Question

In a resonance column, the first and second resonance are obtained at depths 24 cm and 78 cm. The third resonance will be obtained at what depth?

• A. \(160 { cm}\)
• B. \(132 { cm}\)
• C. \(131 { cm}\)
• D. \(152 { cm}\)

Hint:

In resonance column experiments: - The first resonance occurs at \( L_1 = \frac{\lambda}{4} \), - The second resonance occurs at \( L_2 = \frac{3\lambda}{4} \), - The third resonance occurs at \( L_3 = \frac{5\lambda}{4} \). The difference between consecutive resonances gives \( \frac{\lambda}{2} \).

Answer 1

The Correct Option is B

Step 1: Understanding Resonance in a Column of Air In a resonance column, resonance occurs at the positions of the odd harmonics of the fundamental mode. The resonance depths follow the pattern: \[ L_1, L_2, L_3, \dots \] where \( L_n \) corresponds to the \((2n-1)\)th harmonic. 
Step 2: Finding the Wavelength Given: \[ L_1 = 24 { cm}, \quad L_2 = 78 { cm} \] The difference between successive resonance depths gives half of the wavelength: \[ L_2 - L_1 = \frac{\lambda}{2} \] \[ \lambda = 2(L_2 - L_1) = 2(78 - 24) = 108 { cm} \] 
Step 3: Determining the Third Resonance Depth The third resonance depth is given by: \[ L_3 = L_2 + (L_2 - L_1) \] \[ L_3 = 78 + 54 = 132 { cm} \]