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This is a physics problem involving forces and friction, specifically static friction, on a 10-kg box inside a moving train. Let's break it down step by step:

### Problem details:
- **Mass of the box:** \( m = 10 \, \text{kg} \)
- **Acceleration of the train:** \( a = 5.0 \, \text{m/s}^2 \)
- **Coefficient of static friction between the box and the wall:** \( \mu_s = 0.20 \)
- **Gravity:** \( g = 9.8 \, \text{m/s}^2 \)

### Goal:
Find the magnitude of the applied force \( F \) to hold the box in place without sliding. The force \( F \) presses the box against the wall, and static friction keeps it from sliding down.

### Forces acting on the box:
1. **Gravitational force** pulling the box down: 
   \[
   F_g = m \cdot g = 10 \cdot 9.8 = 98 \, \text{N}
   \]
2. **Normal force** is provided by the force \( F \), which is applied horizontally to press the box against the wall.
3. **Static friction force** acts upward to counteract the gravitational force. The maximum static friction force is:
   \[
   f_s = \mu_s \cdot F_N
   \]
   where \( F_N \) is the normal force, and in this case, \( F_N = F \) (the force applied horizontally).

4. The box is also accelerating horizontally with the train, so there is a horizontal force \( F \) required to cause that acceleration.

### Vertical force balance:
For the box to stay in place (not slide down), the static friction must balance the weight:
\[
f_s = F_g
\]
Thus,
\[
\mu_s \cdot F = F_g \implies 0.20 \cdot F = 98
\]
Solving for \( F \):
\[
F = \frac{98}{0.20} = 490 \, \text{N}
\]

### Conclusion:
The required force \( F \) is approximately \( 490 \, \text{N} \). The closest answer is **(C) 500 N**.

Would you like more details on any part of this explanation?

### Follow-up questions:
1. How would the answer change if the coefficient of friction was higher or lower?
2. What if the train were accelerating at a different rate, say 2.5 m/s²?
3. How would the normal force change if the mass of the box increased to 20 kg?
4. What is the role of static friction in other real-world scenarios, such as keeping a book on a sloped surface?
5. How does the direction of the applied force affect the friction and normal force?

### Tip:
Always draw a free-body diagram in physics problems involving forces to visualize how different forces interact.

A 10-kg box is held in place by an applied force F on a rough wall inside a train car. The coefficient of static friction between the box and the wall is 0.20. If the train accelerates at 5.0 m/s², what must the magnitude of the applied force F be to hold the box in place without sliding?

This problem involves applying Newton's laws of motion and the concept of static friction to calculate the applied force required to hold a 10-kg box in place against a rough wall inside a train. With a coefficient of static friction of 0.20 and a train acceleration of 5.0 m/s², the solution provides a step-by-step breakdown using physics principles and formulas.

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