A wheel is at rest. Its angular velocity increases uniformly and becomes 60 rad/sec after 5 sec. The total angular displacement is

Options

A. 600 rad
B. 75 rad
C. 300 rad
D. 150 rad

Explanation

To solve the problem, we need to find the total angular displacement of the wheel as it accelerates uniformly from rest to an angular velocity of 60 rad/sec over a time period of 5 seconds. We can use the equations of motion for rotational motion to find the angular displacement.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Initial angular velocity (\( \omega_0 \)) = 0 rad/sec (the wheel is at rest)
- Final angular velocity (\( \omega_f \)) = 60 rad/sec
- Time (\( t \)) = 5 sec

2. **Calculate Angular Acceleration (\( \alpha \)):**
- Use the formula for angular acceleration:
\[ \alpha = \frac{\omega_f - \omega_0}{t} \]
- Substitute the known values:
\[ \alpha = \frac{60 \, \text{rad/sec} - 0 \, \text{rad/sec}}{5 \, \text{sec}} = \frac{60}{5} = 12 \, \text{rad/sec}^2 \]

3. **Use the Angular Displacement Formula:**
- The formula for angular displacement (\( \theta \)) when starting from rest is:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
- Since \( \omega_0 = 0 \), the equation simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]

4. **Substitute Values into the Displacement Formula:**
- Substitute \( \alpha = 12 \, \text{rad/sec}^2 \) and \( t = 5 \, \text{sec} \):
\[ \theta = \frac{1}{2} \times 12 \, \text{rad/sec}^2 \times (5 \, \text{sec})^2 \]
- Calculate \( (5 \, \text{sec})^2 = 25 \, \text{sec}^2 \):
\[ \theta = \frac{1}{2} \times 12 \times 25 \]
- Now calculate:
\[ \theta = 6 \times 25 = 150 \, \text{radians} \]

5. **Final Answer:**
- The total angular displacement of the wheel is \( \theta = 150 \, \text{radians} \).