The displacement of an oscillating particle varies with time (in seconds ) according to the equation `y=(cm)= sin ((pi)/(2))(t/2 + 1/3)` The maximum acceleration of the particle is approximately

Options

A. `5.21 cm //s^(2)`
B. `3.62 cm //s^(2)`
C. `1.81 cm //s^(2)`
D. `0.62 cm //s^(2)`

Answer

To find the maximum acceleration of the oscillating particle given the displacement equation \( y = \sin\left(\frac{\pi}{2}\left(\frac{t}{2} + \frac{1}{3}\right)\right) \), we can follow these steps:

### Step 1: Identify the Displacement Equation
The displacement of the particle is given by:
\[ y = \sin\left(\frac{\pi}{2}\left(\frac{t}{2} + \frac{1}{3}\right)\right) \]

### Step 2: Simplify the Displacement Equation
We can rewrite the argument of the sine function:
\[ y = \sin\left(\frac{\pi}{4}t + \frac{\pi}{6}\right) \]
This shows that the angular frequency \( \omega \) can be identified from the term multiplying \( t \).

### Step 3: Determine the Angular Frequency \( \omega \)
From the equation \( y = \sin\left(\frac{\pi}{4}t + \frac{\pi}{6}\right) \), we can see that:
\[ \omega = \frac{\pi}{4} \, \text{radians/second} \]

### Step 4: Identify the Amplitude \( A \)
The amplitude \( A \) of the oscillation is the coefficient in front of the sine function. Here, the amplitude is:
\[ A = 1 \, \text{cm} \]

### Step 5: Calculate the Maximum Acceleration \( A_{max} \)
The maximum acceleration \( A_{max} \) of a simple harmonic oscillator is given by the formula:
\[ A_{max} = \omega^2 \cdot A \]
Substituting the values we found:
\[ A_{max} = \left(\frac{\pi}{4}\right)^2 \cdot 1 \]
Calculating this gives:
\[ A_{max} = \frac{\pi^2}{16} \, \text{cm/s}^2 \]

### Step 6: Approximate the Value
Now we can calculate \( \frac{\pi^2}{16} \):
\[ \pi \approx 3.14 \implies \pi^2 \approx 9.86 \]
Thus,
\[ A_{max} \approx \frac{9.86}{16} \approx 0.61625 \, \text{cm/s}^2 \]
Rounding this gives approximately:
\[ A_{max} \approx 0.62 \, \text{cm/s}^2 \]

### Final Answer
The maximum acceleration of the particle is approximately:
\[ \boxed{0.62 \, \text{cm/s}^2} \]