A wheel is a rest. Its angular velocity increases uniformly and becomes `80 rad s^(-1)` after 5 s. The total angular displacement is

Options

A. 800 rad
B. 400 rad
C. 200 rad
D. 100 rad

Explanation

To solve the problem, we need to find the total angular displacement of a wheel that starts from rest and has a uniform increase in angular velocity. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Initial angular velocity, \( \omega_0 = 0 \, \text{rad/s} \) (since the wheel is at rest)
- Final angular velocity, \( \omega = 80 \, \text{rad/s} \)
- Time, \( t = 5 \, \text{s} \)

### Step 2: Calculate the angular acceleration
Since the angular velocity increases uniformly, we can use the formula:
\[ \omega = \omega_0 + \alpha t \]
Substituting the known values:
\[ 80 = 0 + \alpha \cdot 5 \]
This simplifies to:
\[ \alpha = \frac{80}{5} = 16 \, \text{rad/s}^2 \]

### Step 3: Use the angular displacement formula
We can use the following kinematic equation for angular motion:
\[ \omega^2 = \omega_0^2 + 2\alpha \theta \]
Substituting the known values:
\[ 80^2 = 0 + 2 \cdot 16 \cdot \theta \]
This simplifies to:
\[ 6400 = 32\theta \]

### Step 4: Solve for angular displacement \( \theta \)
Now, we can solve for \( \theta \):
\[ \theta = \frac{6400}{32} = 200 \, \text{radians} \]

### Final Answer
The total angular displacement is \( \theta = 200 \, \text{radians} \).