A wheel is at rest. Its angular velocity increases uniformly and becomes 80 radian per second after 5 second. The total angular displacement is :-

Options

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

Explanation

To find the total angular displacement of the wheel, we can use the equations of motion for rotational motion. Given that the wheel starts from rest and its angular velocity increases uniformly, we can apply the following steps:

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

### Step 2: Calculate the angular acceleration (\( \alpha \))
Since the angular velocity increases uniformly, we can use the formula:
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]
Where:
- \( \Delta \omega = \omega - \omega_0 = 80 \, \text{rad/s} - 0 \, \text{rad/s} = 80 \, \text{rad/s} \)
- \( \Delta t = 5 \, \text{s} \)

Plugging in the values:
\[ \alpha = \frac{80 \, \text{rad/s}}{5 \, \text{s}} = 16 \, \text{rad/s}^2 \]

### Step 3: Calculate the total angular displacement (\( \theta \))
We can use the formula for angular displacement when the initial angular velocity, final angular velocity, and time are known:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ \theta = 0 \cdot 5 + \frac{1}{2} \cdot 16 \cdot (5)^2 \]
\[ \theta = 0 + \frac{1}{2} \cdot 16 \cdot 25 \]
\[ \theta = 8 \cdot 25 = 200 \, \text{radians} \]

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

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