This problem involves understanding angular momentum and rotational inertia. Let's analyze it step by step:

Given

1. A circular dartboard spins with constant angular speed.
2. The first dart lands at the bullseye (center).
3. The second dart lands at the outer edge of the dartboard.

We need to determine what happens after each dart hits.

Assumptions

1. The darts stick to the dartboard upon impact.
2. The dartboard is initially spinning in a vacuum (no air resistance or friction).
3. The mass of the dart is non-negligible compared to the dartboard.
4. Conservation of angular momentum applies because there is no external torque.

Analysis

After the first dart hits:

• The dart lands at the bullseye (center), contributing no torque because the lever arm (distance from the axis of rotation) is zero.
• The rotational inertia of the system (dartboard + dart) does not change significantly.
• Outcome: The angular speed of the dartboard remains constant after the first dart hits.

After the second dart hits:

• The second dart lands at the outer edge of the dartboard, increasing the system's rotational inertia. The formula for rotational inertia is $I = I_{\text{board}} + m \cdot r^2$, where $r$ is the distance from the axis of rotation (outer edge here).
• Angular momentum ($L = I \cdot \omega$) must be conserved. As $I$ increases due to the second dart, the angular speed ($\omega$) decreases.
• Outcome: The angular speed of the dartboard decreases after the second dart hits.

Conclusion

1. After the first dart hits: The angular speed of the dartboard remains unchanged.
2. After the second dart hits: The angular speed of the dartboard decreases due to the increase in rotational inertia.

Let me know if you'd like further clarification or additional details!

Related Questions:

1. What would happen if the dartboard had friction at the bearing?
2. How does the mass of the dartboard influence the effect of the second dart?
3. How would angular speed change if multiple darts hit at different radii?
4. What is the relationship between torque and angular momentum in such systems?
5. How can conservation of energy be used to analyze this problem?

Tip:

Always check whether external forces or torques act on a system when applying conservation laws.