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 $50\,cm$. The next larger length of the column resonating with the same tuning fork is :

• A. 100 cm
• B. 150 cm
• C. 200 cm
• D. 66.7 cm

Answer 1

The Correct Option is B

Resonating Length for a Closed Organ Pipe 

The first minimum resonating length for a closed organ pipe is given by:

\(\frac{\lambda}{4} = 50 \, \text{cm}\)

Where: - \( \lambda \) is the wavelength of the sound wave in the pipe.

\(\therefore\) The next larger length of the air column is:

\(\frac{3 \lambda}{4} = 150 \, \text{cm}\)

Conclusion:

The next resonating length of the air column is 150 cm.

Answer 2

Concept: For an air column closed at one end and open at the other, resonance occurs at odd multiples of the quarter wavelength.

So, the resonant lengths are:
\(L_1 = \frac{\lambda}{4},\quad L_2 = \frac{3\lambda}{4},\quad L_3 = \frac{5\lambda}{4},\quad \text{and so on.}\)

Given:

• First resonant length \(L_1 = 50\, \text{cm}\)

Next resonant length:
\(L_2 = \frac{3\lambda}{4} = 3 \times \frac{\lambda}{4} = 3 \times L_1 = 3 \times 50 = 150\, \text{cm}\)

Answer: \(\boxed{150\, \text{cm}}\)