An organ pipe is open at both ends. It is producing sound at its third harmonic, the frequency of which is 262 Hz. The speed of sound is 343 m/s. What is the length of the pipe?

Options

A. 1.96 m
B. 1.31 m
C. 1.85 m
D. 0.926 m

Answer

To solve the problem of finding the length of an organ pipe that is open at both ends and producing sound at its third harmonic, we can follow these steps:

### Step 1: Understand the Harmonics in an Open Organ Pipe
An open organ pipe supports standing waves with nodes at both ends. For the third harmonic, the pattern consists of three segments of a wave, which corresponds to 3/2 wavelengths fitting into the length of the pipe.

### Step 2: Relate the Length of the Pipe to Wavelength
The relationship between the length of the pipe (L) and the wavelength (λ) for the third harmonic can be expressed as:
\[ L = \frac{3}{2} \lambda \]

### Step 3: Calculate the Wavelength
The wavelength can be calculated using the speed of sound (v) and the frequency (f) of the sound wave:
\[ \lambda = \frac{v}{f} \]
Given:
- Speed of sound, \( v = 343 \, \text{m/s} \)
- Frequency, \( f = 262 \, \text{Hz} \)

Substituting these values into the equation for wavelength:
\[ \lambda = \frac{343 \, \text{m/s}}{262 \, \text{Hz}} \]

### Step 4: Calculate the Length of the Pipe
Now substitute the value of λ back into the equation for the length of the pipe:
\[ L = \frac{3}{2} \lambda = \frac{3}{2} \left(\frac{343}{262}\right) \]

### Step 5: Simplify the Calculation
Calculating the value:
1. First, calculate \( \frac{343}{262} \):
\[ \frac{343}{262} \approx 1.309 \]
2. Now, calculate \( L \):
\[ L = \frac{3}{2} \times 1.309 \approx 1.9635 \, \text{m} \]

### Final Answer
Thus, the length of the pipe is approximately:
\[ L \approx 1.96 \, \text{m} \]

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