Elevator Scale Reading During Acceleration
A 62 kg student stands on a bathroom scale inside an elevator. The scale reads in newtons. Starting from rest at the ground floor, the elevator accelerates upward at 1.8 m/s² for 2.0 s, then travels at constant velocity for 6.0 s, then decelerates to rest at the top over 1.5 s. The student keeps a video log and notices the scale reading changes three times during the trip. Later, a classmate claims the scale reads the student's true weight the entire time because 'gravity doesn't change.' Assume g = 9.8 m/s² and neglect air resistance.
1. Determine the scale reading during each of the three phases and explain, using Newton's second law, why the reading differs from the student's weight.
Read the worked answer
Weight: W = mg = (62)(9.8) = 607.6 N.
Phase 1 (accelerating up, a = +1.8 m/s²): ΣF = N − W = ma → N = m(g + a) = 62(9.8 + 1.8) = 62(11.6) = 719.2 N ≈ 720 N. The scale reads the normal force, which must exceed weight to produce upward acceleration.
Phase 2 (constant velocity, a = 0): N = W = 607.6 N ≈ 610 N. Net force is zero, so the reading equals true weight.
Phase 3 (decelerating to rest while moving up, a = −?): The elevator slows while moving upward, so acceleration points down. First find the deceleration magnitude: the elevator was moving at v = (1.8)(2.0) = 3.6 m/s and stops in 1.5 s, so a = 3.6/1.5 = 2.4 m/s² downward. N = m(g − a) = 62(9.8 − 2.4) = 62(7.4) = 458.8 N ≈ 460 N. The scale reads less than weight because the net force must point downward.
The classmate is wrong: the scale does not measure weight directly, it measures the normal force supporting the student, which depends on acceleration.
2. The student claims that during the constant-velocity phase the elevator must be pushing up harder than during the acceleration phase because 'it has to keep the student moving.' Evaluate this claim.
Read the worked answer
The claim is incorrect. During constant velocity the acceleration is zero, so the net force is zero and N = W = 610 N - the smallest of the moving-phase readings (only the deceleration phase reads lower).
Newton's first law: an object in motion stays in motion at constant velocity with no net force. The elevator does not need to 'keep pushing' to maintain motion; it only needs a net force to change motion (accelerate). The acceleration phase requires N = 720 N precisely because a net upward force is needed to speed the student up. Once at constant speed, no net force is required, so the reading drops back to the true weight. The student is confusing the force needed to maintain motion with the force needed to change motion.