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Chapter 12: Problem 27

Two tuning forks, \(A\) and \(B\), excite the next-to-lowest resonant frequency in two air columns of the same length, but A's column is closed at one end and B's column is open at both ends. What is the ratio of A's frequency to B's frequency?

Short Answer

Expert verified
Answer: The ratio of the frequencies of tuning forks A and B is 5/8.

Step by step solution

01

Understanding resonant frequencies in air columns

For each type of air column, we need to find the formula for the resonant frequency. For an open-closed air column, the resonant frequency can be described as: $$ f_A = \frac{2n_A - 1}{4L}v$$ Where \(n_A\) is the harmonic number (an odd integer), \(L\) is the length of the air column, and \(v\) is the speed of sound in the air. For an open-open air column, the resonant frequency can be described as: $$ f_B = \frac{2n_B}{2L}v$$ Where \(n_B\) is the harmonic number (an integer), and the other variables are the same as before.
02

Finding the next-to-lowest resonant frequency

For the open-closed air column, the lowest resonant frequency occurs when \(n_A = 1\). Thus, the next-to-lowest resonant frequency occurs when \(n_A = 3\). For the open-open air column, the lowest resonant frequency occurs when \(n_B = 1\). Thus, the next-to-lowest resonant frequency occurs when \(n_B = 2\).
03

Plugging in values for next-to-lowest resonant frequencies for both columns

Now that we know the values for the harmonic numbers, we can find the next-to-lowest resonant frequencies for each column. For the open-closed column: $$ f_A = \frac{2(3) - 1}{4L}v = \frac{5}{4L}v $$ For the open-open column: $$ f_B = \frac{2(2)}{2L}v = \frac{2}{L}v $$
04

Finding the ratio of A's frequency to B's frequency

Now we need to find the ratio \(\frac{f_A}{f_B}\): $$\frac{f_A}{f_B} = \frac{\frac{5}{4L}v}{\frac{2}{L}v}$$ To simplify this expression, we can cancel variables \(L\) and \(v\) in the numerator and denominator: $$\frac{f_A}{f_B} = \frac{5}{4} \cdot \frac{1}{2}$$ Finally, we find the ratio: $$\frac{f_A}{f_B} = \frac{5}{8}$$ So, the ratio of A's frequency to B's frequency is \(\frac{5}{8}\).

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