Question

A string of length 25 cm and mass $ 10^{-3} $ kg is clamped at its ends. The tension in the string is 2.5 N. The identical wave pulses are generated at one end and at regular interval of time, $ \Delta t $. The minimum value of $ \Delta t $, so that a constructive interference takes place between successive pulses is:

• A. 0.2 s
• B. 1 s
• C. 40 ms
• D. 20 ms

Hint:

For constructive interference in waves, the time interval between successive pulses is related to the wavelength and the wave speed. The condition for the minimum \( \Delta t \) is based on the period of the wave.

Answer 1

The Correct Option is D

To calculate the minimum value of \( \Delta t \) for constructive interference between successive pulses, we use the concept of the wave speed and the condition for constructive interference. The wave speed \( v \) on a string is given by: \[ v = \sqrt{\frac{T}{\mu}} \] Where: 
- \( T \) is the tension in the string (2.5 N), 
- \( \mu \) is the linear mass density of the string, which is given by \( \mu = \frac{m}{L} \), where \( m \) is the mass of the string and \( L \) is its length. 
Substituting the values: - \( m = 10^{-3} \, \text{kg} \), - \( L = 25 \, \text{cm} = 0.25 \, \text{m} \). The linear mass density \( \mu \) is: \[ \mu = \frac{10^{-3}}{0.25} = 4 \times 10^{-3} \, \text{kg/m} \] Now, the wave speed \( v \) is: \[ v = \sqrt{\frac{2.5}{4 \times 10^{-3}}} = \sqrt{625} = 25 \, \text{m/s} \] The time interval \( \Delta t \) for constructive interference is given by the time it takes for the wave pulse to travel a full wavelength. For constructive interference, this time corresponds to the period of the wave: \[ \Delta t = \frac{\lambda}{v} \] Where \( \lambda \) is the wavelength of the wave. For constructive interference, the wavelength corresponds to twice the distance traveled by a wave pulse, which implies the pulse travels half the wavelength in the time \( \Delta t \). Given that the minimum value for constructive interference corresponds to a time of \( \Delta t = \frac{1}{50} \) seconds, we find: \[ \Delta t = 20 \, \text{ms} \] Thus, the minimum value of \( \Delta t \) is 20 ms.