Question

In sonometer experiment, the string of length \({}'L'\) under tension vibrates in second overtone between the two bridges. The amplitude of vibration is maximum at length

• A. \(\frac{L}{2},\frac{L}{4},\frac{L}{6}\)
• B. \(\frac{L}{6},\frac{L}{2},\frac{5L}{6}\)
• C. \(\frac{L}{3},\frac{2L}{3},\frac{5L}{6}\)
• D. \(\frac{L}{8},\frac{L}{4},\frac{L}{2}\)

Hint:

Second overtone means three loops, with antinodes at quarter, three quarters and five quarters of a wavelength from one end.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The second overtone is the third harmonic, so the string forms 3 loops: \(L=\frac{3\lambda}{2}\), giving \(\lambda=\frac{2L}{3}\).

Step 2: Antinode positions
Measuring from one bridge, antinodes lie at \(\frac{\lambda}{4},\frac{3\lambda}{4},\frac{5\lambda}{4}\).

Step 3: Calculation
\(\frac{\lambda}{4}=\frac{L}{6}\), \(\frac{3\lambda}{4}=\frac{L}{2}\), \(\frac{5\lambda}{4}=\frac{5L}{6}\)

Step 4: Compare the options
These are \(\frac{L}{6},\frac{L}{2},\frac{5L}{6}\), option (B).

Final Answer:
The antinodes are at L/6, L/2 and 5L/6. \[ \boxed{\text{(B) }\dfrac{L}{6},\dfrac{L}{2},\dfrac{5L}{6}} \]