Apparent Weight

When you stand on a scale, what does it measure? The scale reads the normal force $N$ that it exerts upward on you. This is your apparent weight.

Why does the scale read the normal force? By Newton's Third Law, you push down on the scale, and the scale pushes back up on you with equal force. The scale's mechanism responds to the force you exert and displays that force as the reading.

When you're at rest or moving at constant velocity, Newton's Second Law tells us the net force is zero:

$N - mg = 0$, so $N = mg$.

The scale reads your actual weight.

Vertical Acceleration (Elevators):

You've probably experienced the sensation of feeling heavier or lighter when riding in an elevator. When an elevator starts moving upward, you feel pressed into the floor—you feel heavier. When it starts moving downward, you feel lighter, almost as if you're floating.

This sensation reflects your apparent weight changing, even though your actual weight $mg$ stays the same.

Let's analyze what happens. When an elevator accelerates vertically with acceleration $a_y$, we apply Newton's second law in the vertical direction.