Anderson Video - How Fast Can I Drive in a Circle

Professor Anderson
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>> Centrifugal force is the following. Let's say we are driving in a car. Here we are driving in a car, you're sitting on a seat, holding onto the steering wheel and you're going to turn left. You, in the car, feel like you're getting pushed to the right. Okay. You feel like there's a force to the right. Why is that? Why would that be? It's because Newton's first law says, "Objects in motion tend to stay in motion." You in the car are tending to stay in motion. This is where you would like to go, in a straight line. But, of course, the car is moving underneath you. And, so, you feel like there's some force that is pushing you to the side of the car as you go around this curve. And this thing is what we call the centrifugal force. It's not a real force. It's fictitious. It's simply the fact that your underlying reference frame now changed direction. So, you feel like you moved relative to that reference frame. Really you were going in a straight line. All right, so that's centrifugal force. So, let's continue with this idea of the car moving in a circle. And let's see if we can analyze the following problem. Let's say you go to a parking lot and you're going to drive your car in a circle and you want to see how fast you can go in a circle and not spin out. So, here's your car. You're going to go around in this circle. And you are going to do it at speed V and we need to give the circle a radius R and you're going to go faster and faster until you just start to skid. What is V max, that's our question? If we are given the following numbers? Okay, and let's come up with some reasonable numbers. So, a car is, you know, a couple thousand pounds if it's like a sports car, which is probably about a thousand kilograms. What's a typical radius of a turn? Well, you know, a football field is a 100 meters roughly in diameter, so if we did half of that, 50 meters in diameter, we could probably turn a car within the goal line to the 50 yard line. So, let's say that R is 25 meters. What else do we need to know? We need to know something about the force that is holding us in this circle. So, what force is actually holding us in that circle? Friction. Static friction. In that case, we need to know what mu S is. What is mu S for rubber on concrete about? It's about 1. Okay. It's about 1 for rubber on concrete. All right, so we're going to drive our car in a circle on the pavement and we're going to see how fast we can go. These are the givens for the problem. This is the top view. Why don't we take a look at the side view. So, the side view, the back of the car is right there, it's the tires. And then there are some forces that are acting on this car. What forces are acting on the car? We just mentioned one of them. Friction. Which way is friction? In this side view, which way should I draw friction, up, down, right or left? Left. That's the force that's keeping us moving in the circle, right? Top view looks like this, side view would be like this. Friction is towards the circle center. What else is acting on this car from the side view? Yes. Gravity. We love gravity. If gravity's down, there's got to be something going up, which is, of course, the normal force. Okay. Looks like the side view is enough information now for us to solve this problem. So, let's see if we can do it. Before we attack the math, let's take a guess. What's a good guess for how fast you could drive your car? I mean, we're putting in some numbers here that seem reasonable, right? In a normal parking lot, a pretty tight radius of curvature, 1000 kilogram car, how fast do you think you guys could do that? Just take any guess. You've probably done this before, right? When you were learning how to drive with your mom or your dad. You probably were in a parking lot going around like that. I know when I was learning to drive my dad used to take me out on these dirt roads and he'd say, "See how fast you can take this turn." Right? So, we'd go around the turn and we'd get about halfway around the turn and he'd pull the parking brake on me. He'd say, "Just want to see if you can control your vehicle." Awesome. Years of therapy, I'm fine now, it's okay, no problem. What's a good guess for how fast you could drive your car? Is it 100 miles per hour? No. Is it 5 miles per hour? No. So, what's a good guess? >> 45. >> 45. Great. Any other guesses? >> No other guesses, okay. 45 sounds perfectly reasonable. When we get an answer we got to make sure we're not off by like an order of magnitude, right? If we get 450 miles per hour that can't be right. If we get 4.5 miles per hour, that can't be right. It's got to be somewhere within the vicinity of that. Okay? All right. What do we do next? We got a picture, we've got our free body diagram, all right? This is our free body right there. What do we do next? What's step three on the list? Newton's Second. >> Okay. Everybody remembers step three, make her open the box. Okay. MV squared over R is the important feature here, because the sum of the forces don't add up to zero in the radial direction, they add up to MV squared over R. What's the only force in the radial direction? It is friction. Friction is towards the center of the circle. So, it's positive. Put a box around that. We'll come back to it. What about N and MG? What do we do with that? Well, why don't we look at the sum of the forces in the vertical direction. >> Okay, there's no reason we can't call this radial and this vertical now. But it looks like we have N minus MG. What's the acceleration in the vertical direction? Zero, right? You're staying on the pavement. You don't go flying up or down, so we get N equals MG. All right, that looks pretty good, except, we're looking for V max and I don't think we have quite enough information now, because we have this frictional force that we need to know a little bit more about. So, what is static friction? Static friction, you'll remember, is this. F sub S is than less than or equal to mu S times the normal force. So, what is the maximum F S max is just when it's equal. If we want to go as fast as possible we're going to take advantage of the maximum static friction, right? Your tires are just about to break free and you're just about to skid out. All right. It looks like we have three equations now that we can put together. Let's take this equation right here. We've got V squared equals F sub S, times R, divided by M. But I know what F sub S is, it is mu S times the normal force. Multiplying by R, dividing by M and now we know exactly what the normal force is as well. So, this becomes mu SMGR divided by M. The Ms cancel out. And we just get mu SG times R and then we, of course, have to take the square root. Let's make sure this works out in terms of units. V we know is meters per second. G is meter per second squared. R is meter, so I get meter squared per second squared. Mu S is, of course, unitless. When I take the square root of that, I get meters per second. We like that. What about the limits? Does it make sense to you that if mu S gets bigger you can go around this circle faster? Yeah, because you are sticking to the road more. Right, if you can stick to the road more you can certainly go around faster. Likewise, if mu S goes way down, like you're on ice, then the maximum speed would be smaller. And you know this. You can't drive around in a circle on ice nearly as fast as you can on dry cement. So, all those limits seem to make sense. And you can think about the other ones if you'd like, the radius and gravity, but I think you get the idea that it works out. So, let's plug in some numbers and let's calculate what we get and see if we're even close to our guess. So, V is equal to square root of mu S, which is 1. G, which is 9.8. R we said was 25. And, I don't know what that is, it's -- what is it? 15.7. And the units are meters per second. So, that's pretty close to 16 meters per second. What would that be in miles per hour? Roughly double it. 32 miles per hour. So, I guess a 45 is pretty good. Right? We're getting an answer that is off by a little bit, right? 1.5 of that. But, we're not off by a factor of 10, we're not 100 miles per hour, we're not 1 mile per hour, we're somewhere right in the zone. So, that seems like a good answer. 15.7 meters per second. All right, questions about that one?
>> Centrifugal force is the following. Let's say we are driving in a car. Here we are driving in a car, you're sitting on a seat, holding onto the steering wheel and you're going to turn left. You, in the car, feel like you're getting pushed to the right. Okay. You feel like there's a force to the right. Why is that? Why would that be? It's because Newton's first law says, "Objects in motion tend to stay in motion." You in the car are tending to stay in motion. This is where you would like to go, in a straight line. But, of course, the car is moving underneath you. And, so, you feel like there's some force that is pushing you to the side of the car as you go around this curve. And this thing is what we call the centrifugal force. It's not a real force. It's fictitious. It's simply the fact that your underlying reference frame now changed direction. So, you feel like you moved relative to that reference frame. Really you were going in a straight line. All right, so that's centrifugal force. So, let's continue with this idea of the car moving in a circle. And let's see if we can analyze the following problem. Let's say you go to a parking lot and you're going to drive your car in a circle and you want to see how fast you can go in a circle and not spin out. So, here's your car. You're going to go around in this circle. And you are going to do it at speed V and we need to give the circle a radius R and you're going to go faster and faster until you just start to skid. What is V max, that's our question? If we are given the following numbers? Okay, and let's come up with some reasonable numbers. So, a car is, you know, a couple thousand pounds if it's like a sports car, which is probably about a thousand kilograms. What's a typical radius of a turn? Well, you know, a football field is a 100 meters roughly in diameter, so if we did half of that, 50 meters in diameter, we could probably turn a car within the goal line to the 50 yard line. So, let's say that R is 25 meters. What else do we need to know? We need to know something about the force that is holding us in this circle. So, what force is actually holding us in that circle? Friction. Static friction. In that case, we need to know what mu S is. What is mu S for rubber on concrete about? It's about 1. Okay. It's about 1 for rubber on concrete. All right, so we're going to drive our car in a circle on the pavement and we're going to see how fast we can go. These are the givens for the problem. This is the top view. Why don't we take a look at the side view. So, the side view, the back of the car is right there, it's the tires. And then there are some forces that are acting on this car. What forces are acting on the car? We just mentioned one of them. Friction. Which way is friction? In this side view, which way should I draw friction, up, down, right or left? Left. That's the force that's keeping us moving in the circle, right? Top view looks like this, side view would be like this. Friction is towards the circle center. What else is acting on this car from the side view? Yes. Gravity. We love gravity. If gravity's down, there's got to be something going up, which is, of course, the normal force. Okay. Looks like the side view is enough information now for us to solve this problem. So, let's see if we can do it. Before we attack the math, let's take a guess. What's a good guess for how fast you could drive your car? I mean, we're putting in some numbers here that seem reasonable, right? In a normal parking lot, a pretty tight radius of curvature, 1000 kilogram car, how fast do you think you guys could do that? Just take any guess. You've probably done this before, right? When you were learning how to drive with your mom or your dad. You probably were in a parking lot going around like that. I know when I was learning to drive my dad used to take me out on these dirt roads and he'd say, "See how fast you can take this turn." Right? So, we'd go around the turn and we'd get about halfway around the turn and he'd pull the parking brake on me. He'd say, "Just want to see if you can control your vehicle." Awesome. Years of therapy, I'm fine now, it's okay, no problem. What's a good guess for how fast you could drive your car? Is it 100 miles per hour? No. Is it 5 miles per hour? No. So, what's a good guess? >> 45. >> 45. Great. Any other guesses? >> No other guesses, okay. 45 sounds perfectly reasonable. When we get an answer we got to make sure we're not off by like an order of magnitude, right? If we get 450 miles per hour that can't be right. If we get 4.5 miles per hour, that can't be right. It's got to be somewhere within the vicinity of that. Okay? All right. What do we do next? We got a picture, we've got our free body diagram, all right? This is our free body right there. What do we do next? What's step three on the list? Newton's Second. >> Okay. Everybody remembers step three, make her open the box. Okay. MV squared over R is the important feature here, because the sum of the forces don't add up to zero in the radial direction, they add up to MV squared over R. What's the only force in the radial direction? It is friction. Friction is towards the center of the circle. So, it's positive. Put a box around that. We'll come back to it. What about N and MG? What do we do with that? Well, why don't we look at the sum of the forces in the vertical direction. >> Okay, there's no reason we can't call this radial and this vertical now. But it looks like we have N minus MG. What's the acceleration in the vertical direction? Zero, right? You're staying on the pavement. You don't go flying up or down, so we get N equals MG. All right, that looks pretty good, except, we're looking for V max and I don't think we have quite enough information now, because we have this frictional force that we need to know a little bit more about. So, what is static friction? Static friction, you'll remember, is this. F sub S is than less than or equal to mu S times the normal force. So, what is the maximum F S max is just when it's equal. If we want to go as fast as possible we're going to take advantage of the maximum static friction, right? Your tires are just about to break free and you're just about to skid out. All right. It looks like we have three equations now that we can put together. Let's take this equation right here. We've got V squared equals F sub S, times R, divided by M. But I know what F sub S is, it is mu S times the normal force. Multiplying by R, dividing by M and now we know exactly what the normal force is as well. So, this becomes mu SMGR divided by M. The Ms cancel out. And we just get mu SG times R and then we, of course, have to take the square root. Let's make sure this works out in terms of units. V we know is meters per second. G is meter per second squared. R is meter, so I get meter squared per second squared. Mu S is, of course, unitless. When I take the square root of that, I get meters per second. We like that. What about the limits? Does it make sense to you that if mu S gets bigger you can go around this circle faster? Yeah, because you are sticking to the road more. Right, if you can stick to the road more you can certainly go around faster. Likewise, if mu S goes way down, like you're on ice, then the maximum speed would be smaller. And you know this. You can't drive around in a circle on ice nearly as fast as you can on dry cement. So, all those limits seem to make sense. And you can think about the other ones if you'd like, the radius and gravity, but I think you get the idea that it works out. So, let's plug in some numbers and let's calculate what we get and see if we're even close to our guess. So, V is equal to square root of mu S, which is 1. G, which is 9.8. R we said was 25. And, I don't know what that is, it's -- what is it? 15.7. And the units are meters per second. So, that's pretty close to 16 meters per second. What would that be in miles per hour? Roughly double it. 32 miles per hour. So, I guess a 45 is pretty good. Right? We're getting an answer that is off by a little bit, right? 1.5 of that. But, we're not off by a factor of 10, we're not 100 miles per hour, we're not 1 mile per hour, we're somewhere right in the zone. So, that seems like a good answer. 15.7 meters per second. All right, questions about that one?