To calculate the normal force acting on a box on a slope, you need to consider the components of the gravitational force acting on the box.

1. Gravitational Force:

The gravitational force acting on the box is given by $F_g = m \cdot g$, where:

• $m$ is the mass of the box,
• $g$ is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$).

2. Components of the Gravitational Force:

When a box is on an inclined slope at an angle $\theta$, the gravitational force can be resolved into two components:

• Parallel Component: $F_{\parallel} = m \cdot g \cdot \sin(\theta)$ (This component acts down the slope.)
• Perpendicular Component: $F_{\perp} = m \cdot g \cdot \cos(\theta)$ (This component acts perpendicular to the slope.)

3. Normal Force Calculation:

The normal force $F_N$ is the force exerted by the surface of the slope that is perpendicular to it. This force balances the perpendicular component of the gravitational force:

$F_N = F_{\perp} = m \cdot g \cdot \cos(\theta)$

Summary of Steps:

• Identify the mass $m$ of the box.
• Determine the angle $\theta$ of the slope.
• Use the formula $F_N = m \cdot g \cdot \cos(\theta)$ to calculate the normal force.

Example:

If the mass of the box is 10 kg and the slope is inclined at $30^\circ$:

1. Calculate the gravitational force: $F_g = 10 \times 9.8 = 98 \, \text{N}$.
2. Resolve this into components:
   • $F_{\parallel} = 98 \times \sin(30^\circ) = 49 \, \text{N}$.
   • $F_{\perp} = 98 \times \cos(30^\circ) \approx 84.87 \, \text{N}$.
3. The normal force $F_N$ is $84.87 \, \text{N}$.

Would you like more details or have any other questions? Here are some related questions you might find helpful:

1. How do frictional forces depend on the normal force?
2. How can the angle of the slope affect the normal force?
3. What happens to the normal force if the slope becomes vertical?
4. How does the normal force change if the surface is not frictionless?
5. What would be the normal force if the box were on a horizontal surface?

Tip: Always resolve forces into components parallel and perpendicular to the slope to analyze problems involving inclined planes accurately.