The "Elevator Problem" is a classic problem in physics. The situation is this:

"You are standing on a bathroom scale in an elevator. You are holding an apple. (Yes, peoplearestaring at you...) You weigh 500 Newtons, so your mass is about 50 kg."

This assignment is a step-by-step analysis of the elevator
problem. A good deal of the work has been done for you - in which
case it is your job to study the answers given and use them as a
pattern and guide for the answers that you supply. Answers
are available.

You have just boarded the elevator, so it (with you inside) is at rest...

There are 2 forces acting on you. (See the diagram at right.) The Earth pulls down on you with the force we call your weight (= mg) of 500 Newtons.

Since the elevator is at rest, your acceleration is 0
m/s^{2}. Since your acceleration is 0 m/s^{2},
**Newton's First Law** says the net force on you must be
0 Newtons.

Since the net force on you is 0 Newtons, the upward forces and downward forces on you must balance exactly. Therefore the scale must push on you with a force of (1a) _______ Newtons, and the scale must read (1b) _______ Newtons.

The apple would be in free fall, so its acceleration relative to
the earth is 10 m/s^{2} downward. Since you are at rest
relative to the earth, the apple's acceleration relative to you would
be10 m/s^{2} also, so the apple would appear to fall just as
it does anywhere else on the earth.

The
elevator, (with you inside) begins to accelerate upward from rest at
2 m/s^{2}.

There are 2 forces acting on you. (Complete the diagram.) Your weight pulls down with a force of 500 Newtons. The scale pushes up with a force of (3a) ______ (see below) Newtons.

Since your acceleration is 2 m/s^{2} upward, Newton's
Second Law says that there must be a net force pushing you upward,
and the net force has a magnitude F_{net} = ma. So the net
force on you,

F_{net} = (50 kg)(2 m/s^{2}) = 100 Newtons
(upward).

Since the net force on you is100 Newtons, the upward forces and downward forces on you must cancel to leave a 100 Newton upward force. Therefore the scale must push on you with a force of (3b) _______ Newtons, and the scale must read (3c) _______ Newtons as the elevator accelerates upward.

The apple would be in free fall, so its acceleration relative to
the earth is 10 m/s^{2} downward. Since you are accelerating
at 2 m/s^{2} upward relative to the earth, the apple's
acceleration relative to you would be 10 m/s^{2} + 2
m/s^{2} = 12 m/s^{2}, so the apple would appear to
fall *faster* inside the elevator than it does in "normal"
free fall on the earth.

To the occupants of the upwardly accelerating elevator, it
*appears that gravity is stronger*, since they *seem to
weigh more* (why?) and objects fall faster than "normal."

The elevator (and you) accelerated for 5 seconds, so it is moving upward with a velocity of 10 m/s. It now moves with this constant upward velocity of 10 m/s.

There are 2 forces acting on you. (Complete the diagram.) Your weight pulls down with a force of 500 Newtons. The scale pushes up with a force of (5a) ________ (see below) Newtons.

Since the elevator is moving with constant velocity, your
acceleration is (5b) ____ m/s^{2}. Since your acceleration is
(5c) ____ m/s^{2}, Newton's First Law says the net force on
you is (5d) _____ Newtons.

Since the net force on you is (5e) ____ Newtons, the scale must push on you with a force of (5f) _______ Newtons, and the scale must read (5g) _______ Newtons.

The
elevator, (with you inside) begins to slow down as it approaches its
destination. Its acceleration (or deceleration) is 2 m/s^{2}
downward.

There are 2 forces acting on you. (Complete the diagram.) The Earth (your weight) pulls down with a force of (7a) _____ Newtons. The scale pushes up with a force of (7b)______ (see below) Newtons.

Since your acceleration is 2 m/s^{2} downward, Newton's
Second Law says that there must be a net force pulling you downward,
and the net force has a magnitude F_{net} = ma. So the net
force on you,

F_{net} = ((7c) _____ kg)((7d) _____ m/s^{2}) = (7e)
______ Newtons .

Since the net force on you is (7f) _____ Newtons downward, the upward forces and downward forces on you must cancel out to leave a (7g) _____ Newton downward force . Therefore the scale must push on you with a force of (7h) _______ Newtons, and the scale must read (7i) _______ Newtons as the elevator accelerates downward.

To the occupants of the downwardly accelerating elevator, it
*appears that gravity is weaker*, since they *seem to weigh
less* (why?) and objects *fall more slowly* than
"normal."

The
elevator, (with you inside) reaches its floor, stops for a while, and
then begins to accelerate downward. Its acceleration is 2
m/s^{2} downward.

There are 2 forces acting on you. (Complete the diagram.) The Earth (your weight) pulls down with a force of (9a) _____ Newtons. The scale pushes up with a force of (9b)______ (see below) Newtons.

Since your acceleration is 2 m/s^{2} downward, Newton's
Second Law says that there must be a net force pulling you downward,
and the net force has a magnitude F_{net} = ma. So the net
force on you,

F_{net} = ((9c) _____ kg)((9d) _____ m/s^{2}) = (9e)
______ Newtons .

Since the net force on you is (9f) _____ Newtons downward, the upward forces and downward forces on you must cancel out to leave a (9g) _____ Newton downward force. Therefore the scale must push on you with a force of (9h) _______ Newtons, and the scale must read (9i) _______ Newtons as the elevator accelerates downward.

To the occupants of the downwardly accelerating elevator, it
*appears that gravity is weaker*, since they *seem to weigh
less* (why?) and objects *fall more slowly* than
"normal."

The elevator (and you) accelerated for 5 seconds, so it is moving downward with a velocity of 10 m/s. It now moves with this constant downward velocity of 10 m/s.

There are 2 forces acting on you. (Complete the diagram.) Your weight pulls down with a force of 500 Newtons. The scale pushes up with a force of (11a) ________ (see below) Newtons.

Since the elevator is moving with constant velocity, your
acceleration is (11b) ____ m/s^{2}. Since your acceleration
is (11c) ____ m/s^{2}, Newton's First Law says the net force
on you is (11d) _____ Newtons.

Since the net force on you is (11e) ____ Newtons, the scale must push on you with a force of (5f) _______ Newtons, and the scale must read (11g) _______ Newtons.

The elevator cable snaps, and the elevator (with you inside!) begins to fall! Perhaps you have time for one last Physics observation!

There are 2 forces acting on you. (Complete the diagram.) The Earth (your weight) pulls down with a force of (13a) _____ Newtons. The scale pushes up with a force of (13b) _____ (see below) Newtons.

Since the elevator and you are in free fall, your acceleration is
(13c) ____ m/s^{2} downward. Newton's Second Law says that
there must be a net force pulling you downward, and the net force has
a magnitude F_{net} = ma. So the net force on you,

F_{net} = ma = ((13d) _____ kg)((13e) _____ m/s^{2})
= (13f) _____ Newtons.

Since the net force on you = (13g) _____ Newtons downward, the upward forces and downward forces on you must cancel out to leave a (13h) _____ Newton downward force . Therefore the scale must push on you with a force of (13i) _______ Newtons, and the scale must read (13j) _______ Newtons as the elevator accelerates downward.

Weightlessness is a phenomenon that we most often associate with
astronauts in space, but it is not necessary to be floating in space
to be "weightless." Describe what you could do, right here and now,
to be "weightless." (Hint: An elevator is not necessary!)

Answers are available (*really,
no tricks* this time - promise!).

last update November 16, 2009 by JL Stanbrough