An air column, closed at one end and open at the other, resonates with a tuning fork when the smallest length

Question

An air column, closed at one end and open at the other, resonates with a tuning fork when the smallest length of the column is $50 	ext{ cm}$. The next larger length of the column resonating with the same tuning fork is

Accepted Answer

1. **Identify the Type of Air Column:** An air column closed at one end and open at the other acts as a closed organ pipe.
2. **Resonance Conditions for Closed Pipe:** For a closed organ pipe, resonance occurs at odd multiples of $\lambda/4$. The resonating lengths are given by $L = (2n-1)\frac{\lambda}{4}$, where $n = 1, 2, 3, \dots$
3. **Find the Smallest Length:** The smallest length (fundamental mode, $n=1$) is given as $L_1 = \frac{\lambda}{4} = 50 	ext{ cm}$.
4. **Calculate the Next Larger Length:** The next larger length (first overtone, $n=2$) will be $L_2 = \frac{3\lambda}{4}$.
   Since $\frac{\lambda}{4} = 50 	ext{ cm}$, we substitute this value into the equation for $L_2$:
   $$L_2 = 3 	imes 50 	ext{ cm} = 150 	ext{ cm}$$

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