## Weight of an Object in an Elevator

The following example is a practical application that shows variations in physical quantities associated with those quantities being measured in a non-inertial frame of reference. Remember, that inertial frames of reference were  studied in the section related to the law of inertia in this notes.

When the elevator is at rest, the weight of the woman is her actual weight , .

What happens with the weight of the woman if the elevator is moving with constant velocity? . Just as in the case when the elevator is at rest, the forces acting on the woman are the pulling of the Earth (weight) and the normal force applied by the scale over the woman. As always, the normal force is the reaction of the surface to the pushing down of the woman on the scale. The value of the normal force is the reading of the scale. From Figure 2, the net force acting on the woman is

Applying Newton's second law to the motion,

Thus, the net force is zero implying that,

An the reading of the scale will correspond to the actual weight of the woman.

Let us know consider the case when the elevator is accelerating upward; notice that the elevator velocity can still be upward or downward .

The elevator can accelerate upward in two cases:

1. Elevator at rest in a lower floor starting moving upward; in this case, the acceleration and velocity are upward.

2. Elevator is arriving to a lower floor; in this case, the acceleration is upward and the velocity is downward.

There is no difference in the equation for the normal force, Equation 1. However, Newton's second law applied to this case provides a different result,

Thus,

Notice that in this case, Equation 3, the normal force is greater than in the previous cases, Equation 2. The scale will indicate a greater mass when accelerating upward than under non-accelerating conditions.

Finally, let us solve the case when the elevator is accelerating downward, of the applet. Just as in the case of elevator accelerating upward, the elevator can accelerate downward when moving upward or downward.

The elevator can accelerate downward in the following two cases:

1. Elevator is at rest in a higher floor than destination and elevator is starting to move downward.

2. Elevator is moving upward and needs to slow down when arriving to destination.

The equation for the normal force still is , while Newton's second law applied to this case changes to

In the previous relations, the acceleration a is considered positive. The negative sign in front  indicates the fact that this acceleration is pointing downward. Thus, the equation equivalent to Equation 3 is

Equation 4

Since the normal force acting on the woman is smaller than the acceleration of gravity, this results correspond to the scale indicating a lower value for the mass compared to the case of the non-accelerating elevator. What should the value of the acceleration be in order for the reading of the scale to be zero? What does this mean?

Calculate the acceleration of the elevator in order for the mass scale read 80 Kg as indicated in Figure 3 of the applet.

 a) 2.7 m/s2 b) 9.8 m/s2 c) 4.9 m/s2 N d) 3.3 m/s2 e) None of the above.
Solution:

The result of Equation 3 should be read as where mInertial is the mass of the person in the inertial frame of reference (non accelerating frame). From this equation, the acceleration of the elevator can be calculated,

Based on the result of Equation 2, the normal force should be replaced by where mNon-Inertial is the mass of the person detected by the scale when the elevator is accelerating. Thus, the acceleration is

This solution is also valid in the case of the elevator is accelerating downward; in which case, the negative sign for the acceleration is just a remainder that the acceleration is pointing down in agreement with our conventions for the signs of vectors. For the present problem, the numerical solution is

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com