5.9: Apparent Weight

True weight is the measure of the gravitational force acting on an object. However, if the object accelerates, its measured weight is different from its true weight. Similar observations can be made when the object is submerged in water. An object's weight in water is its apparent weight, which is equal to the difference between its true weight and the buoyant forces.

Consider a person standing on a bathroom scale inside an elevator. If the scale is accurate at rest, its reading equals the magnitude of the force the person exerts downward on it. The only forces acting on the person are their weight and the upward normal force of the scale. At rest, the net force on the person is zero. According to Newton's third law, weight and normal force are equal in magnitude and opposite in direction, so the normal force equals the true weight of the person. The bathroom scale reads the normal force exerted by the scale on the person.

If the elevator accelerates upward with constant acceleration, the apparent weight is more than the true weight as the net force acting on it is greater than zero. However, if the elevator decelerates, the apparent weight is less than the true weight of the object as the net force acting on the object is negative. If the elevator is in free-fall, the person feels weightless as they and the elevator are falling under gravitational acceleration. This phenomenon is called apparent weightlessness.

The "weightlessness" experienced by people in a satellite orbit close to the Earth is the same apparent weightlessness experienced in a free-falling elevator. The force of gravity causes the satellite to "fall" out of its natural straight-line path. Thus, although the force of gravity acts on objects within the satellite, the objects experience an apparent weightlessness because they and the satellite are accelerating together as in free fall.

Consider an elevator at rest, having a box on a weighing scale. The gravitational force on the box equals the normal force exerted by the scale on the box, such that the magnitude of the net force on the box is zero. The scale reads the magnitude of the normal force, which is the true weight of the box.

If the elevator moves upward with a constant acceleration, the net magnitude of the force on the box equals the difference between the normal force and the gravitational force on it. The scale reads the apparent weight higher than the true weight of the box.

When the elevator accelerates downward, conventionally, the acceleration becomes negative. The apparent weight is now lower than the true weight of the box.

Now, if the elevator's cable breaks, the box with the scale will momentarily hover but eventually accelerate downward due to gravity, experiencing a free fall.

The scale reads zero, and the box appears weightless. This phenomenon is termed apparent weightlessness.