An organ pipe closed at one end has a length 1m and an open organ pipe has a length 1.6 m . The speed of sound in air is 320 m/s . The two pipes can resonates for a sound of frequency

Options

A. 100 Hz
B. 240 Hz
C. 320 Hz
D. 400 Hz

Answer

To solve the problem of finding the resonating frequency for an organ pipe closed at one end and an open organ pipe, we can follow these steps:

### Step 1: Identify the parameters
- Length of the closed organ pipe (L1) = 1 m
- Length of the open organ pipe (L2) = 1.6 m
- Speed of sound in air (v) = 320 m/s

### Step 2: Calculate the resonant frequencies for the closed organ pipe
For a closed organ pipe, the frequency of the nth harmonic is given by the formula:
\[ f_n = \frac{(2n + 1) v}{4L} \]
Where:
- \(n\) = harmonic number (n = 0, 1, 2, ...)

For the closed pipe (L1 = 1 m):
- For n = 0:
\[ f_0 = \frac{(2 \cdot 0 + 1) \cdot 320}{4 \cdot 1} = \frac{320}{4} = 80 \text{ Hz} \]
- For n = 1:
\[ f_1 = \frac{(2 \cdot 1 + 1) \cdot 320}{4 \cdot 1} = \frac{3 \cdot 320}{4} = 240 \text{ Hz} \]
- For n = 2:
\[ f_2 = \frac{(2 \cdot 2 + 1) \cdot 320}{4 \cdot 1} = \frac{5 \cdot 320}{4} = 400 \text{ Hz} \]
- For n = 3:
\[ f_3 = \frac{(2 \cdot 3 + 1) \cdot 320}{4 \cdot 1} = \frac{7 \cdot 320}{4} = 560 \text{ Hz} \]

### Step 3: Calculate the resonant frequencies for the open organ pipe
For an open organ pipe, the frequency of the nth harmonic is given by:
\[ f_n = \frac{n v}{2L} \]
Where:
- \(n\) = harmonic number (n = 1, 2, 3, ...)

For the open pipe (L2 = 1.6 m):
- For n = 1:
\[ f_1 = \frac{1 \cdot 320}{2 \cdot 1.6} = \frac{320}{3.2} = 100 \text{ Hz} \]
- For n = 2:
\[ f_2 = \frac{2 \cdot 320}{2 \cdot 1.6} = \frac{640}{3.2} = 200 \text{ Hz} \]
- For n = 3:
\[ f_3 = \frac{3 \cdot 320}{2 \cdot 1.6} = \frac{960}{3.2} = 300 \text{ Hz} \]
- For n = 4:
\[ f_4 = \frac{4 \cdot 320}{2 \cdot 1.6} = \frac{1280}{3.2} = 400 \text{ Hz} \]

### Step 4: Find the common resonating frequency
Now we will compare the frequencies calculated for both pipes:
- Closed pipe frequencies: 80 Hz, 240 Hz, 400 Hz, 560 Hz
- Open pipe frequencies: 100 Hz, 200 Hz, 300 Hz, 400 Hz

The common frequency for both pipes is:
\[ \text{Common frequency} = 400 \text{ Hz} \]

### Conclusion
The two pipes can resonate at a frequency of **400 Hz**.

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