Understanding how to determine the normal force (\(F_n\)) acting on an object on an inclined plane is essential in physics. The normal force can be calculated using the formula:
\[ F_n = m \cdot g \cdot \cos(\theta) \]
Where:
- \(F_n\) is the normal force (in newtons, N)
- \(m\) is the mass of the object (in kilograms, kg)
- \(g\) is the acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\))
- \(\theta\) is the angle of the slope (in degrees)
Example 1: Calculating the Normal Force on a Car on a Slope
Problem: A car with a mass of 1000 kg is parked on a slope with an angle of 30 degrees. What is the normal force acting on the car?
Calculation:
Given:
- \(m = 1000 \, \text{kg}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(\theta = 30^\circ\)
Using the formula:
\[ F_n = m \cdot g \cdot \cos(\theta) \]
\[ F_n = 1000 \cdot 9.8 \cdot \cos(30^\circ) \]
\[ F_n = 1000 \cdot 9.8 \cdot 0.866 \]
\[ F_n \approx 8486.8 \, \text{N} \]
Answer: The normal force acting on the car is approximately 8486.8 N.
Example 2: Calculating the Normal Force on a Box on a Ramp
Problem: A box with a mass of 50 kg is placed on a ramp inclined at 45 degrees. What is the normal force acting on the box?
Calculation:
Given:
- \(m = 50 \, \text{kg}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(\theta = 45^\circ\)
Using the formula:
\[ F_n = m \cdot g \cdot \cos(\theta) \]
\[ F_n = 50 \cdot 9.8 \cdot \cos(45^\circ) \]
\[ F_n = 50 \cdot 9.8 \cdot 0.707 \]
\[ F_n \approx 346.15 \, \text{N} \]
Answer: The normal force acting on the box is approximately 346.15 N.
Example 3: Calculating the Normal Force on a Skier on a Hill
Problem: A skier with a mass of 70 kg is standing on a hill inclined at 60 degrees. What is the normal force acting on the skier?
Calculation:
Given:
- \(m = 70 \, \text{kg}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(\theta = 60^\circ\)
Using the formula:
\[ F_n = m \cdot g \cdot \cos(\theta) \]
\[ F_n = 70 \cdot 9.8 \cdot \cos(60^\circ) \]
\[ F_n = 70 \cdot 9.8 \cdot 0.5 \]
\[ F_n \approx 343 \, \text{N} \]
Answer: The normal force acting on the skier is approximately 343 N.
