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ï»¿ >> Hello class Professor Anderson here, let's talk a little bit about weight and gravity and these are obviously concepts that you're all familiar with, but let's see how they tie together. You're standing here on the Earth and you feel a force on you, f sub g, which is due to gravity. Okay that's the gravitational force pulling down on you. Now let's say somebody else standing on the other side of the Earth also experiences a force f sub g but of course that f sub g is pointing upwards. This is the way gravity works, it always points towards the center of the Earth, so if you're on the moon it's going to point towards the center of the moon. If you're on Mercury it's going to point towards the center of Mercury. The force is always acting towards the central point. Now when we think about you standing here on the Earth, how do we draw a free body diagram for that? Well we make you a dot and then we identify the forces that are acting on you. You've got f sub g due to gravity and then you have the normal force n which is the Earth pushing back up on you. Okay. And what we know is if you are stationary, then we can say the sum of the force is equal to mass times acceleration, the acceleration is going to be 0, if you're stationary. Now whenever you write down this equation you look at it, you say oh I have a vector there, I have a vector there, I can break it up into components. So sum of the forces in the x direction is equal to the mass and the acceleration in the x direction. Sum of the forces in the y direction is equal to mass times the acceleration in the y direction. And now we just identify what those forces are. So in the x direction do we have any forces? No, we don't have anything pointing in the x direction. All right typically we say x is to the right, y is up. So there's no forces in the x direction which is good because that's equal to the mass times the acceleration in the x direction, which is also 0, 0 equals 0 we're happy. Well what about the y direction? In the y direction we have the normal force from the Earth pushing up on us and we have fg down. But we know exactly what fg is, it's equal to your mass times the gravity, 9.8 meters per second squared. So we put a minus mg right there, that's equal to the mass times the acceleration in the y direction and we said that that has to be equal to 0. And so you just get n equals mg. Okay, how hard is the Earth pushing up on you? It's pushing up with your mass times gravity 9.8 meters per second squared. But let's change the picture just slightly. Here's our Earth, here is a scale and you are going to stand on the scale. Okay we can draw the exact same picture here, the only difference is this normal force n is the force of the scale pushing up on you. Okay this is our scale right there. Okay and you've all done this, you've all stood on a scale, what happens in a scale is there's a little spring that compresses, as it compresses it rotates a needle that shows you different numbers, those numbers correspond to your weight, they correspond to your mg. And so this is really what we call weight, what I like to call weight. How hard is a scale pushing up on you? How hard is the normal force underneath you pushing up on your body? Okay. A lot of times in the homework when they refer to weight, what they refer to is mg. So if you ever see that in the homework, the weight of this thing that's what they mean, they mean mg. But I like to think of weight in terms of this sort of psychological principle, how hard is the scale underneath you pushing up on you? And if I weigh 170 pounds what that means is when I stand on a scale it's pushing up on me with a 170 pounds of force. You guys liking pink? Is pink looking okay? Yeah. Let's take a look at the forces that act on you when you're in an elevator. And let's setup the drawing here, here's our elevator. And just to be tricky let's put a scale inside the elevator. Okay and you are standing on the scale inside the elevator, here are the cables going up. This is the elevator, that's you. What are the forces that are acting on you? Somebody have a thought. Let me know. What is the force that is acting on you? Yeah? >> Gravity. >> Gravity, gravity is of course down f sub g. Okay. If this dot represents us then gravity is going down. Any other forces that are acting on us? Thomas, what do you think is there another force that's acting on us? >> The scale is pushing up on you. >> Okay the scale pushing up on us okay and we can call that the normal force due to the scale. Okay. So if that's the normal force due to the scale, it's pushing up on us, gravity is pushing down on us, what can we say about that normal force from the scale? If I weigh 170 pounds, is the normal force from the scale equal to 170 pounds? Anybody have a thought on that one? Yeah Ryan in the back what do you think? >> Doesn't it depend on the acceleration of the elevator it's going up or down? >> Okay have you experienced this before? >> Yeah. >> Okay you've been in an elevator right? So in an elevator do you feel heavier or do you feel lighter when it starts moving up? >> You feel heavier. >> You feel heavier right when it starts moving up you feel heavier, when it gets to the top and it slows down, you in fact feel a little bit lighter. Okay and so you identified the important parameter here is, is the elevator accelerating? So let's take a look at that. All right we know we don't have any forces in the x direction, we don't really have to worry about that. But we do have forces in the y direction and the forces that we have are n of the scale minus gravity which we know is mg and all of that is equal to the mass times the acceleration of the elevator. Okay we don't know exactly what that a is yet but we drew it going up, so it's the same direction as the normal force from the scale. And now we can write the normal force from the scale. It's just equal to mg plus ma and in fact we can combine the g and the a because they both have a common factor of m and this is what we get for the normal force from the scale. This is your weight, this is your perceived weight, when we said you feel heavier, when the elevator is starting to accelerate upwards, it's because a is a positive number then and I'm adding it to a positive number, remember g is always positive 9.8 meters per second squared. We took into account the negative part of it right here when we said that force is downwards, okay. But g itself is always positive 9.8. All right so if we are accelerating upwards rather quickly, like g which would be way to quick for an elevator, okay elevator would never do that. But let's say it did, right? If it accelerates upwards with a magnitude of g, what do we get? We get n for the scale is g plus a which is now another g and we get 2mg. Okay so if this, if this elevator accelerated up with g, you would suddenly feel twice as heavy. If you were 170 pounds initially you would now be 340 pounds. Okay. You know elevators don't do that right? I mean when elevators accelerate upward it's a small fraction of g, you feel a little bit heavier but really not that much heavier okay. Let's look at the other extreme. Let's say that we cut the cable. Okay if we cut the cable, break it in half and now all of sudden this thing is falling. Then we are in free fall, what is our acceleration? Yeah Austin what do you think, what is our acceleration if we are in free fall? >> It would be a negative g. >> Negative g. The elevator falling at negative g, the scale is falling at negative g, I am falling at negative g, everything is falling at negative g. So what do we get for the force of the scale pushing up on us? Well it is right here, m times g plus a and we just said a was negative g so we get m times g minus g which is of course 0. What do I call that Austin? If the n from the scale is 0, what is that phenomenon? What do I call that? >> Weightless. >> Weightless, exactly right. This is weightless. When there's nothing pushing back up on you, you feel weightless. And this is what happens in roller coasters, remember I talked about going to Knott's Berry Farm and riding the Excelsior, which if you have any kind of heart condition you should never do. It's ridiculously fast and ridiculously high roller coaster. But one of the goals of that roller coaster is to make you feel weightless. And so it launches up and then it goes around in this curve and for that entire curve they arc it to try to make you feel weightless, they try to get the normal force exactly equal to 0 and it feels like you are in free fall the whole time. Okay. And roller coasters like to do this because then all the riders vomit and it's good fun for everyone right? I'm getting old, I go on a roller coaster I'm like nauseous for an hour afterwards. Okay questions about this idea? Weight versus gravity and including acceleration. Everybody okay with that? All right if that's not clear, come see me in office hours. Cheers.

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