A flywheel at rest is reached to an angular velocity of 36 `rad//s` in 6 s with a constant angular accleration. The total angle turned during this interval is

Options

A. 216 rad
B. 144 rad
C. 108 rad
D. 72 rad

Answer

To solve the problem, we need to find the total angle turned by the flywheel during the time interval of 6 seconds while it accelerates from rest to an angular velocity of 36 rad/s with constant angular acceleration.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Initial angular velocity, \( \omega_0 = 0 \, \text{rad/s} \) (since the flywheel is at rest)
- Final angular velocity, \( \omega = 36 \, \text{rad/s} \)
- Time, \( t = 6 \, \text{s} \)

2. **Use the Angular Velocity Equation to Find Angular Acceleration:**
We can use the equation of motion for angular velocity:
\[ \omega = \omega_0 + \alpha t \]
Substituting the known values:
\[ 36 = 0 + \alpha \cdot 6 \]
Solving for \( \alpha \):
\[ \alpha = \frac{36}{6} = 6 \, \text{rad/s}^2 \]

3. **Calculate the Total Angle Turned Using the Angular Displacement Equation:**
The angular displacement \( \theta \) can be calculated using the formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ \theta = 0 \cdot 6 + \frac{1}{2} \cdot 6 \cdot (6)^2 \]
Simplifying:
\[ \theta = 0 + \frac{1}{2} \cdot 6 \cdot 36 \]
\[ \theta = 3 \cdot 36 = 108 \, \text{radians} \]

4. **Final Answer:**
The total angle turned during this interval is \( \theta = 108 \, \text{radians} \).