Two pipes `P_(1)` and `P_(2)` are closed and open respectively. `P_(1)` has a length of `0.3 m`. Find the length of `P_(2)`, if third harmonic of `P_(1)` is same as first harmonic of `P_(2)`.

Answer

To solve the problem, we need to find the length of pipe \( P_2 \) given that the third harmonic of the closed pipe \( P_1 \) is equal to the first harmonic of the open pipe \( P_2 \).

### Step-by-Step Solution:

1. **Understand the Harmonics of the Pipes**:
- For a closed pipe (like \( P_1 \)), the harmonics are given by the formula:
\[ f_n = \frac{nV}{4L} \]
where \( n \) is the harmonic number (odd integers), \( V \) is the speed of sound, and \( L \) is the length of the pipe.
- For an open pipe (like \( P_2 \)), the harmonics are given by the formula:
\[ f_n = \frac{nV}{2L} \]
where \( n \) can be any integer.

2. **Identify the Given Information**:
- Length of pipe \( P_1 \) (closed pipe) is \( L_C = 0.3 \, m \).
- We need to find the length of pipe \( P_2 \) (open pipe) such that the third harmonic of \( P_1 \) equals the first harmonic of \( P_2 \).

3. **Calculate the Third Harmonic of \( P_1 \)**:
- The third harmonic of \( P_1 \) is given by:
\[ f_{3} = \frac{3V}{4L_C} = \frac{3V}{4 \times 0.3} \]

4. **Calculate the First Harmonic of \( P_2 \)**:
- The first harmonic of \( P_2 \) is given by:
\[ f_{1} = \frac{V}{2L_O} \]

5. **Set the Two Frequencies Equal**:
- According to the problem, we have:
\[ f_{3} = f_{1} \]
- Therefore, we can set up the equation:
\[ \frac{3V}{4 \times 0.3} = \frac{V}{2L_O} \]

6. **Cancel \( V \) from Both Sides**:
- Since \( V \) is present in both terms, we can cancel it out:
\[ \frac{3}{4 \times 0.3} = \frac{1}{2L_O} \]

7. **Cross-Multiply to Solve for \( L_O \)**:
- Cross-multiplying gives:
\[ 3 \cdot 2L_O = 4 \cdot 0.3 \]
- Simplifying this:
\[ 6L_O = 1.2 \]
- Now, divide both sides by 6:
\[ L_O = \frac{1.2}{6} = 0.2 \, m \]

8. **Conclusion**:
- The length of pipe \( P_2 \) is \( 0.2 \, m \) or \( 20 \, cm \).

### Final Answer:
The length of pipe \( P_2 \) is \( 0.2 \, m \) or \( 20 \, cm \).