 One of the most common examples of uniform circular motion that you may even have encountered this morning is a vehicle, such as a car, bus, bicycle, quad bike, or tractor, turning around a bend. In this video, we'll look at motion around bends and why the turns on racetracks are banked at such steep angles. Firstly, let's make sure that we're actually dealing with uniform circular motion. It's straightforward to see that a car travelling around a roundabout is following a circular path, but what about a car going around a bend or making a turn? This path is clearly not a circle, but it turns out a section of the turn is approximately circular. So, if the car is going at a constant speed, then we can analyse this part of the turn. We know that when a car turns, something must be causing the centripetal force. That something is the friction between the tyres and the road. The same force that keeps you from pushing a car sideways allows cars to turn around bends. Now, at some point, friction won't be strong enough to oppose the applied force, and the car will start sliding. This occurs at the maximum friction force. The maximum friction force dictates the faster speed a car can travel around a bend. In this example, we'll call the friction force F subscript F. We know that the car experiences a centripetal force, so F subscript C is equal to the mass times the velocity squared divided by the radius. And this centripetal force is provided by friction. So this centripetal force is equal to the friction force. Therefore, the speed is equal to the square root of the friction force times the radius divided by the mass of the vehicle. But we also know that we can write friction in terms of the normal force and the coefficient of friction. So we know that the friction force is equal to the coefficient of friction, or mu, times the normal force. Since the car is not accelerating vertically up into the air or down into the ground, the net vertical force is zero, so vertical forces must balance. This means that the normal force is equal to the weight, on flat ground at least. Rearranging these equations, we find that the speed can be written as the square root of mu times gr. To look at some numbers, a car navigating a bend with a radius of 20 meters and a coefficient of dynamic friction of 0.7 could travel around the bend at a maximum speed of 11.7 meters per second. This is equal to 43.1 kilometers per hour. Of course, if there was no friction, the car wouldn't be able to go around the bend at all. However, by banking the bend, like you see at a velodrome or at a car racing track, cars can more easily travel around the bend, even if there is no friction at all. Intuitively, it can help to imagine a marble rolling around the inside of a cone. There's very little friction between the marble and the cone, however, if you spin the cone, the marble will begin to undergo uniform circular motion. To see this, let's look at a very simple model, a car sliding around a banked track with no friction. Firstly, how do we know when we can use our understanding of uniform circular motion to understand a banked track? If we consider a slope that is curving around a central point, so that, like the marble we discussed before, you can model the car as if we're travelling along the inside of a cone, with the radius being the distance to the central axis of the cone, and the bank angle equal to the angle the cone makes of the horizontal, then, as long as the car is travelling at a constant speed, we can use our uniform circular motion equations. It's difficult to draw both the bank and the forces acting on the car all in the same drawing, so bank tracks are typically shown as simply a vertical cutaway of the problem. In this case, the cone cutaway will resemble a triangle. So what forces are acting on the car? Well, there's gravity pointing downwards, of course, and there will be a normal force applied by the track on the car. This normal force points perpendicularly to the surface, like any normal force. Since there's no friction, that's it. Those are all the forces acting on the car. As the car is following a circular path, there must be a force pointing radially inwards. Looking at our free body diagram of the car, the centripetal force responsible for uniform circular motion must come from the horizontal component of the normal force, so that's what we need to find. Let's start with what we know. We know that the force due to gravity is mg, and it points directly downwards. We also know that the car isn't lifting off into the air or sinking into the road beneath it. That is, the vertical forces must balance. To get the vertical components of the normal force, we can draw out a vector triangle, showing both the vertical component and the horizontal component, which together add up to the total normal force. The angle between the normal force and the vertical is theta, the same as the angle of the banked track. This can be shown using geometry. This means that the vertical component of the vertical force is equal to the normal force multiplied by car's theta, and the horizontal component that points radially inwards is equal to the normal force times sine theta. To get the horizontal component of force, we need to calculate the normal force, and we can do this by balancing the vertical forces acting on the car. As there is no net vertical acceleration, we know that the gravitational force is equal to the normal force times car's theta, and therefore the normal force is equal to mg divided by car's theta. Now let's look at the horizontal components. We know, as the car is undergoing uniform circular motion, that the centripetal force is equal to the horizontal component of the normal force. Calculating, we find that the centripetal force is equal to mg sine theta divided by car's theta, which is equal to mg times tan of theta. If we want to find the velocity of the car, we can plug in our uniform circular motion equation and find that the velocity is equal to the square root of rg tan theta. Let's plug in some values. We'll use a car mass of 1200 kilograms, a banking radius of 20 meters, and a bank angle of 5 degrees to find the centripetal force and the car's velocity. The centripetal force is equal to 1029 newtons, and the velocity is equal to 4.1 meters per second, or in a more typical unit for cars, this is equal to 14.9 kilometers per hour. Now, in reality, you'd have a combination of the normal force and the friction force providing the centripetal acceleration. However, the calculations we've seen in this video give you a worst-case scenario where there's no friction and you have to rely entirely on the banking of the track.