Normal Force in an Elevator

What I want to do
in this video is

think about how the
normal force might

be different in
different scenarios.

And since my 2 and 1/2-year-old
son is obsessed with elevators,

I thought I would
focus on those.

So here I've drawn
four scenarios.

And we could imagine
them almost happening

in some type of a sequence.

So in this first
picture right over here,

I'm going to assume that
the velocity is equal to 0.

Or another way to think about it
is this elevator is stationary.

And everything we're going to
be talking about in this video,

I'm talking about in
the vertical direction.

That's the only dimension
we're going to be dealing with.

So this is 0 meters per second
in the vertical direction.

Or another way to think about
it, this thing is not moving.

Now also it is
also-- and this may

be somewhat obvious to
you-- but its acceleration

is also 0 meters
per second squared

in this picture right over here.

Then let's say that I'm sitting
in this transparent elevator.

And I press the button.

So the elevator begins
to accelerate upwards.

So in this video
right over here,

or in this screen
right over here,

let's say that the acceleration
is 2 meters per second.

And I'll use the convention
that positive means

upwards or negative
means downwards.

We're only going to be operating
in this one dimension right

here.

I could write 2 meters per
second times the j unit vector

because that tells us
that we are now moving.

Why don't we just
leave it like that.

That tells us that we are
moving in the upward direction.

And let's say we do
that for 1 second.

And then we get to this
screen right over here.

So we had no velocity.

We move.

We accelerate.

Let me-- oh, this is 2
meters per second squared.

Let me make sure I--
It's 2 meters per second.

This is acceleration here.

So we do that for 1 second.

And then at the end of 1
second, we stop accelerating.

So here, once we get to this
little screen over here,

our acceleration
goes back to 0 meters

per second squared
in the j direction,

only you don't have to write
that because it's really

just 0.

But now we have some velocity.

We did that just for
the sake of simplicity.

Let's say this screen
lasted for 1 second.

So now our velocity is going
to be 2 meters per second

in the j direction, or
in the upwards direction.

And then let's say we
do that for 10 seconds.

So at least at the
constant velocity,

we travel for 20 meters.

We travel a little bit while
we're accelerating, too.

But we're getting
close to our floor.

And so the elevator
needs to decelerate.

So then it decelerates.

The acceleration here
is negative 2 meters

per second squared times--
in the j direction.

So it's actually
accelerating downwards now.

It has to slow it down to
get it back to stationary.

So what I want to do
is think about what

would be the normal
force, the force

that the floor of the
elevator is exerting

on me in each of
these situations.

And we're going to assume
that we are operating

near the surface of the Earth.

So in every one of
these situations,

if we're operating near
the surface of the Earth,

I have some type of
gravitational attraction

to the Earth and the
Earth has some type

of gravitational
attraction to me.

And so let's say that
I'm-- I don't know.

Let's just make the math simple.

Let's say that I'm
some type of a toddler.

And I'm 10 kilograms.

So maybe this is
my son, although I

think he's 12 kilograms.

But we'll keep it simple.

Oh, let me be clear.

He doesn't weigh 10 kilograms.

That's wrong.

He has a mass of 10 kilograms.

Weight is the force
due to gravity.

Mass of the amount of stuff,
the amount of matter there is.

Although I that's not
a rigorous definition.

So the mass of the individual,
of this toddler sitting

in the elevator,
is 10 kilograms.

So what is the force of gravity.

Or another way to
think about it,

what is this person's weight?

Well, in this vignette
right over here,

in this picture right
over here, its mass

times the gravitational
field near the surface

of the Earth, the 9.8
meters per second squared.

Let me write that over here.

The gravitational field near
the surface of the Earth

is 9.8 meters per
second squared.

And the negative tells
you it is going downwards.

So you multiply this
times 10 kilograms.

The downward force,
the force of gravity,

is going to be 10 times negative
9.8 meters per second squared.

So negative 98 newtons.

And I could say that that's
going to be in the j direction.

Well, what's going to be the
downward force of gravity here?

Well, it's going to
be the same thing.

We're still near the
surface of the Earth.

We're going to assume that the
gravitational field is roughly

constant, although
we know it slightly

changes with the distance
from the center of the Earth.

But when we're dealing
on the surface,

we assume that it's
roughly constant.

And so what we'll assume we have
the exact same force of gravity

there.

And of course, this person's
mass, this toddler's mass,

does not change, depending
on going up a few floors.

So it's going to have the same
force of gravity downwards

in every one of
these situations.

In this first
situation right here,

this person has no acceleration.

If they have no acceleration
in any direction,

and we're only
concerning ourselves

with the vertical
direction right here,

that means that there must
be no net force on them.

This is from Newton's
first law of motion.

But if there's no
net force on them,

there must be some force that's
counteracting this force.

Because if there
was nothing else,

there would be a net force of
gravity and this poor toddler

would be plummeting to
the center of the Earth.

So that net force
in this situation

is the force of the floor
of the elevator supporting

the toddler.

So that force would
be an equal force

but in the opposite direction.

And in this case, that
would be the normal force.

So in this case,
the normal force

is 98 newtons in
the j direction.

So it just completely
bounces off.

There's no net force
on this person.

They get to hold their
constant velocity of 0.

And they don't plummet to
the center of the Earth.

Now, what is the net force
on this individual right

over here?

Well, this individual
is accelerating.

There is acceleration
going on over here.

So there must be some
type of net force.

Well, let's think about
what the net force must

be on this person, or on
this toddler, I should say.

The net force is going to
be the mass of this toddler.

It's going to be 10 kilograms
times the acceleration

of this toddler, times 2 meters
per second squared, which

is equal to 20 kilogram meters
per second squared, which

is the same thing as
20 newtons upwards.

20 newtons upwards
is the net force.

So if we already have
the force due to gravity

at 98 newtons downwards--
that's the same thing here;

that's that one
right over there,

98 newtons downwards-- we need a
force that not only bounces off

that 98 newtons downwards to
not only keep it stationary,

but is also doing
another 20 newtons

in the upwards direction.

So here we need a force
in order for the elevator

to accelerate the toddler
upwards at 2 meters per second,

you have a net force
is positive 20 newtons,

or 20 newtons in the
upward direction.

Or another way to think about
it, if you have negative 98

newtons here, you're going
to need 20 more than that

in the positive direction.

So you're going to need 118
newtons now in the j direction.

So here, where the elevator
is accelerating upward,

the normal force is
now 20 newtons higher

than it was there.

And that's what's allowing
this toddler to accelerate.

Now let's think
about this situation.

No acceleration, but
we do have velocity.

So here we were stationary.

Here we do have velocity.

And you might be
tempted to think,

oh, maybe I still
have some higher force

here because I'm moving upwards.

I have some upwards velocity.

But remember Newton's
first law of motion.

If you're at a
constant velocity,

including a constant
velocity of 0,

you have no net force on you.

So this toddler right over
here, once the toddler

gets to this stage,
the net forces

are going to look
identical over here.

And actually, if you're
sitting in either this elevator

or this elevator, assuming it's
not being bumped around it all,

you would not be able
to tell the difference

because your body is
sensitive to acceleration.

Your body cannot sense its
velocity if it has no air,

if it has no frame of reference
or nothing to see passing by.

So to the toddler
there, it doesn't

know whether it is
stationary or whether it

has constant velocity.

It would be able to
tell this-- it would

feel that kind of
compression on its body.

And that's what its nerves
are sensitive towards,

perception is sensitive to.

But here it's identical
to the first situation.

And Newton's first law tells
there's no net force on this.

So it's just like
the first situation.

The normal force, the force of
the elevator on this toddler's

shoes, is going to be
identical to the downward force

due to gravity.

So the normal force here
is going to be 98 newtons.

Completely nets out the
downward, the negative 98

newtons.

So once again, this
is in the j direction,

in the positive j direction.

And then when we
are about to get

to our floor, what is happening?

Well, once again we
have a net acceleration

of negative 2 meters per second.

So if you have a negative
acceleration, so once again

what is the net force here?

The net force over
here is going to be

the mass of the
toddler, 10 kilograms,

times negative 2
meters per second.

And this was right here
in the j direction.

That's the vertical direction.

Remember j is just
the unit vector

in the vertical
direction facing upwards.

So negative 2 meters per second
squared in the j direction.

And this is equal to negative
20 kilogram meters per second

squared in the j direction, or
negative 20 newtons in the j

direction.

So the net force on this
is negative 20 newtons.

So we have the force of
gravity at negative 98 newtons

in the j direction.

So we're fully
compensating for that

because we're still going
to have a net negative force

while this child
is decelerating.

And that negative net force
is a negative net force

of-- I keep repeating
it-- negative 20.

So we're only going to have
a 78 newton normal force here

that counteracts all but
20 newtons of the force

due to gravity.

So this right over
here is going to be

78 newtons in the j direction.

And so I really want
you to think about this.

And I actually really want you
to think about this next time

you're sitting in the elevator.

The only time that you realize
that something is going on

is when that elevator is
really just accelerating

or when it's just decelerating.

When it's just accelerating,
you feel a little bit heavier.

And when it's just decelerating,
you feel a little bit lighter.

And I want you to think a
little bit about why that is.

But while it's moving
at a constant velocity

or is stationary, you
feel like you're just

sitting on the surface
of the planet someplace.