A wheel starts rotating with an-angular velocity of 2 rad/s. If it rotates with a constant angular acceleration `4 "rad/s"^2` what angle does the wheel rotate through in 2.0 s ? What is the angular speed, after 2.0 s.

Explanation

To solve the problem step by step, we will use the equations of motion for rotational motion.

### Step 1: Identify the given values
- Initial angular velocity, \( \omega_0 = 2 \, \text{rad/s} \)
- Angular acceleration, \( \alpha = 4 \, \text{rad/s}^2 \)
- Time, \( t = 2 \, \text{s} \)

### Step 2: Use the formula for angular displacement
The angular displacement \( \theta \) can be calculated using the formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ \theta = (2 \, \text{rad/s}) \times (2 \, \text{s}) + \frac{1}{2} \times (4 \, \text{rad/s}^2) \times (2 \, \text{s})^2 \]

### Step 4: Calculate the first term
Calculating the first term:
\[ \theta_1 = 2 \, \text{rad/s} \times 2 \, \text{s} = 4 \, \text{rad} \]

### Step 5: Calculate the second term
Calculating the second term:
\[ \theta_2 = \frac{1}{2} \times 4 \, \text{rad/s}^2 \times 4 \, \text{s}^2 = \frac{1}{2} \times 4 \times 4 = 8 \, \text{rad} \]

### Step 6: Add the two terms to find total angular displacement
Now, adding both terms to find the total angular displacement:
\[ \theta = \theta_1 + \theta_2 = 4 \, \text{rad} + 8 \, \text{rad} = 12 \, \text{rad} \]

### Step 7: Calculate the final angular velocity
To find the final angular velocity \( \omega \), we use the formula:
\[ \omega = \omega_0 + \alpha t \]

### Step 8: Substitute the values into the formula
Substituting the known values:
\[ \omega = 2 \, \text{rad/s} + (4 \, \text{rad/s}^2) \times (2 \, \text{s}) \]

### Step 9: Calculate the final angular velocity
Calculating:
\[ \omega = 2 \, \text{rad/s} + 8 \, \text{rad/s} = 10 \, \text{rad/s} \]

### Final Answers
- The angle rotated through in 2.0 s is \( 12 \, \text{rad} \).
- The final angular speed after 2.0 s is \( 10 \, \text{rad/s} \).

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