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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.