A sinusoidal wave travels along a taut string of linear mass density `0.1g//cm`. The particles oscillate along `y-` direction and wave moves in the positive `x-` direction . The amplitude and frequency of oscillation are `2mm` and `50Hz` respectively. The minimum distance between two particles oscillating in the same phase is` 4m`.
The amount of energy transferred `(` in Joules `)` through any point of the string in 5 seconds is

Options

A. `(pi^(2))/(10)`
B. `(pi^(2))/(50)`
C. `(pi^(2))/(5)`
D. Cannot be calculated because area of cross`-` section of string is not give.

Explanation

To solve the problem, we need to calculate the amount of energy transferred through a point on the string in 5 seconds. We will follow these steps:

### Step 1: Identify the given parameters
- Linear mass density, \( \mu = 0.1 \, \text{g/cm} = 0.1 \times 10^{-3} \, \text{kg/m} = 0.0001 \, \text{kg/m} \)
- Amplitude, \( A = 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)
- Frequency, \( f = 50 \, \text{Hz} \)
- Minimum distance between two particles oscillating in the same phase (wavelength), \( \lambda = 4 \, \text{m} \)

### Step 2: Calculate the wave velocity
Using the relationship between frequency and wavelength:
\[ V = f \cdot \lambda \]
Substituting the values:
\[ V = 50 \, \text{Hz} \times 4 \, \text{m} = 200 \, \text{m/s} \]

### Step 3: Calculate the average power transmitted by the wave
The formula for average power transmitted by a wave on a string is:
\[ P_{\text{avg}} = 2 \pi^2 f^2 A^2 \mu V \]
Substituting the known values:
\[ P_{\text{avg}} = 2 \pi^2 (50)^2 (2 \times 10^{-3})^2 (0.0001) (200) \]

Calculating each part:
1. \( (50)^2 = 2500 \)
2. \( (2 \times 10^{-3})^2 = 4 \times 10^{-6} \)
3. \( 2 \pi^2 \approx 19.7392 \)

Putting it all together:
\[ P_{\text{avg}} = 19.7392 \times 2500 \times 4 \times 10^{-6} \times 0.0001 \times 200 \]
Calculating:
\[ P_{\text{avg}} = 19.7392 \times 2500 \times 4 \times 0.00002 = 19.7392 \times 0.2 = 3.94784 \, \text{W} \]

### Step 4: Calculate the energy transferred in 5 seconds
Using the relationship between power and energy:
\[ E = P_{\text{avg}} \times t \]
Substituting the values:
\[ E = 3.94784 \, \text{W} \times 5 \, \text{s} = 19.7392 \, \text{J} \]

### Final Answer
The amount of energy transferred through any point of the string in 5 seconds is approximately:
\[ E \approx 19.74 \, \text{J} \]