A sinusoidal progressive wave is generated in a string. It's equation is given by `y = (2mm) sin (2 pi x - 100 pi t + pi//3)` The time when particle at `x = 4 m` first passes through mean position, will be :-

Options

A. `(1)/(150)`s
B. `(1)/(12)` s
C. `(1)/(300)` s
D. `(1)/(100)` s

Explanation

To solve the problem, we need to determine the time when the particle at \( x = 4 \, \text{m} \) first passes through the mean position. The equation of the wave is given as:

\[ y = (2 \, \text{mm}) \sin(2 \pi x - 100 \pi t + \frac{\pi}{3}) \]

### Step 1: Substitute \( x = 4 \, \text{m} \) into the wave equation

We start by substituting \( x = 4 \, \text{m} \) into the wave equation:

\[ y = (2 \, \text{mm}) \sin(2 \pi (4) - 100 \pi t + \frac{\pi}{3}) \]

Calculating \( 2 \pi (4) \):

\[ 2 \pi (4) = 8 \pi \]

So the equation becomes:

\[ y = (2 \, \text{mm}) \sin(8 \pi - 100 \pi t + \frac{\pi}{3}) \]

### Step 2: Simplify the sine argument

Using the property of sine, \( \sin(\theta + 2\pi n) = \sin(\theta) \) for any integer \( n \), we can simplify:

\[ y = (2 \, \text{mm}) \sin(-100 \pi t + \frac{\pi}{3}) \]

### Step 3: Set the sine argument to zero for mean position

For the particle to be at the mean position, the sine function must equal zero:

\[ -100 \pi t + \frac{\pi}{3} = 0 \]

### Step 4: Solve for \( t \)

Rearranging the equation gives:

\[ -100 \pi t = -\frac{\pi}{3} \]

Dividing both sides by \(-\pi\):

\[ 100 t = \frac{1}{3} \]

Now, dividing both sides by 100:

\[ t = \frac{1}{300} \, \text{s} \]

### Conclusion

The time when the particle at \( x = 4 \, \text{m} \) first passes through the mean position is:

\[ t = \frac{1}{300} \, \text{s} \]