An air column closed at one end and open at the other resonates with a tuning fork when the smallest length…
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 $50\text{ cm}$. The next larger length of the column resonating with the same tuning fork is [NEET 2016]
100 cm
150 cm
200 cm
66.7cm
Solution
The smallest length of the air column is associated with fundamental mode of vibration of the air column as shown in the figure of closed pipes for fundamental and higher modes.
$\because L_{\min} = \frac{\lambda}{4} \Rightarrow 50\text{ cm} = \frac{\lambda}{4} \Rightarrow \lambda = 200\text{ cm}$
The next higher length of the air column,
$L = \frac{\lambda}{4} + \frac{\lambda}{2} = \frac{\lambda + 2\lambda}{4} = \frac{3\lambda}{4} = \frac{3}{4} \times 200 = 150\text{ cm}$