A U-tube having unequal arm-lengths has water in it. A tuning fork of frequency 440 Hz can set up the air in the shorter arm in its fundamental mode of vibration and the same tuning fork can set up the air in the longer arm in its first overtone vibration. Find the length of the air columns Neglect any end effect and assume that the speed of sound in air `= 330 ms^-1`

Explanation

To solve the problem, we need to find the lengths of the air columns in a U-tube with unequal arm lengths, where one arm resonates in its fundamental mode and the other in its first overtone mode. We will use the relationship between frequency, speed of sound, and the length of the air column in an open tube.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Frequency of the tuning fork, \( f = 440 \, \text{Hz} \)
- Speed of sound in air, \( v = 330 \, \text{m/s} \)

2. **Fundamental Mode in the Shorter Arm:**
- For the fundamental mode of vibration in an open tube, the formula for frequency is:
\[ f = \frac{v}{2L_1} \]
- Here, \( L_1 \) is the length of the air column in the shorter arm.

3. **Rearranging the Formula for \( L_1 \):**
- Rearranging the formula gives:
\[ L_1 = \frac{v}{2f} \]
- Substituting the values:
\[ L_1 = \frac{330 \, \text{m/s}}{2 \times 440 \, \text{Hz}} = \frac{330}{880} = 0.375 \, \text{m} \]
- Therefore, \( L_1 = 0.375 \, \text{m} \) or \( 37.5 \, \text{cm} \).

4. **First Overtone in the Longer Arm:**
- For the first overtone in an open tube, the formula for frequency is:
\[ f = \frac{3v}{4L_2} \]
- Here, \( L_2 \) is the length of the air column in the longer arm.

5. **Rearranging the Formula for \( L_2 \):**
- Rearranging the formula gives:
\[ L_2 = \frac{3v}{4f} \]
- Substituting the values:
\[ L_2 = \frac{3 \times 330 \, \text{m/s}}{4 \times 440 \, \text{Hz}} = \frac{990}{1760} = 0.5625 \, \text{m} \]
- Therefore, \( L_2 = 0.5625 \, \text{m} \) or \( 56.25 \, \text{cm} \).

### Final Results:
- Length of the air column in the shorter arm, \( L_1 = 37.5 \, \text{cm} \)
- Length of the air column in the longer arm, \( L_2 = 56.25 \, \text{cm} \)