Question

A sonometer wire resonates with a given tuning fork, forming standing waves with five antinodes between the two bridges, when a mass of \(9\) kg is suspended from the wire. When this mass is replaced by mass \(M\), the wire resonates with the same tuning fork, forming three antinodes for the same positions of the bridges. The value of \(M\) (in kg) is

• A. \(\frac{1}{25}\)
• B. \(5\)
• C. \(\frac{25}{2}\)
• D. \(25\)

Hint:

For a fixed wire length and tuning fork, the number of loops times the square root of the tension is constant.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A stretched wire vibrating in \(p\) loops has frequency \(f=\frac{p}{2l}\sqrt{\frac{T}{\mu}}\).

Step 2: Same fork and bridges
The frequency \(f\), the length \(l\) and the mass per unit length \(\mu\) remain the same, so \(p\sqrt{T}\) is constant.

Step 3: Calculation
The tension is \(T=Mg\), so \(5\sqrt{9g}=3\sqrt{Mg}\). Hence \(5\times 3=3\sqrt{M}\) and \(\sqrt{M}=5\).

Step 4: Result
\(M=25\) kg, which is option (D).

Final Answer:
The required mass is 25 kg. \[ \boxed{\text{(D) }25} \]