The equation of a travelling wave on a string is y = (0.10 mm )` sin [31.4 m^(-1) x + (314s^(-1)) t]`

Options

A. wave is travelling along negative x- axis.
B. The speed of the wave is 100 cm/s
C. The frequency wave is `50 s^(-1)`
D. The maximum speed of a portion of string is = 3.14 cm/s

Answer

To solve the problem, we need to analyze the given wave equation and extract relevant parameters such as wave speed, frequency, and maximum speed of a point on the string. The wave equation given is:

\[ y = (0.10 \, \text{mm}) \sin [31.4 \, \text{m}^{-1} \, x + (314 \, \text{s}^{-1}) \, t] \]

### Step 1: Identify the wave parameters
From the wave equation, we can identify:
- Amplitude \( A = 0.10 \, \text{mm} = 0.10 \times 10^{-3} \, \text{m} = 10^{-4} \, \text{m} \)
- Wave number \( k = 31.4 \, \text{m}^{-1} \)
- Angular frequency \( \omega = 314 \, \text{s}^{-1} \)

### Step 2: Determine the wave speed
The wave speed \( v \) can be calculated using the formula:

\[ v = \frac{\omega}{k} \]

Substituting the values we have:

\[ v = \frac{314 \, \text{s}^{-1}}{31.4 \, \text{m}^{-1}} = 10 \, \text{m/s} \]

To convert this to centimeters per second:

\[ v = 10 \, \text{m/s} \times 100 \, \text{cm/m} = 1000 \, \text{cm/s} \]

### Step 3: Calculate the frequency of the wave
The frequency \( f \) can be found using the relationship between angular frequency and frequency:

\[ \omega = 2\pi f \]

Rearranging gives:

\[ f = \frac{\omega}{2\pi} \]

Substituting the value of \( \omega \):

\[ f = \frac{314 \, \text{s}^{-1}}{2\pi} \approx 50 \, \text{Hz} \]

### Step 4: Find the maximum speed of a point on the string
The maximum speed \( v_{\text{max}} \) of a point on the string is given by:

\[ v_{\text{max}} = \omega A \]

Substituting the values:

\[ v_{\text{max}} = 314 \, \text{s}^{-1} \times (0.10 \times 10^{-3} \, \text{m}) = 314 \times 10^{-4} \, \text{m/s} = 3.14 \, \text{cm/s} \]

### Summary of Results
1. Wave speed \( v = 1000 \, \text{cm/s} \)
2. Frequency \( f = 50 \, \text{Hz} \)
3. Maximum speed \( v_{\text{max}} = 3.14 \, \text{cm/s} \)

### Conclusion
- The wave is traveling in the negative x-direction.
- The speed of the wave is 1000 cm/s (not 100 cm/s).
- The frequency of the wave is 50 Hz.
- The maximum speed of a point on the string is 3.14 cm/s.