An open organ pipe has a fundamental frequency of `300 H_(Z)` . The first overtone of a closed organ pipe has the same frequency as the first overtone of this open pipe . How long is each pipe ? (Speed of sound in air = `330 m//s`)

Explanation

To solve the problem, we need to find the lengths of both the open organ pipe and the closed organ pipe based on the given fundamental frequency and the relationship between their overtones.

### Step 1: Find the length of the open organ pipe

The fundamental frequency (first harmonic) of an open organ pipe is given by the formula:

\[ f_1 = \frac{v}{2L_1} \]

Where:
- \( f_1 \) is the fundamental frequency (300 Hz)
- \( v \) is the speed of sound in air (330 m/s)
- \( L_1 \) is the length of the open organ pipe

Rearranging the formula to solve for \( L_1 \):

\[ L_1 = \frac{v}{2f_1} \]

Substituting the known values:

\[ L_1 = \frac{330 \, \text{m/s}}{2 \times 300 \, \text{Hz}} = \frac{330}{600} = 0.55 \, \text{m} \]

### Step 2: Find the first overtone of the open organ pipe

The first overtone (second harmonic) of an open organ pipe is given by:

\[ f_2 = \frac{v}{L_1} \]

Since we already have \( L_1 \):

\[ f_2 = \frac{330 \, \text{m/s}}{0.55 \, \text{m}} = 600 \, \text{Hz} \]

### Step 3: Find the length of the closed organ pipe

The first overtone (third harmonic) of a closed organ pipe is given by:

\[ f_3 = \frac{3v}{4L_2} \]

Where \( L_2 \) is the length of the closed organ pipe. According to the problem, the first overtone of the closed pipe has the same frequency as the first overtone of the open pipe:

\[ f_3 = f_2 = 600 \, \text{Hz} \]

Setting the two equations equal:

\[ 600 = \frac{3 \times 330}{4L_2} \]

Rearranging to solve for \( L_2 \):

\[ L_2 = \frac{3 \times 330}{4 \times 600} \]

Calculating \( L_2 \):

\[ L_2 = \frac{990}{2400} = 0.4125 \, \text{m} \]

### Final Answer

The lengths of the pipes are:
- Length of the open organ pipe \( L_1 = 0.55 \, \text{m} \) (or 55 cm)
- Length of the closed organ pipe \( L_2 = 0.4125 \, \text{m} \) (or 41.25 cm)