Determining the mass (\(m\)) of an object using the normal force and the angle of the slope is a fundamental concept in physics. The mass can be calculated using the rearranged formula:
\[ m = \dfrac{F_n}{g \cdot \cos(\theta)} \]
Where:
- \(m\) is the mass of the object (in kilograms, kg)
- \(F_n\) is the normal force (in newtons, N)
- \(g\) 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 Mass of a Cart on a Ramp
Problem: A cart on a ramp experiences a normal force of 500 N at an angle of 30 degrees. What is the mass of the cart?
Calculation:
Given:
- \(F_n = 500 \, \text{N}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(\theta = 30^\circ\)
Using the formula:
\[ m = \dfrac{F_n}{g \cdot \cos(\theta)} \]
\[ m = \dfrac{500}{9.8 \cdot \cos(30^\circ)} \]
\[ m = \dfrac{500}{9.8 \cdot 0.866} \]
\[ m \approx 58.55 \, \text{kg} \]
Answer: The mass of the cart is approximately 58.55 kg.
Example 2: Calculating the Mass of a Refrigerator on a Ramp
Problem: A refrigerator experiences a normal force of 600 N on a ramp inclined at 45 degrees. What is the mass of the refrigerator?
Calculation:
Given:
- \(F_n = 600 \, \text{N}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(\theta = 45^\circ\)
Using the formula:
\[ m = \dfrac{F_n}{g \cdot \cos(\theta)} \]
\[ m = \dfrac{600}{9.8 \cdot \cos(45^\circ)} \]
\[ m = \dfrac{600}{9.8 \cdot 0.707} \]
\[ m \approx 87.25 \, \text{kg} \]
Answer: The mass of the refrigerator is approximately 87.25 kg.
Example 3: Calculating the Mass of a Log on a Hill
Problem: A log experiences a normal force of 800 N on a hill inclined at 60 degrees. What is the mass of the log?
Calculation:
Given:
- \(F_n = 800 \, \text{N}\)
- \(g = 9.8 \, \text{m/s}^2\)
- \(\theta = 60^\circ\)
Using the formula:
\[ m = \dfrac{F_n}{g \cdot \cos(\theta)} \]
\[ m = \dfrac{800}{9.8 \cdot \cos(60^\circ)} \]
\[ m = \dfrac{800}{9.8 \cdot 0.5} \]
\[ m \approx 163.27 \, \text{kg} \]
Answer: The mass of the log is approximately 163.27 kg.
