Okay, so here's your problem. I'll post the deep, the question on the website and then what I'm going to do is break it into pieces so you can answer each piece. I'm going to solve the whole thing and give you places that you can pause. Remember, you should do it yourself first. If you get stuck, just look at one piece at a time. So that's what we want to do. Okay, so this is a pretty cool problem. Let me just go ahead and show it to you. This, I don't know if you can see that, I assume you can. This is a picture from Google Maps of the intersection between I12 and 59. And it's got this really nice on ramp right here. So you come down here, if you want to go from 12 East to 59 North, you've got to go around that loop like that. Cool. I picked it because I've been on that road a couple of times and maybe you have too. I don't know where you live, but you probably have something like that too. So what do you want to do is calculate what should be a safe speed for cars to go around that curve? Okay. So we'll be like little city planners. So what's important here? Let's draw a picture. So I'll draw a small picture right here. There's the car. It's going like that and it's going around the curve. Now in this case, if it's going at a constant speed, it's still accelerating, okay, because it's changing directions. And we need to look at centripetal acceleration. And what keeps it, what pushes it towards the center to make it accelerate? Friction. Friction does that. So in this case, I'm going to assume a flat road and I want to calculate the speed that it can go without slipping. So if I look at this car head on, it would look like this. There's someone driving in the wheels. The car is coming out this way. I drew it that way so that we could see all the forces acting on the car. Let me just do this in a blue. And so this is the center of the circle over here. Okay. And let's call that R. Yeah, let's call it R. And I actually looked up on Google Maps, you can use a little measuring tool. For this case, I get R about 57 meters. It's not completely circular. It has different radiuses. There's other places too. And we also need to know the coefficient of friction between the car and the road. Now it depends, right? It depends on what kind of rubber you have, what kind of tires, and the road conditions. So let's do this for a wet road where it's sub-reast. So coefficient of static friction for a wet road is about 0.4. That's what we'll use. Okay. So now what forces are acting on this car at this instant as it's coming out? It's moving this way. Well, I have the gravitational force. I have the normal force, let's call it N. And so that the net acceleration up and down, the net force is 0. So the acceleration is 0 up and down. And then I have these tires have to be pushed in this way. So there's the friction call force. I call it F. F. That way. Okay. So, and let's call this the x direction at this instant. And we'll call that the y direction. Okay. So now I'm going to write down Newton's second law in both the x and the y. Wait. This would be a great place to pause. I've made something to you for you. So this would be a place to pause because to think about what to do next. I mean, what did I do so far? All I did was write down what I had. I drew a picture and I drew the free body diagram. So think about what goes next and then pause and work on it and then come back. And I assume that you've done that. Otherwise, you're really just, I mean, you could be watching something else like, I don't know, TV. Okay. But I assume you came back. I assume you worked on it. So let me write down the f net in the x direction and f net in the y direction. So in the x direction, I have just this frictional force. Now I'll put that m a x and in the y direction, I have n minus m g equals zero. Because in the y direction, the car is not accelerating up or down. So the force is half to add up to zero and that's not true in the x direction. And in fact, I know the acceleration in the x direction. If it's moving, if it's acceleration due to something moving in a circle, this would be v squared over r. That's the acceleration of something moving in a circle. And the direction is towards the center of the circle, which is in the x direction. Can I go ahead and solve this for v? No, I can't because I don't know what that is. I know and I don't know what the mass is. Do you have to look that up? Okay, so here would be another great place to pause. Okay. Because what you want to do is think what do you do next? And once you set up these equations, you really made a lot of progress. But what you need to do next is somehow find this frictional force. Well, we can use the magnitude of the frictional force is a coefficient times the normal force. And this is static friction because the tires and the road are not sliding next to each other, they're stationary. So if I use this up here, then I just need to find n. So I can find n from this. So now I can take this and this and write that up here. Why don't you do that? I know I keep putting up the policy on, but. Okay, you're back. Did you do it? Did you really do it? Okay, I'm just going to trust that you did. Okay, so now I'm going to put these things together. So this is, I'm going to write this as mu S n equals m v squared over r. And then I'm going to use this. I'm going to get mu S m g equals m v squared over r. Okay, I think you should be able to finish it from here. So. And go do that. Okay, so you finished it and you want to check and make sure you did it right. What happened to the mass right here? Look at that. That's awesome because then we don't have to have the speed limit for light cars and speed limit for heavy cars. We have the speed limit. Period. So now I can solve this for v squared equals mu S g r v equals the square root of mu S g r. Okay, so now I can put in my values. So I said 0.4 and then 9.8 Newton per kilogram. And then I said 57 meters. If you do this, you get. For 15 meters per second, which is equal to, you can do, this would be a great place for you to practice your unit conversions. 33 miles per hour, 34. Okay, so which I said this, I'm not going to set 34. I'm not going to set 35. So maybe 30. I would probably set the speed limit at 25 miles per hour. Because that way, what if someone has, what if there's some oil on there or what if the radius isn't completely constant or maybe if I put it at 25, people will just go 30. Okay, so there's your answer. Pretty cool, huh?