flat (unbanked) curve on a highway has a radius of 170.0 m. A car...

Question

flat (unbanked) curve on a highway has a radius of 170.0 m. A car rounds the curve at a speed of 25.0 m/s. (a) What is the minimum coefficient of static friction that will prevent sliding?

Accepted Answer

Welcome back everybody. We have an automobile that is traveling along that curve. All our right here. Now, we're told that the mass of this automobile is kg and that it has a tangential velocity of 14 m per second. We're also told that there is friction between the tires and the car. Now, that means for the car to stay on the flat curve, our friction force will have to act in the negative X direction. And we are tasked with finding what the coefficient of friction is, the minimum coefficient of friction so that the tires will keep the car on the ground. Well, let's draw out a couple of other forces here and complete our our semi free body diagram here, since we are traveling along a curve, we are also going to have a centripetal acceleration or radial acceleration. That is acting in words, in the Y direction. We of course are going to have the normal force of the vehicle as well as the weight of the vehicle given by M. D. Now, let's go ahead and apply Newton's law here in the X direction because we're trying to find this coefficient of friction, which means we have to work with this friction force right here. So Newton's law states that the sum of forces in the X direction is equal to mass times the acceleration in the X direction, which is just our radial acceleration that is also acting negative X direction in some values here, we can tell that our only force in the X direction is our negative friction force. This is equal to mass times are negative operation here. But I'm even gonna break this formula down a little bit further. These negatives are gonna cancel out. And our formula for friction force is our coefficient of friction. Are you times our normal force which is equal to the mass times the radial acceleration given by this formula of our tangential velocity squared divided by our radius, dividing both sides by and here to isolate mu. We get that our desired mu is going to be equal to our mass times Our tangential velocity squared all over our radius times our normal force. But what is our normal force? Well, this time I'm going to actually apply Newton's second law in the Y direction. To see if we can't find our normal force. Newton's second law states that the sum of all forces in the Y direction is going to be equal to mass times our acceleration are wide direction, which is just our acceleration due to gravity. This guy is this guy right here. So then plugging in our forces on the left side, we get that are normal force is equal to our mass times our acceleration due to gravity. I'm actually gonna plug this into here. And then we will have a complete formula for our coefficient of friction. So our coefficient of friction is equal to mass times our attention and general velocity squared all divided by our radius times mass times acceleration due to gravity. This mass is going to cancel out on top and bottom. And then let's go ahead and plug in our values. We have that are tangential velocity 14. We're gonna square that divided by our radius, which we are told we have a radius of m Times our acceleration of gravity, which is 9.8, which when you plug this into your calculator, you get that our coefficient of friction is . corresponding to our answer choice of D. Thank you guys so much for watching. Hope this video helped. We will see you all in the next one.

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