Weight and apparent weight
Weight is the pull of the Earth on you: W = mg. For a 50 kg person, W = 50 × 10 = 500 N (taking g = 10 m/s²).
What a scale shows is different. A scale measures how hard you press on it. By Newton's third law the scale pushes back with the same force. We call that push N (the normal force). The reading on the scale is the apparent weight.
If you are not accelerating, N = mg, so the scale shows your true weight. If you are accelerating, N changes.
Overweight: lift accelerating up
Take up as positive. The lift and the person speed up with acceleration a upward. Newton's second law says the total force upward is ma:
N − mg = ma, so N = m(g + a).
N is bigger than mg, so the scale reading goes up. This is called overweight. It happens when the acceleration is upward. That means the lift is either speeding up while going up, or slowing down while going down.
Feeling lighter: lift accelerating down
Now the acceleration a is downward. Taking down as positive: mg − N = ma, so N = m(g − a). N is smaller than mg, and you feel lighter. This happens when the lift speeds up going down or slows down going up.
Steady speed means a = 0, so N = mg whether the lift goes up or down. What you feel is the start or stop, not the motion.
Weightlessness and free fall
If the lift's downward acceleration is a = g, then N = m(g − g) = 0. The floor falls away as fast as you fall, so you do not press on it. You seem to float. This is weightlessness.
Your weight (the pull of gravity) has not gone to zero. Weightlessness means the apparent weight is zero. This is why astronauts in an orbiting station float: the station and the astronauts are both in free fall around the Earth. Other examples: a person on a drop ride in free fall, a diver in mid-jump, and water in a bottle that you drop (it will not leak while falling).
Try it at home
Make a small hole near the bottom of a plastic bottle and fill it with water. A stream flows out. Now let the bottle fall (outside, or over a tub). The stream stops while it falls, because water and bottle fall together. Second try: stand on a bathroom scale and bend your knees quickly. The reading drops, then rises as you stop.
Key formulas and definitions
- W = mg (true weight)
- Lift accelerating up: N = m(g + a)
- Lift accelerating down: N = m(g − a)
- Free fall (a = g down): N = 0, weightless
- Steady speed: a = 0, N = mg
- Rope tension of a lift with total mass M accelerating up: T = M(g + a)
Worked examples
1. A 60 kg person stands on a scale in a lift that accelerates up at 2 m/s². What does the scale read? (g = 10 m/s²)
N = m(g + a) = 60 × (10 + 2) = 720 N. The person feels heavier.
2. The same person is in a lift that accelerates down at 2 m/s². Find the reading.
N = m(g − a) = 60 × (10 − 2) = 480 N. The person feels lighter.
3. The lift moves up at a steady 3 m/s. What does the 60 kg person's scale read?
Steady speed means a = 0. N = mg = 60 × 10 = 600 N.
4. A 50 kg student's scale reads 400 N in a lift. Find the acceleration and its direction.
N = m(g + a) gives 400 = 50 × (10 + a). So 10 + a = 8 and a = −2 m/s². The minus sign means 2 m/s² downward.
5. A 500 kg lift carries a 60 kg person and speeds up at 1 m/s² upward. Find the rope tension.
Total mass M = 560 kg. T = M(g + a) = 560 × 11 = 6160 N.
6. An astronaut of mass 70 kg orbits at a height where g is 8.7 m/s². What is her weight, and what does a scale read?
Weight = mg = 70 × 8.7 = 609 N, so gravity is still strong. A scale inside the freely falling station reads N = 0, because she and the scale fall together.
Common mistakes
- Saying weight becomes zero in space. Gravity is still there; only the apparent weight is zero in free fall.
- Mixing up speed and acceleration. Steady speed up or down gives N = mg.
- Using N = m(g + a) for a downward acceleration without putting a minus sign on a.
- Thinking a scale measures mass. It measures the push N between you and the scale.