What happens when a spinning ice skater draws in her outstretched arms?

Options

A. Her angular momentum decreases.
B. Her angular momentum increases.
C. Her moment of inertia decreases causing her to speed up
D. Her moment of inertia decreases causing her to slow down.

Answer

To solve the question of what happens when a spinning ice skater draws in her outstretched arms, we will analyze the situation using the principles of angular momentum and moment of inertia.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have an ice skater spinning around an axis. Initially, her arms are outstretched, and we denote her angular velocity as \( \omega \).
- As she draws in her arms, we need to analyze the changes in her moment of inertia and angular velocity.

2. **Moment of Inertia**:
- The moment of inertia (\( I \)) of a body is given by the formula \( I = m r^2 \), where \( m \) is the mass and \( r \) is the distance from the axis of rotation.
- When the skater stretches her arms out, her moment of inertia is greater because \( r \) is larger.
- When she draws in her arms, \( r \) decreases, leading to a decrease in her moment of inertia (\( I' < I \)).

3. **Conservation of Angular Momentum**:
- Angular momentum (\( L \)) is given by the product of moment of inertia and angular velocity: \( L = I \omega \).
- In the absence of external torques, angular momentum is conserved: \( L_{\text{initial}} = L_{\text{final}} \).
- This means \( I \omega = I' \omega' \).

4. **Analyzing Changes**:
- Since \( I' < I \) (moment of inertia decreases), we can use the conservation of angular momentum equation:
\[ I \omega = I' \omega' \]
- Rearranging gives:
\[ \omega' = \frac{I}{I'} \cdot \omega \]
- Since \( I' < I \), it follows that \( \frac{I}{I'} > 1 \), which implies that \( \omega' > \omega \). Therefore, her angular velocity increases.

5. **Conclusion**:
- As the skater draws in her arms, her moment of inertia decreases, and her angular velocity increases. Thus, she spins faster.

### Final Answer:
- The correct option is: **Her moment of inertia decreases causing her to speed up.**