Suppose you are standing on the edge of a spinning platform and step off at right angles to the edge (radially outward). Now consider it the other way. You are standing on the ground next to a spinning carousel and you step onto the platform at right angles to the edge (radially inward).

Options

A. There is no change in rotational speed of the carousel in either situation.
B. There is a change in rotational speed in the first situation but not the second.
C. There is a change in rotational speed in the second situation but not the first.
D. There is a change in rotational speed in both instances

Explanation

To solve the problem, we need to analyze the two scenarios described: stepping off a spinning platform and stepping onto a spinning platform. We will consider the principles of angular momentum and moment of inertia.

### Step-by-Step Solution:

1. **Understanding the First Scenario (Stepping Off)**:
- You are standing on the edge of a spinning platform and step off radially outward.
- When you step off, you are moving away from the center of the platform.
- Since you are stepping off at a right angle to the edge, the line of impulse (the direction of your movement) passes through the center of the platform.
- According to the conservation of angular momentum, if there are no external torques acting on the system, the angular momentum before stepping off must equal the angular momentum after stepping off.

2. **Angular Momentum in the First Scenario**:
- Let \( I_1 \) be the moment of inertia of the platform plus you (when standing on it).
- The angular momentum \( L \) is given by \( L = I \omega \), where \( \omega \) is the angular velocity.
- Since you step off without applying any external torque, the moment of inertia \( I_1 \) remains unchanged, and thus \( \omega \) remains unchanged as well.

3. **Understanding the Second Scenario (Stepping On)**:
- Now consider the second scenario where you are standing on the ground and step onto the spinning carousel radially inward.
- When you step onto the platform, you are moving towards the center, which again passes through the center of the platform.
- This action also does not involve any external torque.

4. **Angular Momentum in the Second Scenario**:
- Let \( I_2 \) be the new moment of inertia when you step onto the platform.
- The moment of inertia increases because you are adding your mass at a distance from the axis of rotation.
- Therefore, \( I_2 = I_{platform} + m_{person} \cdot r^2 \) (where \( r \) is the distance from the axis of rotation).
- Since \( I_2 \) increases and there is no external torque, the angular momentum must be conserved. Thus, if \( I_2 \) increases, \( \omega_2 \) must decrease to keep \( L \) constant.

5. **Conclusion**:
- In the first case (stepping off), the angular velocity remains unchanged because the moment of inertia does not change.
- In the second case (stepping on), the angular velocity decreases because the moment of inertia increases.

### Final Answer:
- The correct conclusion is that the angular speed \( \omega \) remains unchanged in the first case and decreases in the second case.