An organ pipe `P_1` closed at one end is vibrating in its second overtone and another pipe `P_2` open at both ends is vibrating in its fourth overtone are in resonance with a given tuning fork. Then what is the ratio of length of `P_2` to that of `P_1` ?

Answer

To find the ratio of the lengths of the two organ pipes \( P_2 \) and \( P_1 \), we will first determine the lengths of each pipe based on their respective overtones.

### Step 1: Determine the length of pipe \( P_1 \) (closed at one end)

For a pipe closed at one end, the harmonics are given by the formula:
- Fundamental frequency (1st harmonic): \( L_1 = \frac{\lambda}{4} \)
- 1st overtone (2nd harmonic): \( L_1 = \frac{3\lambda}{4} \)
- 2nd overtone (3rd harmonic): \( L_1 = \frac{5\lambda}{4} \)

Since \( P_1 \) is vibrating in its second overtone, we have:
\[ L_1 = \frac{5\lambda}{4} \]

### Step 2: Determine the length of pipe \( P_2 \) (open at both ends)

For a pipe open at both ends, the harmonics are given by the formula:
- Fundamental frequency (1st harmonic): \( L_2 = \frac{\lambda}{2} \)
- 1st overtone (2nd harmonic): \( L_2 = \lambda \)
- 2nd overtone (3rd harmonic): \( L_2 = \frac{3\lambda}{2} \)
- 3rd overtone (4th harmonic): \( L_2 = 2\lambda \)

Since \( P_2 \) is vibrating in its fourth overtone, we have:
\[ L_2 = \frac{5\lambda}{2} \]

### Step 3: Calculate the ratio of lengths \( \frac{L_2}{L_1} \)

Now we can find the ratio of the lengths of \( P_2 \) to \( P_1 \):
\[ \frac{L_2}{L_1} = \frac{\frac{5\lambda}{2}}{\frac{5\lambda}{4}} = \frac{5\lambda}{2} \times \frac{4}{5\lambda} \]
The \( 5\lambda \) cancels out:
\[ \frac{L_2}{L_1} = \frac{4}{2} = 2 \]

### Final Answer
The ratio of the length of \( P_2 \) to that of \( P_1 \) is:
\[ \frac{L_2}{L_1} = 2 \]

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