The equation of a wave is `y=4sin{(pi)/(2)(2t+(x)/(8))}` where `y,x` are in cm and time in seconds . The acceleration of particle located at `x=8cm` and `t=1sec` is

Options

A. `4pi^(2)cm//s^(2)`
B. `-4pi^(2)cm//s^(2)`
C. `16pi^(2)cm//s^(2)`
D. `-16pi^(2)cm//s^(2)`

Explanation

To find the acceleration of a particle located at \( x = 8 \, \text{cm} \) and \( t = 1 \, \text{s} \) for the wave described by the equation:

\[ y = 4 \sin\left(\frac{\pi}{2}\left(2t + \frac{x}{8}\right)\right) \]

we will follow these steps:

### Step 1: Identify the wave parameters
The given wave equation can be compared with the standard wave equation:

\[ y = A \sin(\omega t + kx) \]

From the equation, we can identify:
- Amplitude \( A = 4 \)
- Angular frequency \( \omega = \frac{\pi}{2} \cdot 2 = \pi \)
- Wave number \( k = \frac{\pi}{2 \cdot 8} = \frac{\pi}{16} \)

### Step 2: Calculate the acceleration
The acceleration \( a \) of a particle in a wave can be calculated using the formula:

\[ a = -\omega^2 y \]

### Step 3: Find \( y \) at the given \( x \) and \( t \)
Substituting \( x = 8 \, \text{cm} \) and \( t = 1 \, \text{s} \) into the wave equation:

\[ y = 4 \sin\left(\frac{\pi}{2}\left(2(1) + \frac{8}{8}\right)\right) \]

Calculating the argument of the sine function:

\[ = 4 \sin\left(\frac{\pi}{2}(2 + 1)\right) = 4 \sin\left(\frac{\pi}{2} \cdot 3\right) = 4 \sin\left(\frac{3\pi}{2}\right) \]

Since \( \sin\left(\frac{3\pi}{2}\right) = -1 \):

\[ y = 4 \cdot (-1) = -4 \, \text{cm} \]

### Step 4: Substitute \( y \) into the acceleration formula
Now substituting \( y \) into the acceleration formula:

\[ a = -\omega^2 y = -\pi^2 (-4) = 4\pi^2 \, \text{cm/s}^2 \]

### Final Answer
The acceleration of the particle located at \( x = 8 \, \text{cm} \) and \( t = 1 \, \text{s} \) is:

\[ a = 4\pi^2 \, \text{cm/s}^2 \]

---