Apparent Weight: Person on Scale in
Elevator
A person with mass, m, who is located at or near the surface
of the Earth will always have some
weight W=mg. When a person stands on a scale, the reading
(the number of pounds or newtons)
on the scale is actually the Normal Force that the scale
exerts back towards the person to support
the person's weight. (Note that the person and the scale are
stationary relative to each other, in
other words they are always in contact with each other, so
they always have equal and opposite
action and reaction forces acting between them.)
Things get complicated, though, when the scale and the
person experience acceleration. This
will change the contact force (the Normal Force) between the
person and the scale.
Let's look at several cases. We will assume that Up is the
positive direction and Down is the
negative direction.
Case 1: No
acceleration of elevator
If the acceleration of the elevator is zero, then there are
two possible
scenarios; the elevator can be at rest (stationary, zero
velocity) or moving
with a constant speed (no acceleration if velocity does not
change).
In this case, the action and reaction force pair between the
person and the
scale is just the weight. The person pushes down on the
scale with a force of
-W=-mg (negative direction) and the scale pushes back up
against the man
with a Normal Force of FN = +W = +mg. Because the reading on
the scale is
the magnitude of the normal force, the scale will read the
true weight when
the elevator is NOT accelerating.
Case 2: going up & speeding up
(acceleration a is positive (up))
In this case, the elevator and the person are starting from
rest at a lower
floor. The elevator accelerates upward. The inertia of the
person would
prefer to stay stationary, so the elevator floor and scale
must push up on
the person to accelerate him upward along with the elevator.
(The person
doesn't sink into the floor when the elevator accelerates
up. The elevator
and the scale and the person all move together.)
The scale therefore has to push upward with extra force on
the person to
accelerate the person's mass upward. This results in a
greater contact
force between the scale and the person. Therefore the Normal
Force is
larger, so the reading on the scale is a number that is
GREATER than the
true weight.
Let's consider Newton's 2nd
Law (ΣF=ma) acting on
the person. The overall acceleration of the
person is upward (with the elevator). So ma is positive
(upward). The only external forces acting on the person are the force of
gravity acting down (-W=-mg) and the supporting Normal
Force FN that the scale applies upward on the person. So
ΣF=ma= -mg+FN . We want to know
FN because that is the number that we read off the scale. FN
=mg+ma, which is GREATER than
the true weight.
Case 3: going up & slowing down
(acceleration a is negative (down))
In this case, the elevator and the person are initially
moving upward at a
constant speed and slowing down to rest at a higher floor.
The
acceleration of the elevator is downward (opposite to the
upward motion,
which causes a reduction of the velocity). The inertia of
the person
would prefer to keep moving upward at a constant speed, so
the elevator
floor and scale effectively drop out a little bit from
underneath the person
as the elevator slows down.
The person doesn't float upward, because again the elevator
and the
person move together, but the contact force between the
person and the
scale is reduced. The scale therefore has to push upward
with less force
on the person to support the person's weight. Therefore the
Normal Force is smaller, so the
reading on the scale is a number that is LESS than the true
weight.
Let's consider Newton's 2nd
Law (ΣF=ma) acting on
the person. The overall acceleration of the
person is downward (with the elevator). So ma is negative
(downward). The only external
forces acting on the person are the force of gravity acting
down (-W=-mg) and the supporting
Normal Force FN that the scale applies upward on the person.
So ΣF= -ma= -mg+FN . We want
to know FN because that is the number that we read off the
scale. FN =mg - ma, which is LESS
than the true weight.
Case 4: going down & slowing down
(acceleration a is positive (up))
In this case, the elevator and the person are initially
moving downward at
a constant speed and then slow to rest at a lower floor. The
elevator
accelerates upward (opposite direction to negative/downward
velocity to
reduce velocity magnitude). The inertia of the person would
prefer to
keep moving downward at the constant speed, so the elevator
floor and
scale must push up on the person to accelerate him upward,
slowing him
down. (The person doesn't sink into the floor here either.
Elevator and
scale and person move together.)
The scale therefore has to push upward with extra force on
the person to
accelerate the person's mass upward. This results in a
greater contact
force between the scale and the person. Therefore the Normal
Force is
larger, so the reading on the scale is a number that is
GREATER than the
true weight.Let's consider Newton's 2nd
Law (ΣF=ma) acting on
the person. The overall acceleration of the
person is upward (with the elevator). So ma is positive
(upward). The only external forces
acting on the person are the force of gravity acting down
(-W=-mg) and the supporting Normal
Force FN that the scale applies upward on the person. So
ΣF=ma= -mg+FN . (Note that this is
the same equation as we got in case 2.) We want to know FN
because that is the number that we
read off the scale. FN =mg+ma, which is GREATER than the
true weight.
Case 5: going down & speeding up
(acceleration a is negative (down))
In this case, the elevator and the person are initially at
rest at a higher
floor. The elevator then speeds up in the downward direction
towards a
lower floor. The elevator acceleration of the elevator is
negative/downward (increasing the velocity magnitude in the
downward
direction). The inertia of the person would prefer to stay
at rest, so the
elevator floor and scale effectively drop out a little bit
from underneath
the person as the elevator accelerates down.
The person doesn't float upward here also, because again the
elevator and
the person move together, but the contact force between the
person and
the scale is reduced. The scale therefore has to push upward
with less
force on the person to support the person's weight.
Therefore the Normal Force is smaller, so the
reading on the scale is a number that is LESS than the true
weight.
Let's consider Newton's 2nd
Law (ΣF=ma) acting on
the person. The overall acceleration of the
person is downward (with the elevator). So ma is negative
(downward). The only external
forces acting on the person are the force of gravity acting
down (-W=-mg) and the supporting
Normal Force FN that the scale applies upward on the person.
So ΣF= -ma= -mg+FN . (Note
that this is the same equation that we got for Case 3.) We
want to know FN because that is the
number that we read off the scale. FN =mg - ma, which is
LESS than the true weight.
Case 6: freefall (a = -g)
If the elevator cable were to break, the whole
elevator-scale-person
system would all begin to accelerate downward due to the
force of
gravity. All objects in freefall accelerate downward with
the same
magnitude (acceleration due to gravity, g). The scale and
the person are
freefalling together, so there is NO contact force (Normal
Force)
between the scale and the person. (When they are both
falling together,
there is no way that the scale can support any of the
person's weight.)
Note that this is a special case of downward acceleration,
which we
discussed in Case 3 and Case 5. Just as in Cases 3 and 5,
the apparent
weight (which is zero when a=-g) is less than the true
weight.A pictorial summary of apparent weight: