The Elevator and the Bathroom Scale
A 60 kg student steps onto a calibrated bathroom scale inside an elevator. The scale reads the normal force in newtons (taking g = 9.8 m/s²). The elevator starts at rest on the ground floor and follows this sequence: (1) accelerates upward at 2.0 m/s² for 2.0 s; (2) travels at constant velocity for 5.0 s; (3) decelerates to rest at 2.0 m/s²; (4) waits 3.0 s; (5) accelerates downward at 3.0 m/s². The student notices the reading is not equal to body weight and asks whether the scale is broken. A second student argues that during free fall the scale would read zero because "gravity disappears."
1. Calculate the scale reading during phases (1), (2), (3), and (5).
Read the worked answer
The scale reads the normal force N. Taking up as positive: N − mg = ma, so N = m(g + a).
- Phase (1), a = +2.0 m/s²: N = 60 × (9.8 + 2.0) = 60 × 11.8 = 708 N.
- Phase (2), a = 0: N = 60 × 9.8 = 588 N (true weight).
- Phase (3), decelerating while moving up means a = −2.0 m/s²: N = 60 × (9.8 − 2.0) = 60 × 7.8 = 468 N.
- Phase (5), a = −3.0 m/s²: N = 60 × (9.8 − 3.0) = 60 × 6.8 = 408 N.
Conclusion: the scale is not broken; it measures normal force, which differs from weight whenever the elevator accelerates.
2. Address the second student's claim about free fall and explain the physical meaning of "apparent weightlessness."
Read the worked answer
The claim is wrong. Gravity does not disappear in free fall; the gravitational force mg still acts. What becomes zero is the normal force, because the elevator and the student accelerate downward together at g. The scale reads N = m(g − g) = 0.
Apparent weightlessness therefore means the absence of a contact (support) force, not the absence of gravity. Astronauts in orbit are in continuous free fall around Earth; Earth's gravity still provides the centripetal force keeping them in orbit. The sensation of weight comes from the support force pushing on the body, so when that force vanishes, the body feels weightless even though gravity is present.
Conclusion: the reading is zero in free fall because N = 0, not because g = 0.