Question

A sonometer wire vibrating in two loops is in resonance with a tuning fork. If tension applied to the string is increased to 16 times the original tension and length of the wire is doubled. It can still be in resonance with the tuning fork provided it vibrates in

• A. one loop
• B. two loops
• C. three loops
• D. four loops

Hint:

Frequency of a wire with p loops is \(f=\dfrac{p}{2l}\sqrt{\dfrac{T}{\mu}}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
For a wire vibrating in \(p\) loops, \(f=\dfrac{p}{2l}\sqrt{\dfrac{T}{\mu}}\), so \(f\propto\dfrac{p\sqrt{T}}{l}\).

Step 2: Initial state
\(p=2\), tension \(T\), length \(l\): \(f\propto\dfrac{2\sqrt{T}}{l}\).

Step 3: New state
Tension \(16T\) and length \(2l\): \(f\propto\dfrac{p\times4\sqrt{T}}{2l}=\dfrac{2p\sqrt{T}}{l}\).

Step 4: Result
Resonance with the same fork needs equal frequencies, so \(2p=2\), giving \(p=1\).

Final Answer:
The wire must vibrate in one loop. \[ \boxed{\text{(A) }\text{one loop}} \]