https://file.aiquickdraw.com/mathbot/file/17302530895438v4fb9tt.jpg

To solve this problem, we need to determine the rotational kinetic energy of a solid disk after it has rotated through \( 16.0 \, \text{rad} \) under a constant angular acceleration.

### Given Data
- Mass of the disk (\( m \)) = \( 4.00 \, \text{kg} \)
- Radius of the disk (\( R \)) = \( 0.500 \, \text{m} \)
- Angular acceleration (\( \alpha \)) = \( 2.00 \, \text{rad/s}^2 \)
- Angular displacement (\( \theta \)) = \( 16.0 \, \text{rad} \)

### Solution

1. **Moment of Inertia (\( I \))**:
   For a solid disk rotating about its center, the moment of inertia is:
   \[
   I = \frac{1}{2} m R^2
   \]
   Plugging in the values:
   \[
   I = \frac{1}{2} \times 4.00 \, \text{kg} \times (0.500 \, \text{m})^2 = \frac{1}{2} \times 4.00 \times 0.25 = 0.5 \, \text{kg} \cdot \text{m}^2
   \]

2. **Final Angular Velocity (\( \omega \))**:
   Since the disk starts from rest, we can use the following kinematic equation to find the final angular velocity:
   \[
   \omega^2 = \omega_0^2 + 2 \alpha \theta
   \]
   where \( \omega_0 = 0 \) (initial angular velocity). Substituting the values:
   \[
   \omega^2 = 0 + 2 \times 2.00 \, \text{rad/s}^2 \times 16.0 \, \text{rad} = 64.0 \, \text{rad}^2/\text{s}^2
   \]
   \[
   \omega = \sqrt{64.0} = 8.0 \, \text{rad/s}
   \]

3. **Rotational Kinetic Energy (\( K \))**:
   The rotational kinetic energy is given by:
   \[
   K = \frac{1}{2} I \omega^2
   \]
   Substituting \( I = 0.5 \, \text{kg} \cdot \text{m}^2 \) and \( \omega = 8.0 \, \text{rad/s} \):
   \[
   K = \frac{1}{2} \times 0.5 \, \text{kg} \cdot \text{m}^2 \times (8.0 \, \text{rad/s})^2
   \]
   \[
   K = 0.25 \times 64.0 = 16.0 \, \text{J}
   \]

### Answer
The rotational kinetic energy after the disk has turned through \( 16.0 \, \text{rad} \) is \( 16.0 \, \text{J} \), so the correct answer is:
- **(c) 16.0 J**

Would you like further details on any step, or do you have other questions?

### Follow-up Questions
1. How is the moment of inertia derived for different shapes, like cylinders or spheres?
2. How does angular acceleration affect rotational kinetic energy over time?
3. What is the physical interpretation of rotational kinetic energy in practical applications?
4. How would this problem change if the disk started with an initial angular velocity?
5. How does conservation of angular momentum apply to rotating systems?

### Tip
Remember, rotational kinetic energy depends on both the moment of inertia and the square of angular velocity, similar to how linear kinetic energy depends on mass and the square of velocity.

A uniform solid disk with mass 4.00 kg and radius R = 0.500 m is mounted on an axle at its center. It starts at rest and then starts to rotate with constant angular acceleration α = 2.00 rad/s². What is its rotational kinetic energy after it has turned through 16.0 rad?

Calculate the rotational kinetic energy of a solid disk with mass 4.00 kg and radius 0.500 m after it rotates through 16.0 rad under a constant angular acceleration of 2.00 rad/s². This solution includes key concepts such as moment of inertia, angular kinematics, and rotational kinetic energy, suitable for college-level physics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation