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

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:

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

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,

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:

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

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/s^{2
}

b)

9.8 m/s^{2
}

c)

4.9 m/s^{2
}

N

d)

3.3 m/s^{2
}

e)

None of the above.

Solution:

The result of Equation 3 should be read as
where
m_{Inertial} 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
m_{Non-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