Question

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

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

1. Calculate the speed of sound in air at 20°C and 25°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°C and 25°C: For \(20.0^{\circ} \mathrm{C}\): $$ v = 331\sqrt{1+\frac{293}{273}} = 344\,\mathrm{m/s} $$ For \(25.0^{\circ} \mathrm{C}\): $$ v' = 331\sqrt{1+\frac{298}{273}} = 346\,\mathrm{m/s} $$

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

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

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

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

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 \(2\,\mathrm{Hz}\).