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The problem you have provided involves a Ferris wheel rotating at constant speed with two students, \(A\) and \(B\), riding in opposite carts. The question asks for the **total work done** on student \(A\) as they move from the bottom to the top of the Ferris wheel.

### Key points:
1. The Ferris wheel has a radius \(R\) and rotates at constant velocity.
2. Students \(A\) and \(B\) have masses \(m_A\) and \(m_B\), respectively.
3. You are tasked with finding the total work done on student \(A\) as they move from the bottom to the top of the Ferris wheel.

### Analyzing the problem:
- **Work** is defined as \(W = \sum \mathbf{F} \cdot \Delta \mathbf{s}\), where \(\mathbf{F}\) is the force applied and \(\Delta \mathbf{s}\) is the displacement vector.
- The total work done by all forces includes the work done by gravity and the normal force.
  
#### Work Done by Gravity:
- As student \(A\) moves from the bottom to the top, the gravitational force acts downward, while the displacement is vertical. The work done by gravity is given by:
  \[
  W_{\text{gravity}} = -m_A g h
  \]
  where \(h\) is the vertical displacement, which equals \(2R\), the diameter of the Ferris wheel.

- So the work done by gravity is:
  \[
  W_{\text{gravity}} = -m_A g (2R)
  \]
  This is because gravity is acting against the direction of motion (upward).

#### Work Done by the Normal Force:
- The Ferris wheel’s normal force acts perpendicular to the displacement at all times, so the work done by the normal force over a full half-turn is zero:
  \[
  W_{\text{normal}} = 0
  \]

### Total Work:
Since the normal force does no work, the total work done on student \(A\) is just the work done by gravity:
\[
W_{\text{total}} = -m_A g (2R)
\]

Thus, the magnitude of the total work is:
\[
|W_{\text{total}}| = m_A g (2R)
\]

### Conclusion:
The total work done on student \(A\) as they move from the bottom to the top of the Ferris wheel is \(m_A g (2R)\).

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Let me know if you want more details, or if you have any questions!

Here are five related questions:
1. How would the work change if the Ferris wheel’s speed varied?
2. What is the work done on student \(B\) during this motion?
3. How would air resistance affect the total work on student \(A\)?
4. How does the centripetal force affect the motion of students on the Ferris wheel?
5. What would the total energy of student \(A\) be at the top of the Ferris wheel?

**Tip:** When calculating work, always distinguish between the forces doing positive versus negative work based on the direction of motion.

What is the magnitude of the total work done on student A in moving from the bottom to the top of the ferris wheel?

This problem covers the work-energy principles, calculating the total work done on student A as they move from the bottom to the top of a Ferris wheel. The problem involves concepts like gravitational work and normal forces, suitable for Physics I students.

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