The displacemtent x (in cm ) of an oscillating body varies with time t as
`x=3cos(pit+(pi)/(4))`
The magnittude of maximum acceleration of the particle in `cm//s^(2)` is

Options

A. `pi^(2)//3`
B. `3pi^(2)`
C. `4pi^(2)//3`
D. `pi^(2)/2`

Explanation

To find the magnitude of the maximum acceleration of the oscillating body given the displacement equation \( x = 3 \cos(\pi t + \frac{\pi}{4}) \), we will follow these steps:

### Step 1: Identify the displacement function
The displacement function is given as:
\[ x(t) = 3 \cos\left(\pi t + \frac{\pi}{4}\right) \]

### Step 2: Differentiate to find velocity
To find the velocity \( v(t) \), we differentiate the displacement function with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = -3\pi \sin\left(\pi t + \frac{\pi}{4}\right) \]

### Step 3: Differentiate to find acceleration
Next, we differentiate the velocity function to find the acceleration \( a(t) \):
\[ a(t) = \frac{dv}{dt} = -3\pi^2 \cos\left(\pi t + \frac{\pi}{4}\right) \]

### Step 4: Determine the maximum acceleration
The maximum value of \( a(t) \) occurs when the cosine function equals 1. Therefore, the maximum acceleration is:
\[ a_{\text{max}} = |-3\pi^2| = 3\pi^2 \]

### Step 5: Final result
Thus, the magnitude of the maximum acceleration of the particle is:
\[ \boxed{3\pi^2} \text{ cm/s}^2 \]

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