Understanding Apparent Weight
Have you ever felt 'lighter' or 'heavier' in an elevator? This feeling isn't an illusion—it is a result of Newton's Laws of Motion. When an elevator accelerates, the normal force exerted by the scale (which measures your 'apparent weight') changes compared to your actual weight.
The Problem
A 550 N physics student stands on a bathroom scale in an elevator. As the elevator starts moving, the scale reads 450 N. We need to find the magnitude and direction of the elevator's acceleration.
Step-by-Step Solution
1. Identify the Given Variables
- Actual Weight (W): 550 N
- Apparent Weight (Normal Force, N): 450 N
- Acceleration due to gravity (g): Approximately $9.8 \text{ m/s}^2$
First, find the student's mass (m): $m = \frac{W}{g} = \frac{550 \text{ N}}{9.8 \text{ m/s}^2} \approx 56.12 \text{ kg}$
2. Free Body Diagram (FBD)
To visualize the forces on the student:
- Draw a point mass representing the student.
- Draw a vector downward for the force of gravity ($W = 550 \text{ N}$).
- Draw a vector upward for the Normal force exerted by the scale ($N = 450 \text{ N}$).
Since the scale reads less than the actual weight ($450 < 550$), the net force must be directed downward.
3. Applying Newton's Second Law
Newton’s Second Law states that $F_{\text{net}} = ma$. Taking upward as the positive direction: $\sum F = N - W = ma$
Substitute the values: $450 \text{ N} - 550 \text{ N} = (56.12 \text{ kg}) \cdot a$ $-100 \text{ N} = (56.12 \text{ kg}) \cdot a$ $a = \frac{-100}{56.12} \approx -1.78 \text{ m/s}^2$
Conclusion
- Magnitude: The magnitude of the acceleration is approximately $1.78 \text{ m/s}^2$.
- Direction: The negative sign indicates that the acceleration is directed downward.
Intuition
Whenever the reading on the scale is less than your actual weight, the elevator is either accelerating downward or decelerating while moving upward. In this case, because the scale reading dropped, the elevator is accelerating toward the floor.