The equational of a stationary wave in a string is `y=(4mm) sin [(314m^(-1)x] cos omegat`. Select the correct alternative (s).

Options

A. The amplitude of component waves in 2mm
B. The amplitude of componen wave is 4mm
C. The smallest possible length of string is 0.5m
D. The smallest possible length of string is 1.0cm

Answer

To solve the problem, we need to analyze the given equation of the stationary wave and extract the necessary information to answer the question.

### Step-by-Step Solution:

1. **Identify the Given Equation**:
The equation of the stationary wave is given as:
\[ y = (4 \, \text{mm}) \sin(314 \, \text{m}^{-1} x) \cos(\omega t) \]

2. **Compare with the General Form**:
The general form of a stationary wave is:
\[ y = 2a \sin(kx) \cos(\omega t) \]
where \( a \) is the amplitude and \( k \) is the wave number.

3. **Determine the Amplitude**:
From the given equation, we can see that:
\[ 2a = 4 \, \text{mm} \]
Therefore, the amplitude \( a \) is:
\[ a = \frac{4 \, \text{mm}}{2} = 2 \, \text{mm} \]

4. **Determine the Wave Number**:
The wave number \( k \) is given as:
\[ k = 314 \, \text{m}^{-1} \]

5. **Calculate the Wavelength**:
The wavelength \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{2\pi}{k} \]
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{314} \]

6. **Calculate the Minimum Length of the String**:
The minimum length \( L_{\text{min}} \) of the string for it to vibrate in the fundamental mode is given by:
\[ L_{\text{min}} = \frac{\lambda}{2} \]
Substituting the value of \( \lambda \):
\[ L_{\text{min}} = \frac{1}{2} \cdot \frac{2\pi}{314} = \frac{\pi}{314} \]
To convert this to centimeters, we can use \( \pi \approx 3.14 \):
\[ L_{\text{min}} \approx \frac{3.14}{314} \approx 0.01 \, \text{m} = 1 \, \text{cm} \]

7. **Select the Correct Alternatives**:
From our calculations:
- The amplitude is \( 2 \, \text{mm} \).
- The minimum length of the string is \( 1 \, \text{cm} \).
Therefore, the correct alternatives are those that correspond to these values.

### Final Answer:
- The amplitude of the wave is \( 2 \, \text{mm} \) (Option A).
- The smallest possible length of the string is \( 1 \, \text{cm} \) (Option D).