A person with outstretched arms, is spinning on a rotating stool. He suddenly brings his arms down to his sides. Which of the following is true about his kinetic energy K and angualr momentum L?

Options

A. 1.Both K and L increase
B. 2.Both K and L remain unchanged
C. 3.K remains constant, L increases
D. 4.K increases but L remains constant

Answer

To solve the problem, we need to analyze the situation of a person spinning on a rotating stool with outstretched arms and then bringing their arms down to their sides. We will examine the effects on kinetic energy (K) and angular momentum (L).

### Step-by-Step Solution:

1. **Understanding the Initial and Final States**:
- Initially, the person has their arms outstretched, which increases their moment of inertia (I).
- When the person brings their arms down, the moment of inertia decreases.

2. **Conservation of Angular Momentum**:
- Angular momentum (L) is given by the formula:
\[ L = I \cdot \omega \]
where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.
- Since there is no external torque acting on the system, angular momentum is conserved:
\[ L_{\text{initial}} = L_{\text{final}} \implies I_1 \cdot \omega_1 = I_2 \cdot \omega_2 \]
- Here, \(I_1\) is the moment of inertia when arms are outstretched and \(I_2\) is the moment of inertia when arms are down. Since \(I_1 > I_2\), it follows that \(\omega_2 > \omega_1\).

3. **Analyzing Kinetic Energy**:
- The rotational kinetic energy (K) is given by:
\[ K = \frac{1}{2} I \omega^2 \]
- Initially, the kinetic energy is:
\[ K_1 = \frac{1}{2} I_1 \omega_1^2 \]
- After the person brings their arms down, the kinetic energy becomes:
\[ K_2 = \frac{1}{2} I_2 \omega_2^2 \]
- Since \(\omega_2 > \omega_1\) and \(I_2 < I_1\), we need to analyze how these changes affect kinetic energy.

4. **Comparing Kinetic Energies**:
- We know that \(I_1 \cdot \omega_1 = I_2 \cdot \omega_2\). Rearranging gives:
\[ \omega_2 = \frac{I_1}{I_2} \cdot \omega_1 \]
- As \(I_2 < I_1\), \(\frac{I_1}{I_2} > 1\), which means \(\omega_2\) is significantly greater than \(\omega_1\).
- Thus, even though \(I_2\) is smaller, the increase in \(\omega_2\) leads to an increase in kinetic energy:
\[ K_2 > K_1 \]

5. **Conclusion**:
- Angular momentum (L) remains constant because there is no external torque.
- Kinetic energy (K) increases because the increase in angular velocity outweighs the decrease in moment of inertia.

### Final Answer:
- **Angular Momentum (L)**: Constant
- **Kinetic Energy (K)**: Increases