Question

A sonometer wire resonates with a given tuning fork forming standing wave with \(5\) antinodes between two bridges when mass of \(9\) kg is suspended from the wire. When mass \(M\) is suspended from the wire with same fork and same length between two bridges, \(3\) antinodes are formed. Mass \(M\) is

• A. \(5\) kg
• B. \(14\) kg
• C. \(18\) kg
• D. \(25\) kg

Hint:

For the same fork and length, the number of loops is inversely proportional to the square root of the tension.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The frequency is \(f=\frac{p}{2L}\sqrt{\frac{T}{\mu}}\), where \(p\) is the number of loops. With \(f\), \(L\) and \(\mu\) fixed, \(p\sqrt{\frac{1}{T}}\) is constant.

Step 2: Set up the ratio
\(\frac{p_{1}}{p_{2}}=\sqrt{\frac{T_{2}}{T_{1}}}\)
Here \(p_{1}=5\), \(p_{2}=3\), \(T_{1}=9g\), \(T_{2}=Mg\).

Step 3: Calculation
\(\frac{5}{3}=\sqrt{\frac{M}{9}}\Rightarrow\frac{M}{9}=\frac{25}{9}\Rightarrow M=25\) kg

Step 4: Compare the options
This is option (D).

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