Question

Fundamental frequency of a sonometer wire is \(n\). If the tension is made 3 times and length and diameter are also increased 3 times, what is the new frequency?

• A. \( \frac{n}{3\sqrt{3}} \)
• B. \( 3n \)
• C. \( \sqrt{3}n \)
• D. \( \frac{n}{\sqrt{3}} \)

Hint:

Frequency depends inversely on length and on square root of linear density.

Answer 1

The Correct Option is A

Concept: \[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}, \quad \text{where } \mu \propto d^2 \]

Step 1:Apply changes:
•\(T \to 3T\)
•\(L \to 3L\)
•\(d \to 3d \Rightarrow \mu \to 9\mu\)

Step 2:Substitute: \[ f' = \frac{1}{2(3L)} \sqrt{\frac{3T}{9\mu}} \]

Step 3:Simplify: \[ f' = \frac{1}{3} \cdot \frac{1}{2L} \cdot \sqrt{\frac{T}{3\mu}} = \frac{1}{3\sqrt{3}} \cdot \frac{1}{2L}\sqrt{\frac{T}{\mu}} \] \[ f' = \frac{f}{3\sqrt{3}} \] Since \(f = n\), \[ f' = \frac{n}{3\sqrt{3}} \]