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Chapter 16: Problem 52

A half-open pipe is constructed to produce a fundamental frequency of \(262 \mathrm{~Hz}\) when the air temperature is \(22^{\circ} \mathrm{C} .\) It is used in an overheated building when the temperature is \(35^{\circ} \mathrm{C} .\) Neglecting thermal expansion in the pipe, what frequency will be heard?

Short Answer

Expert verified
Answer: To find the new frequency, follow these steps: 1. Calculate the speed of sound at 22°C: \(v_1 = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{22}{273.15}}\) 2. Calculate the speed of sound at 35°C: \(v_2 = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{35}{273.15}}\) 3. Find the ratio of the speeds of sound: \(ratio = \frac{v_2}{v_1}\) 4. Calculate the new frequency: \(new\_frequency = 262\mathrm{~Hz} \times ratio\) Compute the values and report the new frequency that will be heard in the overheated building.

Step by step solution

01

Calculate the speed of sound at 22°C

To find the speed of sound at the given temperature, we can use the formula: \(v = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{T}{273.15}}\) where \(v\) is the speed of sound and \(T\) is the temperature in Celsius. For \(T = 22^{\circ} \mathrm{C}\): \(v_1 = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{22}{273.15}}\) Calculate the value of \(v_1\).
02

Calculate the speed of sound at 35°C

Now, we will calculate the speed of sound at 35°C using the same formula. For \(T = 35^{\circ} \mathrm{C}\): \(v_2 = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{35}{273.15}}\) Calculate the value of \(v_2\).
03

Find the ratio of the speeds of sound

Find the ratio of the speeds of sound at the two temperatures: \(ratio = \frac{v_2}{v_1}\) Calculate the value of the ratio.
04

Calculate the new frequency

Now, we can find the new frequency by multiplying the original frequency with the ratio: \(new\_frequency = original\_frequency \times ratio\) \(new\_frequency = 262\mathrm{~Hz} \times ratio\) Calculate the value of the new frequency. This is the frequency that will be heard in the overheated building.

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