The vibrations of a string fixed at both ends are represented by `y=16sin((pix)/(15))cos(96pit)`. Where 'x' and 'y' are in cm and 't' in seconds. Then the phase difference between the points at `x=13cm` and `x=16` in radian is

Options

A. `pi//5`
B. `pi`
C. `0`
D. `2pi//5`

Explanation

To find the phase difference between the points at \( x = 13 \, \text{cm} \) and \( x = 16 \, \text{cm} \) for the given wave equation \( y = 16 \sin\left(\frac{\pi x}{15}\right) \cos(96 \pi t) \), we can follow these steps:

### Step 1: Identify the wave equation
The given wave equation is:
\[ y = 16 \sin\left(\frac{\pi x}{15}\right) \cos(96 \pi t) \]
Here, the term \( \sin\left(\frac{\pi x}{15}\right) \) represents the spatial part of the wave.

### Step 2: Determine the wave number \( k \)
The wave number \( k \) can be extracted from the sine term. The general form of the wave function is:
\[ y = A \sin(kx) \cos(\omega t) \]
Comparing with our equation, we have:
\[ k = \frac{\pi}{15} \, \text{(in radians per cm)} \]

### Step 3: Calculate the path difference \( \Delta x \)
The path difference \( \Delta x \) between the two points \( x = 16 \, \text{cm} \) and \( x = 13 \, \text{cm} \) is:
\[ \Delta x = 16 \, \text{cm} - 13 \, \text{cm} = 3 \, \text{cm} \]

### Step 4: Calculate the phase difference \( \Delta \phi \)
The phase difference \( \Delta \phi \) is given by the formula:
\[ \Delta \phi = k \Delta x \]
Substituting the values we have:
\[ \Delta \phi = \left(\frac{\pi}{15}\right) \times 3 \, \text{cm} \]

### Step 5: Simplify the expression
Now, simplifying the expression:
\[ \Delta \phi = \frac{3\pi}{15} = \frac{\pi}{5} \, \text{radians} \]

### Final Result
Thus, the phase difference between the points at \( x = 13 \, \text{cm} \) and \( x = 16 \, \text{cm} \) is:
\[ \Delta \phi = \frac{\pi}{5} \, \text{radians} \]

---