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Anderson Video - Weight in an Elevator

Professor Anderson
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>> Hello, class, Professor Anderson here. Let's take a look at Megan's question, which was, "Can the normal force be in fact bigger than gravity?" What we just saw is that when you're on an incline plane, and you have an object, then the normal force, N, is smaller than gravity. Okay? So if it's equal to gravity at the horizontal and it gets smaller as I tip it up, that thing is always less than gravity. But can you have a situation where the normal force is in fact bigger than gravity? Well, let's think about it. Let's say that we're in an elevator. And now you're just standing in the elevator, and the elevator is not moving, what are the forces that are acting on you on the elevator? Allie, what do you think? >> The force of gravity downwards? >> Force of gravity; mg, okay, and you are in the elevator and the whole elevator is at rest. What else, Allie? >> The normal force? >> Which way? >> Opposite of gravity. >> Okay. How long? >> Same length. >> Same length. Exactly right. Normal force is the floor pushing up on me, gravity is pulling me down. Those exactly balance out if I'm at rest. Good. But now let's change the picture slightly. Let's say that we push the floor number six button, trying to get up to the top of PS1. And so the elevator is in fact accelerating upward, okay? When the elevator first starts moving upward -- you're all familiar with this, what do you feel; do you feel lighter or do you feel heavier? >> Heavier. >> Heavier, right? And in fact, the normal force is now bigger than gravity when it's accelerating upwards, okay? That floor is pushing up on you harder than gravity is pulling down on you; and so you feel a little heavier. How do we say this mathematically? Well, we say that this thing is accelerating upwards with A. What we know is Newton's Second Law says, "The sum of the forces is equal to the mass times acceleration." We are mass, M. This is a vector equation. And so we can write two equations, one for the X direction, one for the Y direction. But the X direction doesn't really tell us anything. Right; there are no forces in the X direction. We're not moving left or right. Zero equals zero, good. What about in the Y direction? Well, here are the forces. N is going up. MG is going down. All of that has to be equal to the mass times the acceleration upward in the elevator. And now look at this equation. I can solve this very quickly, and I get normal force is equal to MA plus MG, or the normal force is equal to M times A plus G, okay? The normal force can absolutely be bigger than gravity, but only in the case where you are accelerating, in this case, accelerating upward. The bigger that acceleration, the bigger the normal force is upwards. If it's zero, obviously, we go back to the case where the normal force is just MG. But let's look at the other case. Let's say you're in this elevator, and all of the sudden, the cord gets cut, right, and you are in freefall. If you are in freefall, what is your acceleration? What was your name again? >> Anthony. >> Anthony, what do you think, what's the acceleration if the cable gets cut? >> It's 9.8 meters per second squared >> Okay. Also known as -- >> Gravity. >> Negative G, right? >> Negative G. >> So look what happens, if I put a negative G in here, what does the normal force become? Well, it's mass times negative G, plus G; it's zero. The normal force is zero. And if you're in freefall, you're not touching the ground anymore. Right; you're floating about in the middle of the elevator, probably screaming rather loudly, but okay? The normal force is in fact zero. So this is a good segue into something called "weight." Yes, Anthony. >> So when you're accelerating at -- you know, I mean, eventually, on Earth you'll hit ground. So -- but if there was -- I don't know if you could go further into the ground or dig a big hole, would you continue to -- what would your ultimate velocity be, or would it be infinite? >> Okay; excellent question. So the question is the following let's say we start with the Earth, and I'm going to dig a big hole in my Earth. Okay; let's dig a really big hole such that it goes all the way through the Earth. And now I'm going to stand here and I'm going to jump into the hole. Okay? When I jump into the hole, what is my acceleration initially? >> Nine point eight meters per second. >> Yes; negative 9.8 meters per second squared. And if I ignore air resistance, then if I accelerated at negative 9.8 meters per second squared, if I did that forever, then I would reach infinite speed. But we know that you can't do that forever. Right; why do we know that, because somebody else that's standing on the other side of the Earth, if they jump in, are they going to come flying out the top of that tube on the other side? That doesn't really make sense, because this problem has to be symmetric. Whatever I do, if I jump in and start from rest, it should be the exact same as the other person that's jumping in and start from rest, that motion should be the same. And so what is in fact going to happen is the following, with something like the Earth, it's sort of interesting. The person that jumps in speeds all the way through the Earth to the other side, just comes up, takes a little peek out the hole, and then flies back in and does it again. Okay? And they would go up and down from one side of the Earth to the other over, and over, and over again. Now, this is, of course, ignoring air resistance so there's nothing to slow them down. They don't hit the sides of the walls as they go through this tunnel, right? That would probably be bad. Gravity is always acting towards the center of the Earth. So it's pulling you towards the center of the Earth for the first half of your journey. And then on the second half of your journey, it's slowing you down, and you just come back to rest, and then you do it again. Now, as we're going to learn, gravity, G, is only at the surface of the Earth. In the center of the Earth, gravity is zero. Right; there's no gravity at the center of the Earth. You would just be floating there weightless. And so in-between the center and the edge, it slowly increases in its value. It gets stronger and stronger as it gets to the edge. And then as you go away from the edge, it again gets weaker. Okay? There's a nice little proof of that using something called "Gauss's Law." And maybe in a more advanced setting we could talk about Gauss's Law. Okay? Good question. [ Silence ]