Question

‘n’ number of waves are produced on a string in 0.5 second. Now the tension in a string is doubled (keeping radius constant). The number of waves produced in 0.5 second for the same harmonic will be

• A. \( \frac{n}{\sqrt{2}} \)
• B. \( n \)
• C. \( \frac{\sqrt{2}}{n} \)
• D. \( \sqrt{2}n \)

Hint:

When the tension in a string is increased, the wave speed increases by the square root of the ratio of the new tension to the old tension, which also increases the frequency.

Answer 1

The Correct Option is D

Step 1: Understanding wave speed in a string.
The wave speed on a string is given by the equation: \[ v = \sqrt{\frac{T}{\mu}} \] where \( T \) is the tension in the string, and \( \mu \) is the mass per unit length (linear density). Step 2: Effect of doubling the tension.
When the tension in the string is doubled, the wave speed increases by a factor of \( \sqrt{2} \). Since the frequency \( f \) of the wave is related to the wave speed and wavelength by \( v = f \lambda \), the frequency will also increase by \( \sqrt{2} \). Step 3: Conclusion.
Thus, the number of waves produced in 0.5 second will increase by \( \sqrt{2} \), so the new number of waves is \( \sqrt{2}n \), corresponding to option (D).