Starting from rest a wheel rotates with uniform angular acceleration `2pirads^(-2)`. After 4s, if the angular acceleration ceases to act, its angular displacement in the next 4s is

Options

A. `8pirad`
B. `16pirad`
C. `24pirad`
D. `32pirad`

Answer

To solve the problem step by step, we will follow the physics principles related to angular motion.

### Step 1: Identify the given values
- Initial angular velocity, \( \omega_0 = 0 \) rad/s (starting from rest)
- Angular acceleration, \( \alpha = 2\pi \) rad/s²
- Time of acceleration, \( t_1 = 4 \) s

### Step 2: Calculate the angular velocity after 4 seconds
We can use the formula for angular velocity under constant angular acceleration:
\[ \omega = \omega_0 + \alpha t \]
Substituting the values:
\[ \omega = 0 + (2\pi)(4) = 8\pi \text{ rad/s} \]

### Step 3: Determine the angular displacement during the first 4 seconds
We can use the formula for angular displacement under constant angular acceleration:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the values:
\[ \theta = 0 \cdot 4 + \frac{1}{2} (2\pi)(4^2) = \frac{1}{2} (2\pi)(16) = 16\pi \text{ rad} \]

### Step 4: Analyze the motion after the angular acceleration ceases
After 4 seconds, the angular acceleration ceases, and the wheel continues to rotate at a constant angular velocity of \( \omega = 8\pi \) rad/s.

### Step 5: Calculate the angular displacement in the next 4 seconds
For the next 4 seconds, we can calculate the angular displacement using:
\[ \theta = \omega t \]
Substituting the values:
\[ \theta = (8\pi)(4) = 32\pi \text{ rad} \]

### Final Result
The total angular displacement in the next 4 seconds is \( 32\pi \) rad.

### Summary
- The angular displacement in the next 4 seconds after the acceleration ceases is \( 32\pi \) rad.