An organ pipe is closed at one end has fundamental frequency of 1500 Hz. The maximum number of overtones generated by this pipe which a normal person can hear is

Options

A. `14`
B. 13
C. 6
D. 9

Answer

To solve the problem of determining the maximum number of overtones generated by an organ pipe closed at one end with a fundamental frequency of 1500 Hz, we can follow these steps:

### Step 1: Understand the Overtone Frequencies
For an organ pipe closed at one end, the frequencies of the harmonics (overtones) are given by:
- Fundamental frequency (1st harmonic): \( f_1 = f \)
- First overtone (3rd harmonic): \( f_2 = 3f \)
- Second overtone (5th harmonic): \( f_3 = 5f \)
- Third overtone (7th harmonic): \( f_4 = 7f \)
- And so on...

### Step 2: Identify the Fundamental Frequency
The fundamental frequency \( f \) is given as 1500 Hz.

### Step 3: Determine the Frequency Range of Human Hearing
A normal human can hear frequencies from 20 Hz to 20,000 Hz (20 kHz). Therefore, we need to find the maximum overtone frequency that is less than or equal to 20,000 Hz.

### Step 4: Set Up the Inequality
The frequency of the nth overtone can be expressed as:
\[ f_n = (2n - 1)f \]
where \( n \) is the number of overtones. We want to find the maximum \( n \) such that:
\[ f_n \leq 20,000 \text{ Hz} \]

Substituting the fundamental frequency:
\[ (2n - 1) \times 1500 \leq 20,000 \]

### Step 5: Solve the Inequality
1. Rearranging the inequality:
\[ 2n - 1 \leq \frac{20,000}{1500} \]
2. Calculate \( \frac{20,000}{1500} \):
\[ \frac{20,000}{1500} = \frac{200}{15} \approx 13.33 \]
3. Now, we have:
\[ 2n - 1 \leq 13.33 \]
4. Adding 1 to both sides:
\[ 2n \leq 14.33 \]
5. Dividing by 2:
\[ n \leq 7.165 \]

### Step 6: Determine the Maximum Integer Value of \( n \)
Since \( n \) must be an integer, the maximum value of \( n \) is 7.

### Step 7: Calculate the Number of Overtones
The number of overtones is given by \( n - 1 \) (since the first harmonic is the fundamental frequency). Therefore:
\[ \text{Number of overtones} = n - 1 = 7 - 1 = 6 \]

### Final Answer
The maximum number of overtones that can be generated by the pipe, which a normal person can hear, is **6**.

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