Question

A transverse wave is travelling on a string with velocity \( V \). The extension in the string is \( x \). If the string is extended by 50%, the speed of the wave along the string will be nearly (Hooke's law is obeyed)

• A. \( 0.7V \)
• B. \( 1.22V \)
• C. \( 1.1V \)
• D. \( 0.9V \)

Hint:

When the string is stretched, the wave velocity increases due to the increase in tension. The relationship between wave velocity and tension is \( V \propto \sqrt{T} \).

Answer 1

The Correct Option is B

Step 1: Understanding the relationship between wave velocity and extension.
The velocity of a wave on a stretched string is given by the formula: \[ V = \sqrt{\frac{T}{\mu}} \] where \( T \) is the tension and \( \mu \) is the linear mass density of the string. When the string is stretched, the tension increases, which leads to an increase in the wave velocity. Step 2: Considering the change in velocity.
If the string is extended by 50%, the tension increases and the wave velocity will increase by a factor of \( \sqrt{1.5} \), which is approximately 1.22. Hence, the new velocity will be \( 1.22V \).
Step 3: Conclusion.
Thus, the correct answer is (B) \( 1.22V \).