Hey, guys. So we've been working with centripetal forces problems in the last couple of videos. And in this video, I want to show you a specific type of that kind of problem called the flat curve. This happens whenever you have an object that's traveling in a circle like this and there's a force that's keeping it going in a circle. You've definitely experienced this in everyday life. If you've been, like, driving your car and you approach a four-way intersection where there's a roundabout, as you're driving around that roundabout, you're basically just doing a flat curve. So I'm actually going to skip this point and just get back to it in just a second here because now we can kind of understand what this example is going to ask of us. So let's check this out. We're supposed to figure out the maximum speed that a car can drive while rounding a flat curve. So the idea is we want to find *v _{max}* for a massive 800 kilograms car that's driving around a flat curve like this. And we have the radius of this curve, which is 50. Now what we're also told is that this car cannot slip. Right? The tires can't slip on the road if the coefficient of static friction is going to be 0.5. So here's what we have. Right? We're trying to figure out this

*v*here, and we know that this car, as it's traveling around the flat curve, has a centripetal acceleration. This is

_{max}*a*and this equals v2/r. So, really, our velocity is actually kind of wrapped up inside this acceleration term. So in order to figure out this velocity, we're actually going to have to figure out the

_{c}*a*. In order to do that, we're going to actually have to stick to our steps. We're going to have to draw our free body diagram and then write

_{c}*f*equals

*ma*. So let's go ahead and do that. What's the free body diagram look like? Well, for the car, we know we're going to have a weight force that's straight down. This is our

*mg*. And then we're also going to have a normal force from the road. Right? So this is the vertical axis. But we're actually missing one force. There has to be a force that is keeping this object, accelerating centripetally. Remember, if there's an acceleration centripetal, there has to be a force centripetal that's causing it to do that. What is that force? And, And basically, what you need to know about these problems is that the force that keeps these objects traveling in circular motion is static friction. That's why we were given information about the tires not slipping on the road. We were also given the coefficient of static friction. So, basically, the idea is that the force that keeps this object through this car going in a circle is actually the static friction force. So if we were to look at, like, a top view of this road, it would basically look like the road like this. We have our car. And the only force that keeps this object centripetal accelerating is going to be the force of friction. Now we want to figure out the maximum speed that we can go before the start the tires actually start to slip. So that's not just any old friction, that's actually going to be static friction maximum. If we drove any faster than this maximum speed, the tires would start to slip and rub, and then therefore, you'd have kinetic friction. Okay? So let's go ahead and start off with our

*f*equals

*ma*. So we have

*f*equals

*ma*in this centripetal axis right here. And there's really only one force that is keeping this object accelerating in this centripetal direction, and that is the force of friction. And because it points along the same direction as the acceleration, it actually picks up a positive sign. So this is positive

*s*equals

_{f}s_{max}*m*. And now we're just going to replace this

*a*with mv2/r. So this is one of those rare situations where our static friction is actually positive because it points in the direction of our acceleration. Okay? So we want to figure out this velocity here. And so we need to figure out everything else about the problem. Unfortunately, we don't know what that static friction maximum is, so we're going to have to expand out that variable. So we know that

_{c}*f*is equal to

_{s}_{max}*μ*times the normal. This is equal to mv2/r. And we also know in these problems that if these are the only two forces in the vertical axis, your weight and your normal force, they have to cancel and be equal to each other. So this is

_{static}*μ*times

_{static}*ng*is equal to mv2/r. So we can see here is that our

*m*'s are actually going to cancel out, and we can write an expression for this

*v*here. So I'm going to do this. I have

^{2}*v*is equal to once I move the

^{2}*r*to the other side. I'm going to write this as

*g*times

*r*times

*μ*static. I like to think of this as

*gerμ*static. Alright. So this is actually going to be a super important equation here. If you know this equation and you get these kinds of flat curves problems like a multiple choice, you'll be able to solve them very quickly using this equation. This is also going to be really important because we're going to talk about this equation in later videos. Okay? So now we just have to go ahead and solve. We have,

*v*is equal to now I'm just going to take the square roots and then plug in all the numbers. So this is 9.8. Our radius is 50, and then our

*μ*is 0.5. If you go ahead and solve for this, you're going to get 15.6 meters per second, and that is your answer. If you were traveling any faster than 15.6, then the tires would start to slip, and you would actually start skidding, and your friction would turn into kinetic. So that's it for this video. Let me know if you guys have any questions.

_{static}