A travelling wave on a string is given by `y = A sin [alphax + betat + (pi)/(6)]`. If `alpha = 0.56 //cm, beta = 12//sec, A = 7.5 cm`, then find the position and velocity of particle at `x = 1 cm` and `t = 1s` is

Options

A. `4.6 cm, 46.5 cm s^(-1)`
B. `3.75 cm, 77.94 cm s^(-1)`
C. `1.76 cm, 7.5 cm s^(-1)`
D. `7.5 cm, 75 cm s^(-1)`

Explanation

To solve the problem step by step, we need to find the position and velocity of a particle on a string described by the wave function:

\[ y = A \sin(\alpha x + \beta t + \frac{\pi}{6}) \]

Given values:
- \( \alpha = 0.56 \, \text{cm}^{-1} \)
- \( \beta = 12 \, \text{s}^{-1} \)
- \( A = 7.5 \, \text{cm} \)
- \( x = 1 \, \text{cm} \)
- \( t = 1 \, \text{s} \)

### Step 1: Calculate the Position of the Particle

1. Substitute the values into the wave function:
\[ y = 7.5 \sin(0.56 \cdot 1 + 12 \cdot 1 + \frac{\pi}{6}) \]

2. Calculate the argument of the sine function:
\[ \alpha x + \beta t + \frac{\pi}{6} = 0.56 \cdot 1 + 12 \cdot 1 + \frac{\pi}{6} \]
\[ = 0.56 + 12 + \frac{\pi}{6} \]
\[ = 12.56 + \frac{\pi}{6} \approx 12.56 + 0.5236 \approx 13.0836 \]

3. Now calculate \( y \):
\[ y = 7.5 \sin(13.0836) \]

4. Use a calculator to find \( \sin(13.0836) \):
\[ \sin(13.0836) \approx 0.375 \]

5. Finally, calculate \( y \):
\[ y = 7.5 \cdot 0.375 \approx 2.8125 \, \text{cm} \]

### Step 2: Calculate the Velocity of the Particle

1. The velocity of the particle is given by the derivative of the displacement with respect to time:
\[ v = \frac{\partial y}{\partial t} = A \beta \cos(\alpha x + \beta t + \frac{\pi}{6}) \]

2. Substitute the values into the velocity equation:
\[ v = 7.5 \cdot 12 \cdot \cos(0.56 \cdot 1 + 12 \cdot 1 + \frac{\pi}{6}) \]

3. We already calculated the argument of the cosine function:
\[ \alpha x + \beta t + \frac{\pi}{6} \approx 13.0836 \]

4. Now calculate \( \cos(13.0836) \):
\[ \cos(13.0836) \approx 0.869 \]

5. Finally, calculate \( v \):
\[ v = 7.5 \cdot 12 \cdot 0.869 \approx 7.5 \cdot 10.428 \approx 78.21 \, \text{cm/s} \approx 7.82 \, \text{cm/s} \]

### Final Answers:
- Position of the particle at \( x = 1 \, \text{cm} \) and \( t = 1 \, \text{s} \): \( y \approx 2.81 \, \text{cm} \)
- Velocity of the particle at \( x = 1 \, \text{cm} \) and \( t = 1 \, \text{s} \): \( v \approx 7.82 \, \text{cm/s} \)