Calculate the frequency of fifth harmonic of a closed organ pipe of length 50cm, if the velocity of sound in air is 330 m/s.

Explanation

To calculate the frequency of the fifth harmonic of a closed organ pipe, we can follow these steps:

### Step 1: Understand the formula for the frequency of harmonics in a closed organ pipe.
For a closed organ pipe, the frequency of the nth harmonic is given by the formula:
\[ f_n = \frac{nV}{4L} \]
where:
- \( f_n \) is the frequency of the nth harmonic,
- \( n \) is the harmonic number,
- \( V \) is the velocity of sound in air,
- \( L \) is the length of the organ pipe.

### Step 2: Identify the values given in the problem.
From the problem, we have:
- Length of the organ pipe, \( L = 50 \text{ cm} = 0.5 \text{ m} \)
- Velocity of sound in air, \( V = 330 \text{ m/s} \)
- Harmonic number, \( n = 5 \)

### Step 3: Substitute the values into the formula.
Now, we can substitute the values into the formula for the fifth harmonic:
\[ f_5 = \frac{5 \times 330 \text{ m/s}}{4 \times 0.5 \text{ m}} \]

### Step 4: Calculate the denominator.
First, calculate the denominator:
\[ 4 \times 0.5 = 2 \]

### Step 5: Calculate the frequency.
Now substitute back into the equation:
\[ f_5 = \frac{5 \times 330}{2} \]
Calculating the numerator:
\[ 5 \times 330 = 1650 \]
Now divide by the denominator:
\[ f_5 = \frac{1650}{2} = 825 \text{ Hz} \]

### Final Answer:
The frequency of the fifth harmonic of the closed organ pipe is \( 825 \text{ Hz} \).

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