A closed organ pipe 150 cm long gives 7 beats per second with an open organ pipe of length 350 cm, both vibrating in fundamental mode. The velocity of sound is ________ m/s.
Correct Answer: 294
Approach Solution - 1
To find the velocity of sound using the given problem, we'll utilize the principles of sound waves in organ pipes. Since both pipes are in fundamental modes, we'll use the fundamental frequency formulas for open and closed pipes.
- For a closed organ pipe:
The fundamental frequency \(f_c\) is given by:
\(f_c = \frac{v}{4L_c}\)
where \(v\) is the velocity of sound, and \(L_c = 150 \, \text{cm} = 1.5 \, \text{m}\) is the length of the closed pipe. - For an open organ pipe:
The fundamental frequency \(f_o\) is given by:
\(f_o = \frac{v}{2L_o}\)
where \(L_o = 350 \, \text{cm} = 3.5 \, \text{m}\) is the length of the open pipe. - Given the beat frequency is 7 beats per second:
The beat frequency is the absolute difference between the two fundamental frequencies, \(f_o - f_c = 7 \, \text{Hz}\). - Express the equations for the frequencies:
\(f_c = \frac{v}{4 \times 1.5} = \frac{v}{6}\)
\(f_o = \frac{v}{2 \times 3.5} = \frac{v}{7}\) - Plug into the beat frequency equation:
\(\left|\frac{v}{7} - \frac{v}{6}\right| = 7\) - Solve for \(v\):
\(\left|\frac{6v - 7v}{42}\right| = 7\)
\(\left|\frac{-v}{42}\right| = 7\)
\(\frac{v}{42} = 7\)
\(v = 42 \times 7 = 294 \, \text{m/s}\)
The calculated velocity of sound \(v = 294 \, \text{m/s}\) is within the specified range (294, 294).
Approach Solution -2
For a closed pipe of length \(L = 150 \, \text{cm} = 1.5 \, \text{m}\):
The fundamental frequency is given by:
\(f_c = \frac{v}{4L}.\)
For an open pipe of length \(L = 350 \, \text{cm} = 3.5 \, \text{m}\):
The fundamental frequency is given by:
\(f_o = \frac{v}{2L}.\)
Given that the beat frequency is:
\(|f_c - f_o| = 7 \, \text{Hz}.\)
Substituting:
\(\left| \frac{v}{4 \cdot 1.5} - \frac{v}{2 \cdot 3.5} \right| = 7.\)
Simplifying:
\(\left| \frac{v}{6} - \frac{v}{7} \right| = 7.\)
Solving for \(v\):
\(\frac{v}{42} = 7 \implies v = 42 \cdot 7 = 294 \, \text{m/s}.\)
The Correct answer is: 294
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