The rotational kinetic energy is -

Options

A. `m omega^2`
B. `1/2 I alpha^2`
C. ` 1/2 I omega^2`
D. `1/2 mv^2`

Answer

To solve the question regarding rotational kinetic energy, we can break it down into clear steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Rotational Kinetic Energy**:
- Rotational kinetic energy is the energy possessed by an object due to its rotation. It is analogous to translational kinetic energy, which is due to linear motion.

2. **Formula for Rotational Kinetic Energy**:
- The formula for rotational kinetic energy (KE_rot) is given by:
\[ KE_{\text{rot}} = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia of the object and \( \omega \) is the angular velocity.

3. **Identifying the Components of Motion**:
- When an object rolls, it undergoes both translational motion (moving forward) and rotational motion (spinning around its axis). The total kinetic energy of a rolling object is the sum of its translational and rotational kinetic energies.

4. **Total Kinetic Energy in Rolling Motion**:
- For an object rolling without slipping, the total kinetic energy (KE_total) can be expressed as:
\[ KE_{\text{total}} = KE_{\text{trans}} + KE_{\text{rot}} = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2 \]
where \( m \) is the mass of the object and \( v \) is the linear velocity.

5. **Conclusion**:
- Since the question specifically asks for the rotational kinetic energy, we focus on the formula:
\[ KE_{\text{rot}} = \frac{1}{2} I \omega^2 \]
- Therefore, the correct answer to the question is option C, which represents the formula for rotational kinetic energy.

### Final Answer:
The rotational kinetic energy is given by:
\[ \frac{1}{2} I \omega^2 \]
Thus, the correct option is C.

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