The angular speed of a star spinning about its axis increases as the star begins to contract to a smaller radius. Which of the following quantities associated with the star must decrease as this occurs?

Options

A. Moment of inertia
B. Angular momentum
C. Rotational kinetic energy
D. Net external torque

Answer

To solve the problem, we need to analyze the relationship between the angular momentum, moment of inertia, angular speed, and rotational kinetic energy of the star as it contracts.

### Step-by-Step Solution:

1. **Understanding Angular Momentum**:
The angular momentum \( L \) of a rotating object is given by the formula:
\[ L = I \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular speed.

2. **No External Torque**:
The problem states that there is no external torque acting on the star. This means that the angular momentum \( L \) remains constant:
\[ \frac{dL}{dt} = 0 \implies L = \text{constant} \]

3. **Moment of Inertia and Angular Speed**:
As the star contracts, its radius decreases, which leads to a decrease in the moment of inertia \( I \). The moment of inertia for a solid sphere, for example, is given by:
\[ I = \frac{2}{5} m r^2 \]
where \( m \) is the mass and \( r \) is the radius. As \( r \) decreases, \( I \) decreases.

4. **Relationship Between Angular Speed and Moment of Inertia**:
Since \( L \) is constant and \( I \) is decreasing, we can see that \( \omega \) must increase to keep the product \( I \omega \) constant:
\[ L = I \omega \implies \text{If } I \text{ decreases, then } \omega \text{ must increase.} \]

5. **Rotational Kinetic Energy**:
The rotational kinetic energy \( K \) is given by:
\[ K = \frac{1}{2} I \omega^2 \]
Since \( I \) is decreasing and \( \omega \) is increasing, we need to analyze how these changes affect \( K \).

6. **Analyzing the Kinetic Energy**:
We can express \( \omega \) in terms of \( L \):
\[ \omega = \frac{L}{I} \]
Substituting this into the kinetic energy formula gives:
\[ K = \frac{1}{2} I \left(\frac{L}{I}\right)^2 = \frac{L^2}{2I} \]
Since \( I \) is decreasing, \( K \) must increase because \( L \) is constant.

7. **Conclusion**:
The only quantity that must decrease as the star contracts and angular speed increases is the moment of inertia \( I \).

### Final Answer:
The quantity that must decrease as the star contracts is the **moment of inertia (I)**.