hz\)

Determine the correct answer

Comparing the calculated fundamental frequency (\(f_2 = 1024 Hz\)) to the given options: (A) 256 (B) 1024 (C) 128 (D) 512 The correct answer is (B) 1024 Hz.