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

Explanation

To solve the problem step by step, we will follow the wave motion equations and the given parameters.

### Step 1: Understand the wave equation
The general equation for a wave moving in the positive x-direction is given by:
\[ y(x, t) = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( x \) is the position,
- \( t \) is the time.

### Step 2: Identify the parameters from the given equation
From the problem, we have the equation at \( x = 0 \):
\[ y(0, t) = 0.04 \sin(600 \pi t) \]
This tells us:
- Amplitude \( A = 0.04 \, \text{m} \)
- Angular frequency \( \omega = 600 \pi \, \text{rad/s} \)

### Step 3: Calculate the wave number \( k \)
The speed of the wave \( v \) is given as \( 300 \, \text{m/s} \). The relationship between speed, angular frequency, and wave number is:
\[ v = \frac{\omega}{k} \]
Rearranging gives:
\[ k = \frac{\omega}{v} \]
Substituting the values:
\[ k = \frac{600 \pi}{300} = 2 \pi \, \text{rad/m} \]

### Step 4: Write the complete wave equation
Now we can write the complete wave equation using the values of \( A \), \( \omega \), and \( k \):
\[ y(x, t) = 0.04 \sin(600 \pi t - 2 \pi x) \]

### Step 5: Substitute the values for \( x \) and \( t \)
We need to find the displacement at \( x = 75 \, \text{cm} = 0.75 \, \text{m} \) and \( t = 0.01 \, \text{s} \):
\[ y(0.75, 0.01) = 0.04 \sin(600 \pi (0.01) - 2 \pi (0.75)) \]

### Step 6: Simplify the equation
Calculating the argument of the sine function:
1. Calculate \( 600 \pi (0.01) = 6 \pi \)
2. Calculate \( 2 \pi (0.75) = 1.5 \pi \)

Now substitute these values:
\[ y(0.75, 0.01) = 0.04 \sin(6 \pi - 1.5 \pi) \]
\[ = 0.04 \sin(4.5 \pi) \]

### Step 7: Evaluate the sine function
Using the property of sine:
\[ \sin(4.5 \pi) = \sin(4\pi + 0.5\pi) = \sin(0.5\pi) = 1 \]

### Step 8: Final displacement calculation
Thus, we have:
\[ y(0.75, 0.01) = 0.04 \cdot 1 = 0.04 \, \text{m} \]

### Conclusion
The displacement of the point at \( x = 75 \, \text{cm} \) at \( t = 0.01 \, \text{s} \) is:
\[ y = 0.04 \, \text{m} \]

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