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The problem presents different scenarios in which a 50 kg box is either pushed or pulled on a carpeted platform. We are asked to rank the setups based on the magnitude of the friction force between the box and the platform.

### Understanding the Problem:
The friction force depends on two factors:
1. **Normal Force**: The force perpendicular to the surface, which is affected by both the weight of the box and the applied force's vertical component.
2. **Coefficient of Friction**: The friction force increases with the normal force. Friction is given by \( f = \mu N \), where \( \mu \) is the coefficient of friction and \( N \) is the normal force.

### Analyzing the Setups:

1. **Setup A (horizontal push with 200 N)**:
   - The normal force is simply the weight of the box, which is \( N = mg = 50 \times 9.8 = 490 \, \text{N} \), since there is no vertical component of the applied force.
   - Friction force \( f_A = \mu N = \mu \times 490 \).

2. **Setup B (horizontal pull with 200 N)**:
   - Like A, the force is horizontal, so the normal force is \( N = 490 \, \text{N} \).
   - Friction force \( f_B = \mu N = \mu \times 490 \), which is the same as in setup A.

3. **Setup C (pull at 15° upward with 200 N)**:
   - The vertical component of the pulling force reduces the normal force.
   - The vertical component is \( F_{y} = 200 \sin(15^\circ) \), reducing the normal force to \( N = 490 - 200 \sin(15^\circ) \).
   - Since the normal force is reduced, the friction force \( f_C \) is less than \( f_A \) and \( f_B \).

4. **Setup D (push at 15° downward with 200 N)**:
   - The vertical component increases the normal force.
   - The vertical component is \( F_{y} = 200 \sin(15^\circ) \), increasing the normal force to \( N = 490 + 200 \sin(15^\circ) \).
   - Since the normal force is greater, the friction force \( f_D \) is higher than \( f_A \) and \( f_B \).

5. **Setup E (pull on a 15° incline upward with 200 N)**:
   - The normal force is reduced due to both the incline and the upward force component.
   - The combined effects result in the smallest normal force and, consequently, the smallest friction force.

6. **Setup F (pull on a 15° incline with 100 N)**:
   - This has less applied force (100 N) than setup E, reducing both the horizontal and vertical components of the force.
   - This will result in a normal force similar to or slightly lower than in setup E, leading to the same or slightly smaller friction force.

### Conclusion:
- **Setup D** has the largest friction force, as the downward push increases the normal force.
- **Setup E and F** have the smallest friction forces due to the combination of incline and upward pulling.
- **Setups A, B, and C** have the same or similar friction force, with **C** being slightly less because of the vertical force component.

### Correct Answer: 
The ranking should be:
\[ D < E = F < A = B < C \]

Would you like a detailed breakdown of the calculations involved?

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**5 Related Questions:**
1. How do normal forces affect the magnitude of friction?
2. Why does pushing downward increase friction compared to pulling upward?
3. How does the incline affect the friction force in setup E?
4. What would happen to the friction force if the applied force in setup A increased?
5. How does the friction coefficient influence the result?

**Tip:** Always resolve the vertical and horizontal components of a force when dealing with inclined or angled setups.

A student is either pushing or pulling a 50 kg box along a carpeted platform. The box, however, remains at rest, even though the student tests a number of different setups. The same box and carpet are used for each case. The setups involve different angles and forces, and the goal is to rank the setups based on the magnitude of the friction force between the box and the platform.

Learn how to rank friction forces in different setups where a 50 kg box is either pushed or pulled across a carpeted platform. This problem covers the effects of applied forces at different angles on normal force and friction, with a detailed breakdown of each setup. Suitable for students in Grades 11-12 studying physics.

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