The displacement y (in cm ) produced by a simple harmonic wave is
`y=(10)/(pi) sin (2000 pi t -(pi x)/(17))` . The periodic time and maximum velocity of the particles in the medium will respectively be

Options

A. `10^(-3) ` sec and 330 m/sec
B. `10^(-4) ` sec and 20 m/sec
C. `10^(-3) ` sec and 200 m/sec
D. `10^(-2) ` sec and 2000 m/sec

Answer

To solve the problem, we need to extract the periodic time and maximum velocity from the given wave equation. The wave equation is given as:

\[ y = \frac{10}{\pi} \sin(2000 \pi t - \frac{\pi x}{17}) \]

### Step 1: Identify the Angular Frequency
The angular frequency \( \omega \) can be identified from the wave equation. In this case, we have:

\[ \omega = 2000 \pi \, \text{rad/s} \]

### Step 2: Calculate the Periodic Time
The periodic time \( T \) is related to the angular frequency \( \omega \) by the formula:

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

Substituting the value of \( \omega \):

\[ T = \frac{2\pi}{2000 \pi} = \frac{2}{2000} = \frac{1}{1000} \, \text{s} \]

### Step 3: Identify the Amplitude
The amplitude \( A \) of the wave is given by the coefficient of the sine function in the wave equation:

\[ A = \frac{10}{\pi} \, \text{cm} \]

### Step 4: Calculate the Maximum Velocity
The maximum velocity \( v_{\text{max}} \) of the particles in the medium can be calculated using the formula:

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

Substituting the values of \( A \) and \( \omega \):

\[ v_{\text{max}} = \left(\frac{10}{\pi}\right) \cdot (2000 \pi) = 10 \cdot 2000 = 20000 \, \text{cm/s} \]

### Final Answer
The periodic time \( T \) and the maximum velocity \( v_{\text{max}} \) of the particles in the medium are:

- Periodic Time \( T = \frac{1}{1000} \, \text{s} \)
- Maximum Velocity \( v_{\text{max}} = 20000 \, \text{cm/s} \)

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