A wheel rotates with a constant acceleration of `2.0rad/s^2`. If the wheel starts from rest, how many revolutions will it make in the first 10 seconds?

Options

A. `3`
B. `6`
C. `9`
D. `12`

Explanation

To solve the problem of how many revolutions a wheel makes in the first 10 seconds with a constant angular acceleration of \(2.0 \, \text{rad/s}^2\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Angular acceleration (\(\alpha\)) = \(2.0 \, \text{rad/s}^2\)
- Initial angular velocity (\(\omega_0\)) = \(0 \, \text{rad/s}\) (since the wheel starts from rest)
- Time (\(t\)) = \(10 \, \text{s}\)

2. **Use the Angular Displacement Formula**:
The angular displacement (\(\theta\)) for an object starting from rest with constant angular acceleration is given by the equation:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Since \(\omega_0 = 0\), the equation simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]

3. **Substitute the Values**:
Now, substituting the values into the equation:
\[ \theta = \frac{1}{2} \times 2.0 \, \text{rad/s}^2 \times (10 \, \text{s})^2 \]
\[ \theta = \frac{1}{2} \times 2.0 \times 100 \]
\[ \theta = 100 \, \text{radians} \]

4. **Convert Radians to Revolutions**:
To find the number of revolutions, we need to convert radians to revolutions. The relationship between radians and revolutions is:
\[ 1 \, \text{revolution} = 2\pi \, \text{radians} \]
Therefore, the number of revolutions (\(N\)) is given by:
\[ N = \frac{\theta}{2\pi} \]
Substituting the value of \(\theta\):
\[ N = \frac{100}{2\pi} \]
Using \(\pi \approx 3.14\):
\[ N \approx \frac{100}{6.28} \approx 15.92 \]

5. **Final Answer**:
Rounding to the nearest whole number, the wheel makes approximately \(16\) revolutions in the first 10 seconds.