A 73.8-kg person stands on a scale in an elevator. What is the apparent weight when the elevator is (a) accelerating upward with an acceleration of 1.85 m/s^2, (b) moving upward at a constant speed, and (c) accelerating downward with an acceleration of 1.49 m/s^2?
The Correct Answer and Explanation is:
To determine the apparent weight of a person in an elevator, we must consider the effect of the elevator’s motion on the normal force the scale reads. Apparent weight is essentially the normal force NNN acting on the person. The actual weight is the force due to gravity, given by:W=mgW = mgW=mg
where
m=73.8 kgm = 73.8 \, \text{kg}m=73.8kg (mass of the person)
g=9.8 m/s2g = 9.8 \, \text{m/s}^2g=9.8m/s2 (acceleration due to gravity)
Step-by-step Calculations
Actual weight:
W=73.8×9.8=723.24 NW = 73.8 \times 9.8 = 723.24 \, \text{N}W=73.8×9.8=723.24N
(a) Elevator accelerating upward with a=1.85 m/s2a = 1.85 \, \text{m/s}^2a=1.85m/s2
When accelerating upward, the apparent weight increases because the normal force must not only balance gravity but also provide upward acceleration:N=m(g+a)=73.8×(9.8+1.85)=73.8×11.65=859.47 NN = m(g + a) = 73.8 \times (9.8 + 1.85) = 73.8 \times 11.65 = 859.47 \, \text{N}N=m(g+a)=73.8×(9.8+1.85)=73.8×11.65=859.47N
Apparent weight = 859.47 N
(b) Elevator moving upward at constant speed
At constant speed, acceleration is zero. The only force to balance is gravity:N=mg=73.8×9.8=723.24 NN = mg = 73.8 \times 9.8 = 723.24 \, \text{N}N=mg=73.8×9.8=723.24N
Apparent weight = 723.24 N
(c) Elevator accelerating downward with a=1.49 m/s2a = 1.49 \, \text{m/s}^2a=1.49m/s2
When accelerating downward, the apparent weight decreases:N=m(g−a)=73.8×(9.8−1.49)=73.8×8.31=613.28 NN = m(g – a) = 73.8 \times (9.8 – 1.49) = 73.8 \times 8.31 = 613.28 \, \text{N}N=m(g−a)=73.8×(9.8−1.49)=73.8×8.31=613.28N
Apparent weight = 613.28 N
Explanation
In all cases, the apparent weight is determined by the net acceleration experienced by the person. When accelerating upward, the scale must push harder, making the reading higher. When moving at a constant velocity, only gravity acts. When accelerating downward, the scale pushes less, so the reading drops. This principle is rooted in Newton’s Second Law of Motion F=maF = maF=ma, where the direction and magnitude of acceleration change the required normal force.
