Video Transcript
Fill in the blank with less than,
greater than, equal to, less than or equal to, or greater than or equal to. When a body of mass 𝑚 is suspended
from a spring scale fixed at the ceiling of an elevator and the elevator descends
with uniform acceleration, then the apparent weight what the real weight.
Let’s begin by thinking about what
is actually happening here. We have a body suspended from a
spring scale fixed to a ceiling. The reading on the scale gives the
apparent weight. And we can think about this as the
upward force in newtons. Then the real weight is the
downward force of the mass of the body due to acceleration. That real weight is generally
calculated by multiplying mass times gravity, 𝑚𝑔. Then we can calculate the net force
by finding the sum in a given direction of the apparent weight and the real
weight.
The direction in which we choose to
find the sum is somewhat arbitrary, but it can be easier to think about the
direction in which the object is accelerating. In this case, we’re told that the
elevator is descending with uniform acceleration. So, we might define the downwards
direction in this case to be positive. Then we can link all of these
elements using Newton’s third law of motion: net force is equal to mass times
acceleration. Since we define downwards to be
positive, we can say here that the real weight minus the apparent weight is the net
force. This is equal to mass times
acceleration. So, if we define the acceleration
to be 𝑎, this net force is equal to 𝑚𝑎.
But remember, the elevator is
descending with uniform acceleration, so we can say that 𝑎 must be positive since
we defined the downwards direction to be positive. Since mass, of course, is also
positive, we can say that mass times acceleration must also be positive. It’s greater than zero. If 𝑚𝑎 is positive, then we can
also say that the net force must also be greater than zero. In other words, the real weight
minus the apparent weight must be greater than zero. We can rearrange this inequality by
adding apparent weight to both sides. When we do, we get the following
inequality. The real weight must be greater
than the apparent weight. In other words, the apparent weight
is going to be less than the real weight. And so, we can fill in the
blank.
This means that when a body of mass
𝑚 is suspended from a spring scale fixed at the ceiling of an elevator and the
elevator descends with uniform acceleration, then the apparent weight must be less
than the real weight. We fill in the blank with the “less
than” symbol.
