Normal Force in an Elevator

How the normal force changes when an elevator accelerates

What I want to do in this video is

think about how the normal force might be different in different scenarios

Since my 2.5-year-old son is obsessed with elevators

I thought I would focus on those

So here I've drawn 4 scenarios

We can imagine them almost happening in some type of the sequence

So in this first picture right over here

I am going to assume that the velocity is equal to zero

or another way to think about this is, the elevator is stationary

And everything we're gonna be talking about in this video is the vertical direction

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

So 0 m/s in the vertical dimension

Another way to think about it, this thing is not moving

Now it is also--this may be somewhat obvious to you, but its acceleration is also 0 m/s^2

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 screen right over here, let's say the acceleration is 2 m/s^2

I'll use a convention that positive means upwards or negative means downward

So we're only going to be operating in this one dimension right here

I could write 2 m/s times the j unit vector

because that tells us where we're moving. I'll just live it like that

That tells us we're moving in the upward direction

Let's say we do that for one second and then we get to this screen right over here

So we have no velocity; we accelerate--

oh, this is 2 m/s squared. This is acceleration here

So we do that for one second and at the end of one second, we stop accelerating

So here in this screen over here, our acceleration goes back to 0 m/s^2

in the j direction. That's really just zero

But now we have some velocity

Just for the sake of simplicity, let's say this screen lasted for 1 second

So now our velocity's going to be to 2 m/s in the j direction or in the upward direction

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

So with that constant velocity, we travel for 20 meters

We traveled a little bit when we were accelerating too

When we're getting close to our floor

the elevator needs to decelerate

So it decelerates. The acceleration here is -2 m/s^2 times--in the j direction

It's actually accelerating downward now. 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

We're going to assume that we're 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

I'll just make the math simple--let's say that I am some type of a toddler and I'm 10 kg

So maybe this is my son, although he's 12 kg, but we'll keep it simple

So let me make it clear. He doesn't weigh 10 kg. That's wrong

He has the mass of 10 kg; weight is the force due to the gravity

mass is the amount of matter there is

Although I haven't found the rigorous definition

So the mass of the individual, of this toddler sitting at the elevator is 10 kg

So what is the force of gravity, or another way to think about it, what is this person's weight?

Well, in this picture right over here, it's

mass times the gravitational field near the surface of the earth 9.8 m/s squared

And the negative tells you it's going downwards

So you multiply this times 10 kg

The downward force, the force of gravity is going to be 10 kg times -9.8 m/s squared

So -98 newtons. That's in the j direction

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

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 we'll assume we have the exact same force of gravity there and of course

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

the 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 N in the j direction

So just completely balances off. There's no net force on this person

they get to hold their constant velocity of zero; 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

Let's think about what the net force must be on this person, on this toddler I should say

The net force is going to be the mass of this toddler, 10 kg

times acceleration of this toddler, times 2 m/s squared which is equal to

20 kg m / s squared, which is the same thing as 20 N, 20 N upwards

20 N upwards is the net force

So if we already have the force due to gravity at 98 N downwards

That is the same thing here

90 N downwards

We need a force that not only balances off that 90 N downwards

not only keep stationary, but also doing another 20 N in the upward direction

Here we need a force in order for the elevator to accelerate this toddler upwards at 2 m/s

You have a net force of positive 20 N, or 20 N in the upward direction

Or another way to think about it

if you have -98 N here

you're going to need 20 more than that in the positive direction

So you're going to need 118 N, now in the j direction

So here where the elevator is accelerating upward

the normal force the is now 20 N higher than it was there

and that's what's allowing this toddler to accelerate

Now 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 had a constant velocity

including a constant velocity of zero, you have no net force on you

So this toddler right over here, the net forces is gonna look identical over here

And actually if you're sitting in either this elevator

or this elevator

assuming it's not being bumped around at all

you would not be able to tell the difference because there's no--

your body is sensitive to acceleration

your body cannot sense velocity that has no frame of reference or nothing to see passing by

So the toddler there doesn't know whether it is stationary or there's constant velocity

It would be able tell this; it would feel that kind of compression on its body

and that's what its nerves or sensitive perception is sensitive to

But here it's identical to the 1st situation

and Newton's first law tells us that 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 N; completely nets out the downward the -98 N

It's in the j direction, in the positive direction

And then when we're about to get our floor, what is happening?

Well, once again we have a net acceleration

We have a net acceleration of -2 m/s^2

So once again what is the net force here?

The net force over here is going to be the mass of the toddler 10 kg times -2 m/s^2

This is in the j direction, the vertical direction

Remember j is just the unit vector in the vertical direction facing upwards

So -2 m/s^2 in the j direction

And this is equal to -20 kg m / s^2 in the j direction

or -20 N in the j direction

So the net force on this is -20 N

So we have the force of gravity -98 N in the j direction

So we we're not fully compensating for that

because we're still gonna have a net negative force while this child is decelerating

And that negative net force is -20

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

that counteracts all but 20 N of the force due to the gravity

This right over here is going to be 78 N in the j direction

I 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 one that elevator's really just accelerating or when it's just decelerating

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

When it's just decelerating, you feel a little bit lighter

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

While it's moving at a constant velocity or stationary

you feel like you're just sitting on the surface of the planet someplace