A flywheel at rest is to reach an angular velocity of 24 rad/s in 8 second with constant angular acceleration. The total angle turned through during this interval is

Options

A. 24 rad
B. 48 rad
C. 72 rad
D. 96 rad

Answer

To solve the problem, we need to find the total angle turned by the flywheel as it accelerates from rest to an angular velocity of 24 rad/s in 8 seconds with constant angular acceleration.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Initial angular velocity (\( \omega_0 \)) = 0 rad/s (since the flywheel is at rest)
- Final angular velocity (\( \omega \)) = 24 rad/s
- Time (\( t \)) = 8 s

2. **Calculate Angular Acceleration (\( \alpha \)):**
We can use the formula for angular acceleration:
\[ \alpha = \frac{\omega - \omega_0}{t} \]
Substituting the known values:
\[ \alpha = \frac{24 \, \text{rad/s} - 0 \, \text{rad/s}}{8 \, \text{s}} = \frac{24}{8} = 3 \, \text{rad/s}^2 \]

3. **Calculate the Total Angle Turned (\( \theta \)):**
We can use the equation for angular displacement:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Since \( \omega_0 = 0 \):
\[ \theta = 0 \cdot t + \frac{1}{2} \alpha t^2 = \frac{1}{2} \cdot 3 \, \text{rad/s}^2 \cdot (8 \, \text{s})^2 \]
Calculating \( (8 \, \text{s})^2 \):
\[ (8 \, \text{s})^2 = 64 \, \text{s}^2 \]
Now substituting back into the equation:
\[ \theta = \frac{1}{2} \cdot 3 \cdot 64 = \frac{192}{2} = 96 \, \text{radians} \]

4. **Final Answer:**
The total angle turned through during this interval is \( \theta = 96 \, \text{radians} \).