Apparent weight Knowledge-Based Study

Study notes

1. True Weight versus Apparent Weight

   In physics, an object's true weight ($w$) is defined as the gravitational force exerted on it by a celestial body (such as Earth):

   $$w = mg$$

   where $m$ is the object's mass and $g$ is the local acceleration due to gravity.

   In contrast, apparent weight ($w_{\text{app}}$) is the magnitude of the supporting contact force acting on the object. When you stand on a bathroom scale, the scale does not directly measure gravity; it measures the normal contact force ($N$) that it exerts upward on your feet to support you. Therefore, your apparent weight equals the normal force:

   $$w_{\text{app}} = N$$

   When an object is in static equilibrium or moving with a constant velocity, the net acceleration is zero, which means $N = mg$. In this specific case, apparent weight equals true weight. However, whenever the system accelerates vertically, apparent weight changes.

   Analyzing an Elevator Scenario

   Consider a person of mass $m$ standing on a scale inside an elevator. Taking the upward direction as positive, two vertical forces act on the person:

   • The upward normal force $N$ exerted by the scale.
   • The downward gravitational force $mg$.

   Applying Newton's second law along the vertical axis:

   $$\Sigma F_y = ma_y$$

   $$N - mg = ma_y$$

   Solving for the normal force, which represents the apparent weight:

   $$N = m(g + a_y)$$

   This fundamental relation reveals how the sensation of heaviness or lightness depends strictly on the vertical acceleration $a_y$, not on the direction of velocity.

   Common Motion States in an Elevator

   It is a common misconception that apparent weight depends on whether an elevator is moving upward or downward. Instead, apparent weight depends solely on the direction of acceleration:

   • Upward Acceleration ($a_y > 0$): If the elevator speeds up while moving upward, or slows down while moving downward, $a_y$ is positive. Then $N = m(g + a_y) > mg$. The scale reading increases, and the person feels heavier than normal.
   • Constant Velocity ($a_y = 0$): Whether moving upward, downward, or at rest, the net force is zero. Then $N = mg$. Apparent weight equals true weight.
   • Downward Acceleration ($a_y < 0$): If the elevator slows down while moving upward, or speeds up while moving downward, $a_y$ is directed downward (negative). Then $N = m(g - |a_y|) < mg$. The scale reading decreases, and the person feels lighter than normal.

   Apparent Weightlessness

   What happens if the elevator cable snaps and the elevator falls freely under gravity?

   In free fall, the downward acceleration of the elevator and the person equals $g$. Setting $a_y = -g$ in Newton's second law:

   $$N = m(g + (-g)) = 0$$

   Because the scale accelerates downward at the same rate as the person, it exerts no normal force on the person's feet. As a result, the scale reads zero, and the person experiences apparent weightlessness.

   Key Takeaway: True weight $mg$ is still fully present because gravity does not turn off. Weightlessness simply means that contact support forces are absent ($N = 0$). This is the exact condition experienced by astronauts in orbit: they are in continuous free fall around Earth, so no normal force is required to support them against a surface.