Video Transcript
A woman with a mass of 55 kilograms
stands on a weighing scale that is on the floor of a descending elevator as shown in
the diagram. The elevator is descending with an
acceleration of 9.8 meters per second squared. What is the reading on the weighing
scale?
The diagram shows a woman standing
on an elevator with a scale at the bottom on the floor and an arrow representing the
acceleration downwards. Let’s label the information from
our problem on our diagram. The problem told us that the woman
on the elevator has a mass of 55 kilograms and that the value of the acceleration is
9.8 meters per second squared in a downward direction. We’re being asked to find the
reading on the scale, which is labeled as our normal force in the diagram. This is the reaction force of the
scale on the woman. And it’s pointing in an upward
direction. We have also included the weight of
the woman or the force due to gravity pointing downwards as it points towards the
center of the Earth.
To determine what the reading is on
the weighing scale, we need to begin with Newton’s second law. Newton’s second law says that the
net force on an object is equal to the mass of the object times the acceleration of
the object. To find the net force, we add our
two forces together as represented by the left side of our equation. The normal force or the reading on
the scale is positive as it’s in an upward direction. And we typically choose upwards to
be positive. The force of gravity is negative as
it is in the opposite direction pointing downwards.
On the right side of the equation,
our mass was given to us as 55 kilograms and our acceleration was negative 9.8
meters per second squared, where the negative is because the acceleration is in a
downward direction. Multiplying out the right-hand side
of the equation, we get negative 539 newtons. To continue solving our problem,
we’ll need to replace the weight of the woman with an expression based on the
values. We need to remember the weight or
the force due to gravity is equal to the mass of the object times the acceleration
due to gravity. For this problem, the mass of the
woman is 55 kilograms and the acceleration due to gravity is 9.8 meters per second
squared.
Multiplying out our two values, we
once again get 539 newtons. To solve for the reaction force of
the scale on the woman, we need to add 539 newtons to both the left side of the
equation as well as the right side of the equation. This cancels out the 539 newtons on
the left side equation as well as on the right side of the equation, leaving us with
zero newtons. The reading on the weighing scale
on the floor of a descending elevator at a rate of 9.8 meters per second squared is
zero newtons. As the elevator is falling at the
same rate as 𝑔, it is in free fall, and the woman inside is weightless, which is
why the contact force 𝐹 𝑁 is measured to be zero.