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A $1125-\mathrm{kg}$ car and a $2250-\mathrm{kg}$ pickup truck approach a curve on the expressway that has a radius of 225 $\mathrm{m}$ . (a) At what angle should the highway engineer bank this curve so that vehicles traveling at 65.0 $\mathrm{mi} / \mathrm{h}$ can safely round it regard-less of the condition of their tires? Should the heavy truck go slower than the lighter car? (b) As the car and truck round the curve at 65.0 $\mathrm{mi} / \mathrm{h}$ , find the normal force on each one due to the highway surface.

See solution.

Physics 101 Mechanics

Chapter 6

Circular Motion and Gravitatio

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

Cornell University

University of Washington

Hope College

University of Sheffield

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A $13500 \mathrm{N}$ car t…

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A car drives around a curv…

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A highway curve with radiu…

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A $1250-\mathrm{kg}$ car r…

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A curve in a stretch of hi…

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because they're before solving the problem. We should look a few questions that we need for this problem. So first of all, if we look at example six point for, we see that we have a road and then there's a banking off angle, Peter, and then there's a car and ah, we are. We figured out the angle better from this diagram. And if we look att, the previously diagram of this car, we see that the normal force since ah, this road is bank, it has some angle which speeder. So the normal force acting on the car is in this direction, which is in between X and y. So there is two components off this and normal force one is on the wind direction and we call it in cause better now the reason we're calling cause better is because we see that there's a little triangle here and the angle between the base and ah, the hyper cuteness is better. So that's why this line right here is in cause better. And similarly, since this is thie ah, perpendicular to the base. So that's why this accesses and signed it. Also, that's the projection these two access of the projection off this and normal force and in the negative y direction Are you in the negative y direction? We have the weight of the car. So if we actually job is over here, we have the normal force. Then we have the component of the normal force which is counterbalancing the weight of the car and then in this side, we have thie way have another component of the normal course. So, um and also too if we again consult examples X point for we see that the banking angle Ah beta is equal to V squared over tee times are were these thie speed off the car and artist the radius that the car or the artist making. So if this is the whole circle than ours the radius then also as we mentioned that the normal forces mg over cause beta right and we're converting the speed from my spur our tow 29.1 meters for second because it's easier to do calculation in meters per second unit. So, in the first problem, we need to figure out whether the truck on the car will have different speeds or not. So we can use. Ah, this equation right here, which is 10 better, equals the speed over gravity Gravitation Force by 25. That's the Radius. So I actually made a mistake and there will be a 10 data in front of me. Better so sorry. There'll be a 10 media, so we write that down one more time. So this is 10 better because the squared over G times are straight. And from here, we get the banking and this is wrong. So yeah, we use this equation to figure out the banking angle on DH. We know that the velocity is 29.1 meters per second squared and then G's 9.8 and the radio star is to 25 meters. So if we use, that's all for better. We get 20 one degree as the banking angle. Now we see that this equation doesn't have independence on him. So that means no matter how heavy the vehicle is, it won't depend on the mass of the vehicle, so it doesn't depend on the mass of the bagel. So the truck and car should travel at the same speed. So the speed with the same for both of them and for the second. But we need to figure out the normal force and as we mentioned that normal forces mg over cost later so you can use the question directly here. So for the car, it's the mass of the car times gravity, gravitational force developed by the banking. The coastline of the banking angle, which gives us gives us 1.18 times into a four Newtons and for the truck, since it's twice off the car. And the reason it's twice is because these two are constant. And the only thing that's changing is thie. Wait and the weight is or the mass and the mass is twice the mass of the truck is twice of the car. So that's where we just if we actually take twice the value ofthe this in car, we get the normal force off the N truck or normal force of the truck. So, yeah, if we multiply to this number, we get 2.36 times 10 to the bar for Nunes. So that's the Norman force for the truck. Now, Um, there's one more thing that we should know This here we see that the vertical normal force must be equal to the weight of the vehicle. So the normal force is proportional to em. Um, what I mean by that is the Norman force, which is in the white direction. We see that it's and co sign beta, which is equal to M times C because this for should counterbalance the way that's going down. So And from there we see that normal forces mg over cause beta. So this end is proportionate am. And ah, this G and co sign beta is constant. So depending on ah, so this is constant. So depending on the weight off the vehicle or the mass of the vehicle are normal force will go ups. So if the mass is going up, a normal course should go up. Thank you.

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