A little girl of mass m is standing on a bathroom scale inside an elevator. Determine which “weight” the scale will indicate in the following situations:
- When the elevator goes up with an acceleration of magnitude a.
- When the elevator goes down with an acceleration of magnitude a.
- When the elevator is at rest.
Solve the problem with respect to an observer at rest and later with respect to an observer who moves with the elevator.
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First we will solve the problem with respect to an inertial observer O (resting on the ground).
In the statement of the problem the word weight has been quoted because what the scale really measures is the normal force that the girl does on it. The weight is the gravitational force that the Earth exerts on the bodies that are close to its surface. Therefore, the girl’s weight will be applied to her, not to the scale.
On the other hand, the normal that the girl does on the scale (N12) is the normal reaction of the scale on the girl N21 (Newton’s third law). So we will calculate the latter using Newton’s second law.
In the following figure are represented the forces that act on the girl:
With respect to the inertial observer, the girl ascends with the same acceleration as the elevator, so Newton’s second law is:
And projecting on the y-axis:
Note that, although we have not explicitly represented the vertical axis in the drawing, the unit vector j associated with the observer O indicates the positive direction of that axis.
From equation (1) we can isolate the magnitude of the normal N21:
This is the force that the scale will read.
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If the elevator descends with an acceleration of the same magnitude but opposite direction as in the previous scenario, Newton’s second law is the same, but the projection on the y-axis changes, since now the acceleration has a negative projection on this axis:
From equation (2) we can isolate the magnitude of the normal N21 in this scenario:
If the elevator is at rest the acceleration of the girl is zero, so the projection of Newton’s second law on the y-axis will be:
And the magnitude of the normal N21 for this case is:
Observe that the scale will read a different “weight” depending on the acceleration of the elevator. If it is at rest, the scale will indicate the value of the weight (gravitational force). However, if the elevator ascends with acceleration the scale will read a greater “weight” and if it descends with acceleration the “weight” will be lower.
If the elevator ascended with constant speed the scale would read the same weight as if the elevator was at rest.
Next we will solve the problem regarding an observer O’ that moves with the elevator.
When the elevator has acceleration, O’ is a non-inertial observer.
When Newton deduced his three laws, he did so with respect to a reference frame at rest (inertial). Therefore, in principle we could not apply Newton’s second law to solve a problem from the point of view of an accelerated (non-inertial) observer. However, we can do it if we include the apparent forces that act on the mass due to the observer’s own movement. These are the so-called inertial force (or fictitious forces or pseudo forces).
The inertial forces are always proportional to the mass on which they act.
Returning to the resolution of the problem, we start by drawing the forces that act on the little girl:
If you compare this drawing with the one we made in the case of the inertial reference frame, you can see that now a new force Fi is acting on the girl, which did not appear before. This inertial force does not correspond to any physical interaction that the girl is experiencing; it is simply a consequence of the movement of the observer O’.
With respect to O’ the acceleration of the girl is zero, so Newton’s second law is written:
Projecting on the y-axis we get:
The force magnitude Fi is proportional to the mass of the girl and to the acceleration of the observer:
After substituting this magnitude in equation (4) we obtain:
And after isolating we obtain the magnitude of the normal force:
If you compare equation (1) with (5) you will see that the results we have obtained for the two observers are identical.
If the elevator is at rest or moves with a constant speed, the observer that moves with it is inertial and therefore the inertial forces do not appear. The problem would be resolved as we did with respect to O.
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