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Let's break down the problem into the relevant parts.

1. **System Description:**
   - There is a pulley system with a mass attached to a rope. The person is pulling the rope horizontally to lift the mass vertically.
   
2. **Subpart a**:
   - We are asked to find the work done by a person pulling the rope with a force of \( 40.0 \, \text{N} \) to lift the mass from a depth of \( 12.0 \, \text{m} \) to \( 9.00 \, \text{m} \). The mass only is considered.

### a. Work Done on the Mass:
- The **work done** on an object is given by the formula:
  \[
  W = F \cdot d \cdot \cos(\theta)
  \]
  where:
  - \( F = 40.0 \, \text{N} \) is the force applied,
  - \( d \) is the distance the object is moved,
  - \( \theta \) is the angle between the force and the direction of displacement.
  
  Here, the angle is \( 0^\circ \) because the force is in the same direction as the displacement, so \( \cos(0^\circ) = 1 \).

  The displacement of the mass is from 12.0 m deep to 9.0 m deep, so the vertical displacement \( d = 12.0 \, \text{m} - 9.0 \, \text{m} = 3.0 \, \text{m} \).

  Hence, the work done is:
  \[
  W = 40.0 \, \text{N} \cdot 3.0 \, \text{m} = 120.0 \, \text{J}
  \]

### b. Work Done on the System (Mass and Rope):
- In this part, we are considering both the mass and the rope as part of the system. The rope is being pulled horizontally while lifting the mass.

  The key concept here is that **the work done** by the person on the system depends on the **horizontal displacement** of the rope, not just the vertical displacement of the mass. If the pulley is ideal (i.e., frictionless), the rope will need to be pulled a greater distance than the mass rises because of the way the pulley redirects the force.

  In an ideal pulley, if the mass rises by \( 3.0 \, \text{m} \), the length of the rope that needs to be pulled is greater, often by a factor of 2 (depending on the pulley configuration). So, the horizontal displacement could be:

  \[
  \text{Horizontal distance} = 2 \times 3.0 \, \text{m} = 6.0 \, \text{m}
  \]

  The work done by the person pulling the rope is then:
  \[
  W = 40.0 \, \text{N} \times 6.0 \, \text{m} = 240.0 \, \text{J}
  \]

---

### Summary of Answers:
- **a.** The work done by the person on the mass is \( 120.0 \, \text{J} \).
- **b.** The work done by the person on the system (mass and rope) is \( 240.0 \, \text{J} \).

Would you like more detailed explanations, or do you have any specific questions?

Here are some related questions:
1. How does the mechanical advantage of a pulley system affect the force needed to lift a mass?
2. What would change in the work calculation if friction were introduced in the pulley?
3. How would the calculation differ if the angle between the applied force and the displacement was not 0°?
4. What is the significance of energy conservation in pulley systems?
5. How does an ideal pulley compare to a real-world pulley in terms of efficiency?

**Tip:** Always remember that in pulley systems, the distance you pull the rope is often different from the height the mass is lifted, due to mechanical advantage.

Consider the pulley machine shown in the figure, where the rope is attached to the mass. a) Consider the system as only the mass. If someone were to pull the long rope along the horizontal at a force of 40.0 [N] to lift that mass up from 12.0 [m] deep to 9.00 [m] deep, how much work is done by the person on the system? Explain your answer. b) Consider the system as both the mass and rope. What is now the work done by the person on the system? Explain your answer.

Explore the calculation of work done in a pulley system when a horizontal force of 40.0 N is applied to lift a mass from a depth of 12.0 m to 9.0 m. Understand the physics concepts and formulas involved, such as work, force, and displacement, suitable for high school students in Grades 9-12.

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