A skater rotating about a vertical axis pulls her arms inward. Ignoring all frictional effects, which of the following statements are true? Denote the magnitude of her angular velocity by `omega`, the magnitude of her angular momentum by `L`, and her kinetic energy by `E_(k)`.

Options

A. `L` is constant, `E_(k)` increases
B. `L` is constant, `omega` increases
C. `L` and `E_(k)` are both constant
D. `L` and `omega` are both constant

Explanation

To solve the problem regarding the skater rotating about a vertical axis while pulling her arms inward, we will analyze the situation step by step, focusing on the conservation of angular momentum, the relationship between angular momentum and angular velocity, and the effect on kinetic energy.

### Step-by-Step Solution:

1. **Understanding Angular Momentum Conservation**:
- The skater is rotating about a vertical axis and pulls her arms inward. Since we are ignoring all frictional effects, there are no external torques acting on the skater. Therefore, the angular momentum (L) of the system is conserved.
- **Conclusion**: \( L = \text{constant} \)

2. **Relationship Between Angular Momentum and Angular Velocity**:
- The angular momentum (L) is related to the moment of inertia (I) and angular velocity (ω) by the equation:
\[ L = I \omega \]
- As the skater pulls her arms inward, her moment of inertia (I) decreases because moment of inertia is proportional to the square of the radius (r) of the arms from the axis of rotation.
- Since L is constant and I decreases, it follows that ω must increase to keep the product \( I \omega \) constant.
- **Conclusion**: \( \omega \text{ increases} \)

3. **Kinetic Energy Calculation**:
- The kinetic energy (E_k) of the skater is given by:
\[ E_k = \frac{1}{2} I \omega^2 \]
- We can also express ω in terms of L:
\[ \omega = \frac{L}{I} \]
- Substituting this into the kinetic energy formula gives:
\[ E_k = \frac{1}{2} I \left(\frac{L}{I}\right)^2 = \frac{L^2}{2I} \]
- Since L is constant and I decreases as the skater pulls her arms in, the kinetic energy \( E_k \) will increase because it is inversely proportional to I.
- **Conclusion**: \( E_k \text{ increases} \)

### Summary of Conclusions:
- **Angular Momentum (L)**: Constant
- **Angular Velocity (ω)**: Increases
- **Kinetic Energy (E_k)**: Increases

### True Statements:
- The true statements regarding the skater's motion are:
- A) Angular momentum (L) is constant.
- B) Angular velocity (ω) increases.
- C) Kinetic energy (E_k) increases.