A string of length `0.4 m` and mass `10^(-2)kg` is tightly clamped at its ends. The tension in the string is `1.6 N`. Identical wave pulse are produced at one end at equal intervals of time, `Deltat`. The minimum value of `Deltat` which allows constructive interference of successive pulse is

Options

A. 0.05 s
B. 0.10 s
C. 0.20 s
D. 0.40 s

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Calculate the mass per unit length (μ) of the string
The mass per unit length (μ) is calculated using the formula:
\[ \mu = \frac{m}{L} \]
where \( m = 10^{-2} \, \text{kg} \) (mass of the string) and \( L = 0.4 \, \text{m} \) (length of the string).

\[ \mu = \frac{10^{-2} \, \text{kg}}{0.4 \, \text{m}} = 0.025 \, \text{kg/m} \]

### Step 2: Calculate the wave speed (v) on the string
The wave speed (v) on a string under tension can be calculated using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T = 1.6 \, \text{N} \) (tension in the string).

Substituting the values:
\[ v = \sqrt{\frac{1.6 \, \text{N}}{0.025 \, \text{kg/m}}} = \sqrt{64} = 8 \, \text{m/s} \]

### Step 3: Determine the wavelength (λ) for constructive interference
For constructive interference of successive pulses, the path difference must be a multiple of the wavelength (λ). The minimum path difference for constructive interference occurs when it is equal to the wavelength.

In the fundamental mode, the wavelength (λ) can be related to the length of the string (L) as:
\[ \lambda = \frac{2L}{n} \]
where \( n \) is the harmonic number. For the fundamental frequency, \( n = 1 \):
\[ \lambda = 2L = 2 \times 0.4 \, \text{m} = 0.8 \, \text{m} \]

### Step 4: Relate the time interval (Δt) to the wave speed and wavelength
The time interval (Δt) between successive pulses can be related to the wave speed and wavelength by:
\[ \Delta t = \frac{\lambda}{v} \]

Substituting the values of λ and v:
\[ \Delta t = \frac{0.8 \, \text{m}}{8 \, \text{m/s}} = 0.1 \, \text{s} \]

### Final Answer
The minimum value of \( \Delta t \) which allows constructive interference of successive pulses is:
\[ \Delta t = 0.1 \, \text{s} \]

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