Find the angular velocity of a body rotating with an acceleration of `2rev/s^2` as it completes the 5th revolution after the start

Explanation

To find the angular velocity of a body rotating with an angular acceleration of \(2 \, \text{rev/s}^2\) after it completes the 5th revolution, we can follow these steps:

### Step 1: Identify the given values
- Angular acceleration (\(\alpha\)) = \(2 \, \text{rev/s}^2\)
- Total revolutions (\(\theta\)) = \(5 \, \text{rev}\)
- Initial angular velocity (\(\omega_0\)) = \(0 \, \text{rev/s}\) (assuming the body starts from rest)

### Step 2: Convert revolutions to radians
Since \(1 \, \text{rev} = 2\pi \, \text{radians}\), we can convert the total revolutions into radians:
\[ \theta = 5 \, \text{rev} \times 2\pi \, \text{rad/rev} = 10\pi \, \text{rad} \]

### Step 3: Use the angular kinematics equation
We can use the angular kinematics equation:
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]
Substituting the known values:
\[ \omega^2 = 0 + 2 \times (2 \, \text{rev/s}^2) \times (5 \, \text{rev}) \]
\[ \omega^2 = 2 \times 2 \times 5 = 20 \, \text{rev}^2/\text{s}^2 \]

### Step 4: Calculate the final angular velocity
Taking the square root to find \(\omega\):
\[ \omega = \sqrt{20} \, \text{rev/s} = 2\sqrt{5} \, \text{rev/s} \]

### Step 5: Convert to radians per second
To convert \(\omega\) from revolutions per second to radians per second:
\[ \omega = 2\sqrt{5} \, \text{rev/s} \times 2\pi \, \text{rad/rev} = 4\pi\sqrt{5} \, \text{rad/s} \]

### Final Answer
The angular velocity of the body after completing the 5th revolution is:
\[ \omega = 4\pi\sqrt{5} \, \text{rad/s} \]