The standing wave in a medium is expressed as `y=0.2 sin (0.8x) cos (3000 t ) m`. The distance between any two consecutive points of minimum of maximum displacement is

Options

A. `(pi)/(2)m`
B. `(pi)/(4) m`
C. `(pi)/(6) m`
D. None of these

Answer

To find the distance between any two consecutive points of minimum or maximum displacement in the standing wave given by the equation \( y = 0.2 \sin(0.8x) \cos(3000t) \), we can follow these steps:

### Step 1: Identify the wave equation parameters
The given wave equation can be compared with the general form of a standing wave:
\[ y = 2a \sin(kx) \cos(\omega t) \]
From the equation \( y = 0.2 \sin(0.8x) \cos(3000t) \), we can identify:
- \( k = 0.8 \)
- \( \omega = 3000 \)

### Step 2: Calculate the wavelength (\( \lambda \))
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ 0.8 = \frac{2\pi}{\lambda} \]
Rearranging this gives:
\[ \lambda = \frac{2\pi}{0.8} = \frac{2\pi}{\frac{8}{10}} = \frac{2\pi \cdot 10}{8} = \frac{5\pi}{2} \]

### Step 3: Determine the distance between consecutive points of maximum or minimum displacement
The distance between any two consecutive points of maximum or minimum displacement in a standing wave is given by:
\[ \frac{\lambda}{4} \]
Substituting the value of \( \lambda \):
\[ \frac{\lambda}{4} = \frac{5\pi/2}{4} = \frac{5\pi}{8} \]

### Conclusion
Thus, the distance between any two consecutive points of minimum or maximum displacement is:
\[ \frac{5\pi}{8} \text{ meters} \]