Standing waves are produced by the superposition of
two waves
`y_(1)=0.05 sin (3 pit-2x) and y_(2) = 0.05 sin (3pit+2x)`
Where x and y are in metres and t is in second. What
is the amplitude of the particle at `x = 0.5` m ? (Given,
`cos 57. 3 ^(@) = 0. 54)`

Options

A. `2.7 ` cm
B. `5.4` cm
C. `8.1` cm
D. `10. 8` cm

Explanation

To find the amplitude of the particle at \( x = 0.5 \) m for the standing waves produced by the superposition of the two waves \( y_1 = 0.05 \sin(3\pi t - 2x) \) and \( y_2 = 0.05 \sin(3\pi t + 2x) \), we can follow these steps:

### Step 1: Write the superposition of the two waves
The resultant wave \( y \) from the superposition of \( y_1 \) and \( y_2 \) can be expressed as:
\[ y = y_1 + y_2 = 0.05 \sin(3\pi t - 2x) + 0.05 \sin(3\pi t + 2x) \]

### Step 2: Use the sine addition formula
Using the sine addition formula, \( \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \), we can rewrite the expression:
\[ y = 0.05 \left( \sin(3\pi t - 2x) + \sin(3\pi t + 2x) \right) = 0.1 \sin(3\pi t) \cos(2x) \]

### Step 3: Substitute \( x = 0.5 \) m into the equation
Now, we substitute \( x = 0.5 \) m into the equation:
\[ y = 0.1 \sin(3\pi t) \cos(2 \times 0.5) \]
This simplifies to:
\[ y = 0.1 \sin(3\pi t) \cos(1) \]

### Step 4: Find the maximum amplitude
The maximum amplitude occurs when \( \sin(3\pi t) \) is at its maximum value of 1. Thus, we have:
\[ y_{\text{max}} = 0.1 \cos(1) \]

### Step 5: Calculate \( \cos(1) \)
Given that \( \cos(57.3^\circ) = 0.54 \), we can use this value to find \( y_{\text{max}} \):
\[ y_{\text{max}} = 0.1 \times 0.54 = 0.054 \text{ m} \]

### Step 6: Convert to centimeters
To convert meters to centimeters:
\[ y_{\text{max}} = 0.054 \text{ m} = 5.4 \text{ cm} \]

### Final Answer
The amplitude of the particle at \( x = 0.5 \) m is \( 5.4 \) cm.

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