The speed of propagation of a wave in a medium is `300m//s`. The equation of motion of point at `x = 0` is given by `y = 0.04 sin 600 pit(metre)`. The displacement of a point `x=75cm` at `t=0.01 s` is

Options

A. `0.02m`
B. `0.04m`
C. `zero`
D. 0.028 m

Explanation

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

### Step 1: Understand the given information
We have the following information:
- Speed of wave propagation, \( v = 300 \, \text{m/s} \)
- Equation of motion at \( x = 0 \): \( y = 0.04 \sin(600 \pi t) \)
- We need to find the displacement at \( x = 75 \, \text{cm} \) (which is \( 0.75 \, \text{m} \)) at \( t = 0.01 \, \text{s} \).

### Step 2: Identify the angular frequency (\( \omega \))
From the equation \( y = 0.04 \sin(600 \pi t) \), we can identify:
- \( \omega = 600 \pi \, \text{rad/s} \)

### Step 3: Calculate the wave number (\( k \))
Using the relationship between wave speed (\( v \)), angular frequency (\( \omega \)), and wave number (\( k \)):
\[ v = \frac{\omega}{k} \implies k = \frac{\omega}{v} \]
Substituting the values:
\[ k = \frac{600 \pi}{300} = 2 \pi \, \text{rad/m} \]

### Step 4: Write the general wave equation
Since the wave is traveling in the positive x-direction, the equation can be written as:
\[ y(x, t) = 0.04 \sin(600 \pi t - kx) \]
Substituting \( k \):
\[ y(x, t) = 0.04 \sin(600 \pi t - 2 \pi x) \]

### Step 5: Substitute the values of \( x \) and \( t \)
We need to find \( y \) at \( x = 0.75 \, \text{m} \) and \( t = 0.01 \, \text{s} \):
\[ y(0.75, 0.01) = 0.04 \sin(600 \pi (0.01) - 2 \pi (0.75)) \]

### Step 6: Calculate the arguments of the sine function
Calculating \( 600 \pi (0.01) \):
\[ 600 \pi (0.01) = 6 \pi \]
Calculating \( 2 \pi (0.75) \):
\[ 2 \pi (0.75) = 1.5 \pi \]
Now substituting these values:
\[ y(0.75, 0.01) = 0.04 \sin(6 \pi - 1.5 \pi) = 0.04 \sin(4.5 \pi) \]

### Step 7: Simplify the sine function
Since \( 4.5 \pi \) can be simplified:
\[ 4.5 \pi = 4 \pi + 0.5 \pi \]
Thus, \( \sin(4.5 \pi) = \sin(0.5 \pi) = 1 \).

### Step 8: Calculate the final displacement
Substituting back into the equation:
\[ y(0.75, 0.01) = 0.04 \times 1 = 0.04 \, \text{m} \]

### Final Answer
The displacement of the point at \( x = 75 \, \text{cm} \) at \( t = 0.01 \, \text{s} \) is \( 0.04 \, \text{m} \).

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