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.