Anderson Video - Car on a Banked Curve

Professor Anderson
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>> Okay, first off let's say good morning to all your friends at home right now. Good Morning [good morning]. >> Morning Darrin. >> Morning [inaudible]. >> Say hi to your friends, if you got some people out there you want to say hi to go ahead. >> Shout out to Austin. >> Hi Garin [inaudible]. >> Hey Shawn. >> Let's leave the mics on for a sec, and let's have a conversation about this one, and let's see what the people at home said first. Let's go back to this problem and see if we can understand how to do the full kinematics. You guys can turn your mics off, so you can chat again. And, feel free to talk over there on that side of the glass. Nobody's going to hear you. The only mic is right here now, so. If you guys want to talk to each other, go ahead. Alright. How do we do this banked curve problem? We kind of set it up last time, but let's go through it in a little more full detail. So, here's your banked curve. This is the, sort of side view, or the end view. Our car is right here [drawing]. There's the wheels, here's the headlights. Here's the top of the car, okay, and this thing is coming towards you like that which means it's going around a radius, r, like so. Okay, this is a car on the banked curve going around like that. Let's see if we can figure out what the relationship is between the v, between the theta, between the r. How do we do that? Alright, first off, we got a picture. It's not the best picture ever, but you guys get the point. We go to the free body diagram. What forces are acting on the car? Gravity. What other forces are acting on the car? Normal force. Anything else? Okay. You guys just pictured it as that. Static friction. So, let's try an experiment. Let's say that you are driving on ice such that there is no friction anymore. If there is no friction, all we have to do is erase that. So, the only thing left are those two forces, mg down, normal force up at an angle. Alright, that looks pretty good, except nothing is orthogonal here, and we would really like to have orthogonal forces, so let's redraw it. mg is down, that seems okay. But how do I break up n into components? I know that there's some component of n to the right. There's some component of n going up, but I don't know which is which. We know that it's got to be related to theta somehow. So, it's either n sine theta here, and n cosine theta there, or it's flipped. So, how do we figure out which is which? You can go through a little proof and use a little bit of trig and figure it out, or you can just look at the limits. So, let theta go to zero degrees. If theta goes to zero degrees, it's now a flat surface. My car would be right on top of that flat surface. And, the normal force on this car going up would just be equal to what? mg. Okay? We know there's still going to be a normal force holding the car up. We know it has to be equal to mg. We need our normal force to not go away if theta goes to zero. So, which of these is zero and which of them is one? What's the sine of zero? >> Zero. >> Zero. Cosine of zero is one. So, do I have this right, or do I have to switch it? We have to switch. Okay? So, we switch it. And, so we redraw this as mg down, n sine theta to the right, and cosine theta going up. This is now the correct free body diagram once you break it in to components. Alright, so this is actually one of the hard parts of the problem, is just getting that right. Okay, now that we have that right, what do we do next? We've got our picture. We got our free body diagram. Now, we go to Newton's Second Law. And, Newton's Second in circular motion says sum of the forces in the radial direction equals mv squared over r. There's only one force in the radial direction. It's n sine theta. So, we get n sine theta equals mv squared, divided by the radius which we said was r [writing]. We have vertical forces here that we need to worry about, and so the forces in the y direction have to add up to m times the acceleration in the y direction. We have n cosine theta going up. We have mg going down. And all of that is equal to what? Zero. There's no acceleration in the y direction. This car is going around in a horizontal circle. It's not going up or down. And so, we get n cosine theta equals mg. So, let's say that we want to solve for theta. Okay, how do we solve this thing for theta? Well, we've got equation A right there. We've got equation B right there. If I write equation A over here, we said it is n sine theta equals mv squared over r. If I write equation B, I get n cosine theta equals mg, and now here's the cool trick. If I have two equations, I can divide equation one by equation two, and so the whole thing becomes n sine theta over n cosine theta equals mv squared over r divided by mg. Anytime you have two equations, you can always divide those equations. The reason you do that, of course, is n drops out. The whole left side just becomes tangent theta. m drops out over here, and we get v squared over g times r. So, if you're trying to figure out what angle theta, you can use this equation, right. If you know this other stuff, you just take the arc tangent and you're done. So, let's try this for a real setup, and let's just make up some numbers that we think are reasonable. Let's say this is a freeway off-ramp. Okay, we know that freeway off ramps are banked, right? When you get off the freeway, they're banked. What is a typical speed that you might see on a freeway off-ramp? Forty-five miles per hour, okay? Forty-five miles per hour which is approximately 20 meters per second, okay. It's roughly a factor of two, so it's probably a little bit off, but let's just say that's a good number, okay? What is the radius of curvature of that off-ramp, in meters? Any thoughts? Is it five meters? Is it 500 meters? Five meters sounds way too small, right? That's only 15 feet. Five hundred meters, that's like five football fields. That sounds way too long too. So, somewhere, maybe, I don't know 50 meters? Does that sound good? Fifty meters in radius? Perfect. Okay, we're just taking some guesses from our everyday life. Okay, we know g, of course, that's 9.8. So, let's calculate what theta is. Tangent of theta so we need to take the arc tangent, and if we take the arc tangent of v squared which we said was 20 squared, and we're going to divide by g, 9.8, and r we said was 50, and why don't you guys punch that in to your calculator and tell me what you get. We've got the arc tangent of 20 squared which is 400, and in the bottom, we have 50 times 9.8 which is pretty close to 500, alright? It's 490 or something. What do you guys get? Thirty-nine degrees. Okay. What is this number now? This number is theta. How steep do you need to make that bank, right? So, why do you care? Some of you guys are engineers in here, right? Probably a lot of you guys are engineers in here. Why do you care about this number? Okay. Use your mic and let's hear that one again. If it's snowing, you don't want to slide off the road. Okay? When you drive on the freeway and you see that exit sign that says 45 miles per hour, and you notice, oh that road is banked pretty steeply, it's because somebody went through these calculations to figure out how steep they should bank that curve such that you don't require any friction to get around the curve. In other words, if it's snowing, or if it's icy, you can still make the turn. Okay? No friction, no problem. Anybody driven on black ice before? So, black ice which I first discovered in Oregon when I was in grad school, is when it rains and it hits the pavement, and the pavement is really cold and so it immediately turns to ice, and it makes this like invisible thin layer of ice, and that's why they call it black ice, because you're actually just looking directly through it at the pavement. So, you can't even see it, and it's like nearly frictionless. You know, you hit this stuff and you have no friction between your tires and the road anymore. And, I remember like driving along in my four-wheel drive truck, it doesn't matter if all four wheels are going, if there's no friction between you and the road, there's nothing you can do. And, people are just like [sound] just, you know, like these slow-motion slides, you know, off the road. So, the idea for you engineers is if you're going to build that freeway off-ramp, and you want to know how steep to make it, go through this calculation knowing that it has to be the right angle such that if there's no friction at all somebody could still make the turn. Now, nobody ever does this, right? Nobody ever actually takes the exit at 45 miles per hour, right? You guys are heading down the freeway at 80 and you're like, whew, going on the off-ramp, okay. Try it sometime. Go on the off-ramp and actually go at the speed that's posted. Every off-ramp it says, exit 45 miles per hour. If you actually go at that speed, your car nearly steers itself around the curve. You don't have to crank your wheel to the right, or crank your wheel to the left. It just goes around the curve almost naturally. It's kind of a cool experiment. Try it out. You guys are, you know, scientists now. Go do it. Everybody behind you is going to honk like crazy. Just ignore them. Focus on the steering of your car. Alright? It's a fun little experiment.
>> Okay, first off let's say good morning to all your friends at home right now. Good Morning [good morning]. >> Morning Darrin. >> Morning [inaudible]. >> Say hi to your friends, if you got some people out there you want to say hi to go ahead. >> Shout out to Austin. >> Hi Garin [inaudible]. >> Hey Shawn. >> Let's leave the mics on for a sec, and let's have a conversation about this one, and let's see what the people at home said first. Let's go back to this problem and see if we can understand how to do the full kinematics. You guys can turn your mics off, so you can chat again. And, feel free to talk over there on that side of the glass. Nobody's going to hear you. The only mic is right here now, so. If you guys want to talk to each other, go ahead. Alright. How do we do this banked curve problem? We kind of set it up last time, but let's go through it in a little more full detail. So, here's your banked curve. This is the, sort of side view, or the end view. Our car is right here [drawing]. There's the wheels, here's the headlights. Here's the top of the car, okay, and this thing is coming towards you like that which means it's going around a radius, r, like so. Okay, this is a car on the banked curve going around like that. Let's see if we can figure out what the relationship is between the v, between the theta, between the r. How do we do that? Alright, first off, we got a picture. It's not the best picture ever, but you guys get the point. We go to the free body diagram. What forces are acting on the car? Gravity. What other forces are acting on the car? Normal force. Anything else? Okay. You guys just pictured it as that. Static friction. So, let's try an experiment. Let's say that you are driving on ice such that there is no friction anymore. If there is no friction, all we have to do is erase that. So, the only thing left are those two forces, mg down, normal force up at an angle. Alright, that looks pretty good, except nothing is orthogonal here, and we would really like to have orthogonal forces, so let's redraw it. mg is down, that seems okay. But how do I break up n into components? I know that there's some component of n to the right. There's some component of n going up, but I don't know which is which. We know that it's got to be related to theta somehow. So, it's either n sine theta here, and n cosine theta there, or it's flipped. So, how do we figure out which is which? You can go through a little proof and use a little bit of trig and figure it out, or you can just look at the limits. So, let theta go to zero degrees. If theta goes to zero degrees, it's now a flat surface. My car would be right on top of that flat surface. And, the normal force on this car going up would just be equal to what? mg. Okay? We know there's still going to be a normal force holding the car up. We know it has to be equal to mg. We need our normal force to not go away if theta goes to zero. So, which of these is zero and which of them is one? What's the sine of zero? >> Zero. >> Zero. Cosine of zero is one. So, do I have this right, or do I have to switch it? We have to switch. Okay? So, we switch it. And, so we redraw this as mg down, n sine theta to the right, and cosine theta going up. This is now the correct free body diagram once you break it in to components. Alright, so this is actually one of the hard parts of the problem, is just getting that right. Okay, now that we have that right, what do we do next? We've got our picture. We got our free body diagram. Now, we go to Newton's Second Law. And, Newton's Second in circular motion says sum of the forces in the radial direction equals mv squared over r. There's only one force in the radial direction. It's n sine theta. So, we get n sine theta equals mv squared, divided by the radius which we said was r [writing]. We have vertical forces here that we need to worry about, and so the forces in the y direction have to add up to m times the acceleration in the y direction. We have n cosine theta going up. We have mg going down. And all of that is equal to what? Zero. There's no acceleration in the y direction. This car is going around in a horizontal circle. It's not going up or down. And so, we get n cosine theta equals mg. So, let's say that we want to solve for theta. Okay, how do we solve this thing for theta? Well, we've got equation A right there. We've got equation B right there. If I write equation A over here, we said it is n sine theta equals mv squared over r. If I write equation B, I get n cosine theta equals mg, and now here's the cool trick. If I have two equations, I can divide equation one by equation two, and so the whole thing becomes n sine theta over n cosine theta equals mv squared over r divided by mg. Anytime you have two equations, you can always divide those equations. The reason you do that, of course, is n drops out. The whole left side just becomes tangent theta. m drops out over here, and we get v squared over g times r. So, if you're trying to figure out what angle theta, you can use this equation, right. If you know this other stuff, you just take the arc tangent and you're done. So, let's try this for a real setup, and let's just make up some numbers that we think are reasonable. Let's say this is a freeway off-ramp. Okay, we know that freeway off ramps are banked, right? When you get off the freeway, they're banked. What is a typical speed that you might see on a freeway off-ramp? Forty-five miles per hour, okay? Forty-five miles per hour which is approximately 20 meters per second, okay. It's roughly a factor of two, so it's probably a little bit off, but let's just say that's a good number, okay? What is the radius of curvature of that off-ramp, in meters? Any thoughts? Is it five meters? Is it 500 meters? Five meters sounds way too small, right? That's only 15 feet. Five hundred meters, that's like five football fields. That sounds way too long too. So, somewhere, maybe, I don't know 50 meters? Does that sound good? Fifty meters in radius? Perfect. Okay, we're just taking some guesses from our everyday life. Okay, we know g, of course, that's 9.8. So, let's calculate what theta is. Tangent of theta so we need to take the arc tangent, and if we take the arc tangent of v squared which we said was 20 squared, and we're going to divide by g, 9.8, and r we said was 50, and why don't you guys punch that in to your calculator and tell me what you get. We've got the arc tangent of 20 squared which is 400, and in the bottom, we have 50 times 9.8 which is pretty close to 500, alright? It's 490 or something. What do you guys get? Thirty-nine degrees. Okay. What is this number now? This number is theta. How steep do you need to make that bank, right? So, why do you care? Some of you guys are engineers in here, right? Probably a lot of you guys are engineers in here. Why do you care about this number? Okay. Use your mic and let's hear that one again. If it's snowing, you don't want to slide off the road. Okay? When you drive on the freeway and you see that exit sign that says 45 miles per hour, and you notice, oh that road is banked pretty steeply, it's because somebody went through these calculations to figure out how steep they should bank that curve such that you don't require any friction to get around the curve. In other words, if it's snowing, or if it's icy, you can still make the turn. Okay? No friction, no problem. Anybody driven on black ice before? So, black ice which I first discovered in Oregon when I was in grad school, is when it rains and it hits the pavement, and the pavement is really cold and so it immediately turns to ice, and it makes this like invisible thin layer of ice, and that's why they call it black ice, because you're actually just looking directly through it at the pavement. So, you can't even see it, and it's like nearly frictionless. You know, you hit this stuff and you have no friction between your tires and the road anymore. And, I remember like driving along in my four-wheel drive truck, it doesn't matter if all four wheels are going, if there's no friction between you and the road, there's nothing you can do. And, people are just like [sound] just, you know, like these slow-motion slides, you know, off the road. So, the idea for you engineers is if you're going to build that freeway off-ramp, and you want to know how steep to make it, go through this calculation knowing that it has to be the right angle such that if there's no friction at all somebody could still make the turn. Now, nobody ever does this, right? Nobody ever actually takes the exit at 45 miles per hour, right? You guys are heading down the freeway at 80 and you're like, whew, going on the off-ramp, okay. Try it sometime. Go on the off-ramp and actually go at the speed that's posted. Every off-ramp it says, exit 45 miles per hour. If you actually go at that speed, your car nearly steers itself around the curve. You don't have to crank your wheel to the right, or crank your wheel to the left. It just goes around the curve almost naturally. It's kind of a cool experiment. Try it out. You guys are, you know, scientists now. Go do it. Everybody behind you is going to honk like crazy. Just ignore them. Focus on the steering of your car. Alright? It's a fun little experiment.