The elevator in a hotel starts moving up, and for a moment Imre feels heavier.
The feeling is honest physics. For that moment, the floor's normal force on him really is larger than usual. A bathroom scale under his feet would show the extra.
A scale does not read your weight. It reads the normal force it exerts on you.
Standing on solid ground, the two match. In an accelerating elevator they do not, and a = F_net/m says exactly what the scale shows.
Here is the scene: a 60 kg person stands on a scale in an elevator. Take g = 9.8 m/s². Two forces act on the person: the normal force from the scale, upward, and the weight, downward.
The person's weight is w = mg = 60 × 9.8 = 588 N, downward.
Now the elevator accelerates upward at 1.0 m/s². The person accelerates with it, so the net force on the person points upward.
Rearranging a = F_net/m gives F_net = m × a, so the net force is 60 × 1.0 = 60 N, upward.
An upward net force of 60 N means the upward force beats the downward force by 60 N. So the normal force is 588 N + 60 N = 648 N.
The scale reads that normal force: 648 N, more than the 588 N weight. Imre feels heavier because the scale really does exert more force on him.
Accelerating downward at 1.0 m/s² flips the difference. The net force points downward, so the weight must win by those 60 N, and the normal force drops to 588 N − 60 N = 528 N.
The same recipe runs in every vertical case. Find the weight, w = mg. Find the net force's size, F_net = m × a, pointing the way the acceleration points. Then size the supporting force so that the winning side wins by exactly that much.
In short: accelerating upward, the supporting force is w + ma. Accelerating downward, it is w − ma. At rest, or moving at a steady speed, it equals w.
The supporting force can be a normal force or a tension. A crate hanging from a cable in the same elevator follows the same rule, with the cable's tension playing the supporting part.
A 50 kg crate hangs from a cable in an elevator accelerating upward at 2.0 m/s². Calculate the tension in the cable. Take g = 9.8 m/s².
Write down the values in the question:
m = 50 kg
a = 2.0 m/s², upward
g = 9.8 m/s²
tension = ?
Write down the equation:
tension = mg + ma (accelerating upward: supporting force = w + ma)
Substitute in the values:
w = 50 × 9.8 = 490 N
ma = 50 × 2.0 = 100 N
tension = 490 N + 100 N
Simplify:
tension = 590 N
The tension in the cable is 590 N.
An elevator accelerates upward at 1.2 m/s². A 70 kg person stands on a scale on its floor. Enter the scale reading in N. Take g = 9.8 m/s².
An 80 kg person stands on a scale in an elevator accelerating downward at 1.5 m/s². Calculate the scale reading. Take g = 9.8 m/s².
Write down the values in the question:
m = 80 kg
a = 1.5 m/s², downward
g = 9.8 m/s²
scale reading (normal force) = ?
Write down the equation:
reading = mg − ma (accelerating downward: supporting force = w − ma)
Substitute in the values:
w = 80 × 9.8 = 784 N
ma = 80 × 1.5 = 120 N
reading = 784 N − 120 N
Simplify:
reading = 664 N
The scale reads 664 N.
An elevator accelerates downward at 2.3 m/s². A 40 kg box of books rests on a scale on its floor. Enter the scale reading in N. Take g = 9.8 m/s².
A 25 kg crate hangs from a rope in an elevator accelerating upward at 1.8 m/s². Enter the tension in the rope in N. Take g = 9.8 m/s².
An elevator accelerates upward at 1.4 m/s². A 55 kg student stands on a scale on its floor. Enter the scale reading in N. Take g = 9.8 m/s².
An elevator accelerates downward at 0.8 m/s². A 90 kg crate rests on its floor. Enter the normal force from the floor on the crate in N. Take g = 9.8 m/s².
A 30 kg toolbox hangs from a cord in an elevator accelerating downward at 1.3 m/s². Enter the tension in the cord in N. Take g = 9.8 m/s².