A travelling wave travelled in string in `+x` direction with `2 cm//s`, particle at `x = 0` oscillates according to equation y (in mm) `= 2 sin (pi t+pi//3)`. What will be the slope of the wave at `x = 3cm` and `t = 1s` ?

Options

A. `-sqrt(3)pi//2`
B. `tan^(-1)(-sqrt(3)pi//2)`
C. `-sqrt(3)pi//20`
D. `-sqrt(3)pi`

Explanation

To find the slope of the wave at \( x = 3 \, \text{cm} \) and \( t = 1 \, \text{s} \), we will follow these steps:

### Step 1: Write down the wave equation
The equation of the wave is given as:
\[ y = 2 \sin\left(\pi t + \frac{\pi}{3}\right) \]

### Step 2: Determine the wave speed and modify the wave equation
The wave travels in the positive \( x \) direction with a speed of \( v = 2 \, \text{cm/s} \). The general form of a traveling wave can be expressed as:
\[ y = A \sin\left(\omega t - kx + \phi\right) \]
where \( \omega = 2\pi f \) is the angular frequency and \( k = \frac{2\pi}{\lambda} \) is the wave number.

Given that the speed \( v = \frac{\omega}{k} \), we can express \( k \) in terms of \( \omega \):
\[ k = \frac{\omega}{v} \]
Substituting \( v = 2 \, \text{cm/s} \) into the equation, we can rewrite the wave equation as:
\[ y = 2 \sin\left(\pi t + \frac{\pi}{3} - \frac{\pi}{2} x\right) \]

### Step 3: Differentiate the wave equation to find the slope
To find the slope of the wave, we need to compute \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}\left[2 \sin\left(\pi t + \frac{\pi}{3} - \frac{\pi}{2} x\right)\right] \]
Using the chain rule:
\[ \frac{dy}{dx} = 2 \cos\left(\pi t + \frac{\pi}{3} - \frac{\pi}{2} x\right) \cdot \left(-\frac{\pi}{2}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = -\pi \cos\left(\pi t + \frac{\pi}{3} - \frac{\pi}{2} x\right) \]

### Step 4: Substitute \( x = 3 \, \text{cm} \) and \( t = 1 \, \text{s} \)
Now, substitute \( x = 3 \, \text{cm} \) and \( t = 1 \, \text{s} \):
\[ \frac{dy}{dx} = -\pi \cos\left(\pi(1) + \frac{\pi}{3} - \frac{\pi}{2}(3)\right) \]
Calculating the argument of the cosine:
\[ \frac{dy}{dx} = -\pi \cos\left(\pi + \frac{\pi}{3} - \frac{3\pi}{2}\right) \]
Simplifying the expression:
\[ \frac{dy}{dx} = -\pi \cos\left(-\frac{1}{6}\pi\right) \]

### Step 5: Calculate the cosine value
Using the property of cosine:
\[ \cos(-\theta) = \cos(\theta) \]
Thus:
\[ \frac{dy}{dx} = -\pi \cos\left(\frac{\pi}{6}\right) \]
Since \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \):
\[ \frac{dy}{dx} = -\pi \cdot \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}\pi \]

### Final Answer
The slope of the wave at \( x = 3 \, \text{cm} \) and \( t = 1 \, \text{s} \) is:
\[ \frac{dy}{dx} = -\frac{3\pi}{2} \]