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Chapter 12: Problem 32

An auditorium has organ pipes at the front and at the rear of the hall. Two identical pipes, one at the front and one at the back, have fundamental frequencies of \(264.0 \mathrm{Hz}\) at \(20.0^{\circ} \mathrm{C} .\) During a performance, the organ pipes at the back of the hall are at $25.0^{\circ} \mathrm{C},\( while those at the front are still at \)20.0^{\circ} \mathrm{C} .$ What is the beat frequency when the two pipes sound simultaneously?

Short Answer

Expert verified
Based on the given information, and after analyzing and solving the question step by step, we could determine that the beat frequency when the two organ pipes sound simultaneously is 2 Hz.

Step by step solution

01

1. Calculate the speed of sound in air at 20掳C and 25掳C

Using the formula \(v = v_0\sqrt{1+\frac{T}{T_0}}\), where \(v_0 = 331 \mathrm{m/s}\) and \(T_0 = 273 \mathrm{K}\), we can find the speed of sound for 20掳C and 25掳C: For \(20.0^{\circ} \mathrm{C}\): $$ v = 331\sqrt{1+\frac{293}{273}} = 344\,\mathrm{m/s} $$ For \(25.0^{\circ} \mathrm{C}\): $$ v' = 331\sqrt{1+\frac{298}{273}} = 346\,\mathrm{m/s} $$
02

2. Calculate the wavelength of the sound produced by the organ pipe at 20掳C

We know the frequency of the organ pipe (\(f = 264\,\mathrm{Hz}\)) and the speed of sound at 20掳C (\(v = 344\,\mathrm{m/s}\)). Now, we can calculate the wavelength using the formula \(位 = \frac{v}{f}\): $$ 位 = \frac{344}{264} = 1.30\,\mathrm{m} $$
03

3. Calculate the new frequency of the organ pipe at 25掳C

Now, we will calculate the frequency of the organ pipe at 25掳C (\(f'\)). We know the speed of sound at 25掳C (\(v' = 346\,\mathrm{m/s}\)) and the wavelength (\(位 = 1.30\,\mathrm{m}\)). We can calculate the new frequency using the formula \(f' = \frac{v'}{位}\): $$ f' = \frac{346}{1.30} = 266\,\mathrm{Hz} $$
04

4. Calculate the beat frequency

Finally, we can calculate the beat frequency by taking the absolute difference between the frequencies of the pipes: $$ f_{beat} = |f' - f| = |266 - 264| = 2\,\mathrm{Hz} $$ The beat frequency when the two organ pipes sound simultaneously is \(2\,\mathrm{Hz}\).

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