Understanding the Elevator Paradox
When you are inside an elevator and drop an object, you expect it to hit the floor immediately. However, if the object stays suspended in mid-air, it implies a very specific physical condition. This phenomenon is a classic example of apparent weightlessness.
The Physics of Free Fall
To understand why the briefcase does not fall, we must look at Newton's Second Law: $$F_{net} = ma$$
In the frame of the elevator, the briefcase experiences two main forces: gravity acting downward ($mg$) and the normal force from the floor (if it were resting). However, if the briefcase is not falling relative to the woman, it means both the woman and the briefcase are accelerating at the same rate as the elevator.
The Condition: Free Fall
If the elevator cable snaps or the elevator is descending with an acceleration exactly equal to the acceleration due to gravity ($g \approx 9.8 \, m/s^2$), we have a case of free fall:
- The elevator accelerates downward at $a = g$.
- The briefcase is also in free fall, accelerating downward at $g$.
- Since both the floor and the briefcase accelerate downward at the same rate, the distance between them remains constant. To the observer inside, the briefcase appears to hover.
Conclusion
The elevator is moving with a downward acceleration equal to the acceleration due to gravity ($a = g$). This state is often called "weightlessness" because the normal force exerted by the floor on any object inside the elevator becomes zero ($N = m(g - a) = m(g - g) = 0$).