Understanding Forces and Motion in Elevators

When analyzing a person in an elevator, we choose the person as our "system of interest." For this person, there are typically two primary vertical forces:

1. Weight ($w$): The force of gravity pulling the person downward, calculated as $w = mg$.
2. Force of the scale ($F_{\text{s}}$): The upward normal force exerted by the scale on the person.

Fig 4.30: (b) The free-body diagram shows only the external forces acting on the designated system of interest—the person.

By focusing only on the person, we ignore the complex forces acting on the elevator car itself (like the tension in the cables or the weight of the car) and simplify the problem into a one-dimensional analysis.

Exam Tip: When drawing a free-body diagram, only include forces acting on the object you are studying. Do not include forces the object exerts on its surroundings.

Calculating Scale Readings

A common misconception is that a scale measures your actual weight ($mg$). In reality, a scale measures the normal force it must exert upward to support you. If the elevator is accelerating, the scale must push harder (or softer) than usual to satisfy Newton's Second Law.

Case 1: Upward Acceleration

If the elevator accelerates upward, the net force must be directed upward. Using our FBD, where upward is positive:

Solving for the scale reading ($F_{\text{s}}$):

Because $ma$ is positive, the scale reading $F_{\text{s}}$ will be greater than your actual weight. This explains why you feel "heavier" when the elevator starts its ascent.

Case 2: Constant Velocity

When the elevator moves at a constant speed (whether upward or downward), the acceleration $a$ is exactly zero. Substituting this into our equation:

In this state, the scale reading is exactly equal to your weight. You will feel "normal" during any period of constant motion.

Case 3: Downward Acceleration

If the elevator accelerates downward, the acceleration $a$ is negative relative to our upward-positive coordinate system.

Here, the scale reading is less than your actual weight. In the extreme case of free-fall (where $a = g$), the scale reading becomes zero, and you would experience weightlessness.

Worked Example: The 75 kg Passenger

To solidify these concepts, let's apply the formulas to a specific scenario.

Problem: A 75.0 kg man stands on a scale in an elevator. Calculate the scale reading if: (a) The elevator accelerates upward at $1.20 \text{ m/s}^2$. (b) The elevator moves upward at a constant speed of $1 \text{ m/s}$.

Solution for (a): Using $F_{\text{s}} = ma + mg$:

The scale reads 825 N, which is significantly higher than his stationary weight of 735 N.

Solution for (b): Since the velocity is constant, $a = 0$:

The scale reads exactly his weight.

Summary Table of Elevator Dynamics