A wheel has angular acceleration of `3.0 rad//s^2` and an initial angular speed of `2.00 rad//s`. In a time of `2 s` it has rotated through an angle (in radian) of

Options

A. `6`
B. `10`
C. `12`
D. `4`

Answer

To solve the problem of finding the angle through which the wheel has rotated, we can use the formula for angular displacement in rotational motion. Here’s a step-by-step solution:

### Step-by-Step Solution

1. **Identify the given values**:
- Angular acceleration (\(\alpha\)) = \(3.0 \, \text{rad/s}^2\)
- Initial angular speed (\(\omega_0\)) = \(2.0 \, \text{rad/s}\)
- Time (\(t\)) = \(2.0 \, \text{s}\)

2. **Use the formula for angular displacement**:
The formula for angular displacement (\(\theta\)) when there is an initial angular speed and angular acceleration is given by:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

3. **Substitute the known values into the formula**:
- Substitute \(\omega_0 = 2.0 \, \text{rad/s}\), \(\alpha = 3.0 \, \text{rad/s}^2\), and \(t = 2.0 \, \text{s}\):
\[ \theta = (2.0 \, \text{rad/s}) \cdot (2.0 \, \text{s}) + \frac{1}{2} (3.0 \, \text{rad/s}^2) (2.0 \, \text{s})^2 \]

4. **Calculate each term**:
- First term:
\[ (2.0 \, \text{rad/s}) \cdot (2.0 \, \text{s}) = 4.0 \, \text{rad} \]
- Second term:
\[ \frac{1}{2} (3.0 \, \text{rad/s}^2) (2.0 \, \text{s})^2 = \frac{1}{2} (3.0) (4.0) = 6.0 \, \text{rad} \]

5. **Add the results from both terms**:
\[ \theta = 4.0 \, \text{rad} + 6.0 \, \text{rad} = 10.0 \, \text{rad} \]

6. **Final answer**:
The angle through which the wheel has rotated is \(\theta = 10.0 \, \text{rad}\).