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?

Video transcript

Welcome back everybody. We are making observations about an automobile, which I'm going to represent by this box here. It is traveling along a flat curve. And we are told a couple other different things about the system. We are told that the mass of the car is 1200 kg and that it's traveling at a velocity of m/s. And we are tasked with finding what the minimum coefficient of friction is to keep the tires on the ground. All right, So what we are going to do since we are trying to look for the coefficient of friction, we're gonna be dealing with a bunch of different forces here. So it might behoove us to draw these forces out. So of course acting towards the sky and towards the ground for the vehicle. We are going to have the normal force and the force due to gravity now acting inwards towards the curve, Not only are we going to have the friction force, we are also going to have a radial acceleration which, which is important here, that itself is not a force. I just wanted to establish the direction of that acceleration. So let's use Newton's laws here in both the X and Y direction and we'll be able to come up with this coefficient of friction here, let's first start in the X direction. We have that the sum of all forces in the X direction is going to equal a mass time. Some sort of acceleration, but as you can see here, I have already labeled that acceleration and it's going in the negative X direction. So this will just be our negative radial acceleration. Right? So some of our forces here, uh the only force we have acting in this direction is our negative friction force. This is going to be time equal to our mass times our radial acceleration. But we have formulas for all of these terms here. But what are they write well for the friction force, we have the coefficient of friction times. The normal force is equal to mass times our tangential velocity squared divided by the radius. If you'll notice these negative signs cancel out and make everything positive here. Now we want to solve for this coefficient of friction. So I'm gonna divide both sides by our normal force and we get that our coefficient of friction is equal to this equation right here. Now we have all of these terms except what is the normal force? Well, in order to get the normal force, we are going to have to look in the y direction of forces here. So we have that. The sum of all forces in the Y direction is equal to some mass times some acceleration here, acting in the y direction. We have the normal force and then in the negative direction we have the force due to gravity. But what about this right side of the equation here? Well, the car's not lifting on off of the ground and it's not barreling into the ground, meaning that our acceleration is just going to be zero, making the right side of this zero. But this implies that the normal force is equal to mass times gravity so we can sub this into this formula right here in order to find out Coefficient of friction. So let's go ahead and do that. That our coefficient of friction is equal to mass. Times our tangential velocity squared all divided by the radius, times our mass times the acceleration due to gravity will notice the mass cancels out on top and bottom and we can fill in the rest with the terms we know are tangential velocity is 14 and we will square that times are radius of 50m. And then our acceleration due to gravity of 9.8, which when you plug all of this into your calculator, you get a coefficient of friction of . corresponding to our answer choice of D. Thank you all so much for watching. Hope this video helped. We will see you all in the next one