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Anderson Video - Roller Coaster Loop the Loop

Professor Anderson
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>> So everybody's probably been on a roller coaster before, and probably been on a roller coaster that has gone upside down. What does a roller coaster look like? Well, it's a track, goes along and then it goes around this circular loop, and then it continues on its merry way. And you have a car that you're sitting in. You've got your arms up, and you go around this loop. And when you're up here at the top, you are, of course, upside down. Let me ask you a question. Where do you feel heaviest? Do you feel heaviest at the top or at the bottom? The bottom, okay? Everybody knows this. You feel heaviest at the bottom, right? When you get down to the bottom of this circular loop, you kind of get pushed into your seat a little bit more. In other words, that seat is pushing up on you a little bit more. So how do we figure out how to analyze this in terms of rotational motion -- what we know about rotational motion? Well, to do this we need to draw our free body diagram. Let's think about the free body at the top and at the bottom. So here's our point at the top and here's our point at the bottom as we go around this loop. And let's think about the forces that are acting on you okay? What is the force that is acting on you at the bottom? What's the first force? Gravity, MG. What else is acting on you at the bottom? Normal force, right? The seat of the roller coaster car is pushing up on you. There's the normal force. We're not really sure about the lengths yet, but just worry about the orientation. What about at the top? What forces are acting on you at the top of this loop? Gravity. Down, right? Gravity is always down. What else? Anything else? >> Normal force? >> Normal force from the seat acting on you. Which way? Also down, because you're upside down. So if you go around that loop, the normal force from the seat is pushing down on you because you're upside down. So we've got our picture, we've got our free body diagram. Let's go to Newton's second. Newton's second, of course, says sum of the forces in the radial direction is equal to MV squared over R. When we're at the top, we have two forces to worry about, but they are both towards the center of the circle and so they both have the same positive sign. MG plus N equals MV squared over R. And we should be specific that this is N at the top. The other one is going to be N at the bottom. All right? So we can solve this for NT. We get NT equals MV squared over R minus MG. Put a box around that one. And then at the bottom, what do we get? Remember, when we deal with these radial forces, we always say that positive is towards the center of the circle. So we have N sub B going towards the center of the circle. MG is away from the center of the circle, so we put a minus sign right on it. That has to still equal MV squared over R. And now we get NB equals MV squared over R plus MG. So you guys were absolutely right. You are heavier at the bottom of the loop because that thing is bigger. If the normal force is bigger, that's what you're perceiving as your weight, and so you feel heavier at the bottom. We're adding two quantities here, whereas in the top one we are subtracting two quantities. So let's ask a little bit of a follow-up question. Let's say that you want to design the roller coaster such that you feel weightless at the top. Okay? If you are weightless at the top, how do I figure out what the speed should be? What parameter should I set to zero over here? >> Normal force? >> The normal force, right? Weightless means the normal force equals zero, right? If you're weightless, you're just starting to come off that seat. And if you're just losing contact with the seat, then you can't have any normal force. Normal force is a contact force. If they just lose contact, normal force goes to zero. And so, this is where you feel weightless. Okay, but we know exactly what the normal force is. It's right there. NT, we said, was equal to MV squared over R minus MG. We're going to say that's equal to zero. And now we can solve for V. I add MG to the other side. The M cancels out on both sides. And I can solve for V very quickly. V equals G times R square root. This is the speed at which you feel weightless at the top of the roller coaster. Now, why do you want to know this? As an engineer, why would you want to know that? Anybody have a thought? Yeah? >> You want to be [inaudible]. >> Okay, you want to know this speed such that you don't fall. If the roller coaster came up to the top and came to a stop, you would fall. You would fall out of your seat. And so then you're relying on the harness and you're relying on the wheels on the bottom side of the track of the roller coaster to hold you in, and it gets a little bit more dangerous, right? In fact, any speed less than this, you start to fall out of your seat. So this speed or higher is the speed that will keep you safely in your seat. So yes, you want to design the roller coaster such that people couldn't fall out and you're relying on that safety belt. But more importantly, when you design a roller coaster you want to make it fun, right? You want the roller coaster to be exciting, and it's really exciting when you have that feeling of weightlessness. When you go around the roller coaster and you're up at the top, you feel a little bit more weightless. Maybe not entirely, but close to it. And that is what roller coaster designers are trying to do. Why does it feel weird to be weightless? You guys have all done roller coasters before. You've all been in an environment where you feel a little bit weightless. And people always say one thing after they go on rides like this. What do they say? They always talk about one part of their body. What part of their body are they talking about? Their stomach, right? You get off the ride and you go, "Ah, felt like my stomach was in my throat," right? Why is that? Why is that? Because when you're walking around on the earth, your stomach, like everything else in your body, is being pulled down by gravity. The floor is pushing up on your feet. And so your stomach is actually low in your abdomen. But when you're weightless, now all of a sudden everything in your body is being pulled on by gravity, but nothing is pushing back up on it. And so, your whole body relaxes and your stomach comes up a little higher in your abdomen, and that's what feels weird, okay? It's because those organs in you sort of moved to a different position. You're like, "Oh, what's happening," right? You feel a little strange. That's why a lot of people just can't be astronauts, right? You go on these weightless rides. You get off and you go, "Oh, I'm sick." They don't like people puking all over their spaceships.