The acceleration and velocity maximum of simple harmonically oscillating system are `8m//sec^(2) and `8m//sec`respectively. What is the angular frequency? (in rad//sec)

Explanation

To find the angular frequency of a simple harmonic oscillator given the maximum acceleration and maximum velocity, we can follow these steps:

### Step 1: Write down the formulas for maximum acceleration and maximum velocity.

The maximum acceleration (A_max) and maximum velocity (V_max) in a simple harmonic motion are given by the following equations:

1. Maximum Acceleration:
\[ A_{max} = \omega^2 A \]
(where \( \omega \) is the angular frequency and \( A \) is the amplitude)

2. Maximum Velocity:
\[ V_{max} = \omega A \]

### Step 2: Substitute the given values.

From the problem, we know:
- Maximum Acceleration, \( A_{max} = 8 \, \text{m/s}^2 \)
- Maximum Velocity, \( V_{max} = 8 \, \text{m/s} \)

### Step 3: Set up the equations.

Using the formulas from Step 1, we can write:

1. \( 8 = \omega^2 A \) (Equation 1)
2. \( 8 = \omega A \) (Equation 2)

### Step 4: Solve for amplitude (A) from Equation 2.

From Equation 2, we can express \( A \) in terms of \( \omega \):

\[ A = \frac{8}{\omega} \]

### Step 5: Substitute \( A \) into Equation 1.

Now, substitute \( A \) from Step 4 into Equation 1:

\[ 8 = \omega^2 \left(\frac{8}{\omega}\right) \]

### Step 6: Simplify the equation.

This simplifies to:

\[ 8 = 8\omega \]

### Step 7: Solve for angular frequency (\( \omega \)).

Now, divide both sides by 8:

\[ 1 = \omega \]

Thus, we find:

\[ \omega = 1 \, \text{rad/s} \]

### Final Answer

The angular frequency is \( \omega = 1 \, \text{rad/s} \).

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