Question

A half-open pipe is constructed to produce a fundamental frequency of \(262 \mathrm{~Hz}\) when the air temperature is \(22.0^{\circ} \mathrm{C} .\) It is used in an overheated building when the temperature is \(35.0^{\circ} \mathrm{C} .\) Neglecting thermal expansion in the pipe, what frequency will be heard?

Short answer

Answer: To solve this problem, follow the steps provided: 1. Calculate the initial speed of sound at 22.0°C: \(v_1 = 331.4 \mathrm{~m/s} \times \sqrt{1 + \frac{22.0}{273.15}}\) 2. Calculate the final speed of sound at 35.0°C: \(v_2 = 331.4 \mathrm{~m/s} \times \sqrt{1 + \frac{35.0}{273.15}}\) 3. Determine the length of the pipe using the initial frequency: \(L = \frac{v_1}{4f_1}\) 4. Calculate the final frequency, \(f_2\), produced by the pipe at 35.0°C: \(f_2 = \frac{v_2}{4L}\) With the provided values, perform the calculations and find the final frequency, \(f_2\), of the sound produced by the pipe when the temperature is raised to 35.0°C.

Step-by-step solution

Identify the formulas needed

We will need the following formulas: 1. Speed of sound in air as a function of temperature: \(v = 331.4 \mathrm{~m/s} \times \sqrt{1 + \frac{T}{273.15}}\) 2. Frequency of a half-open pipe: \(f = \frac{v}{4L}\), where L is the length of the pipe.

Calculate the initial speed of sound

First, we need to find the speed of sound in air at the initial temperature, which is \(22.0^{\circ} \mathrm{C}\). Using the speed of sound formula: \(v_1 = 331.4 \mathrm{~m/s} \times \sqrt{1 + \frac{22.0}{273.15}}\)

Calculate the final speed of sound

Next, we need to find the speed of sound in air at the final temperature, which is \(35.0^{\circ} \mathrm{C}\). Using the speed of sound formula: \(v_2 = 331.4 \mathrm{~m/s} \times \sqrt{1 + \frac{35.0}{273.15}}\)

Determine the length of the pipe

Now, we need to find the length of the half-open pipe. We know the initial frequency, \(f_1 = 262 \mathrm{~Hz}\) and we can use the half-open pipe frequency formula: \(L = \frac{v_1}{4f_1}\)

Calculate the final frequency of the pipe

Finally, we will use the length of the pipe and the final speed of sound in air to find the final frequency, \(f_2\), produced by the pipe. Using the half-open pipe frequency formula with \(v_2\): \(f_2 = \frac{v_2}{4L}\) Now you can calculate all the values to find the final frequency, \(f_2\), of the pipe.