To keep going in a circular path, you must always have a net force equal to mv 2 / r pointing towards the center of the circle. The normal force changes as you travel around a vertical loop it changes because your speed changes and because your weight has a different effect at each part of the circle. The critical factor in determining whether you make it completely around is the normal force if the track has to exert a downward normal force at the top of the track to keep you moving in a circle, you're fine, but if the normal force drops to zero you're in trouble. To do this without falling off the track, you have to be traveling at least a particular minimum speed at the top. Some roller-coasters have loop-the-loop sections, where you travel in a vertical circle, so that you're upside-down at the top. Note that this does not apply to standard highway off-ramps! The appropriate conditions for no maximum safe speed (or at least a very high maximum) would be found at racetracks like the Indianapolis Speedway, for example. Note that when the coefficient of friction is zero (i.e., the road is very slippery), the maximum speed reduces to the design speed,Īnd for certain combinations of theta and the coefficient of friction (both large, in general) the denominator turns out to be negative, implying that there is no maximum speed in those cases, you could drive as fast as you wanted without worrying about skidding. The equation can then be rearranged to solve for the maximum speed: There is an m in every term, so the mass cancels out. Substituting, and plugging in the equation for the normal force, gives: In the x-direction, the force equation is: Note that we're solving for the maximum speed at which the car can go around the curve, which will correspond to the static force of friction being a maximum, which is why it's valid to say that. This can be rearranged to solve for the normal force: Moving from the free-body diagram to the force equations gives: In this case the coordinate system is horizontal and vertical, because the centripetal acceleration points horizontally in towards the center of the circle and there is no vertical component of the acceleration. There is a critical difference, however for the box on an inclined plane, the coordinate system was parallel and perpendicular to the slope, because the box was either moving, and/or accelerating, up or down the slope. Note that the diagram looks similar to that of a box on an inclined plane. The diagram, and a free-body diagram, of the situation is shown here. Also, if we're worried about the maximum speed at which we can go around the banked turn, if there was no friction the car would tend to slide towards the outside of the curve, so the friction opposes this tendency and points down the slope. Even though the car is moving, the car tires are not slipping on the road surface, so the part of tire in contact with the road is instantaneously at rest with respect to the road. The first thing to realize is that the frictional force is static friction. The speed is known as the design speed of the curve (the speed at which you're safest negotiating the curve) and is given by:Ĭonsider now the role that friction plays, and think about how to determine the maximum speed at which you can negotiate the curve without skidding. The textbook does a good analysis of a car on a banked curve without friction, arriving at a connection between the angle of the curve, the radius, and the speed. Going too fast is another story, however. In most cases the coefficient of friction is sufficiently high, and the angle of the curve sufficiently small, that going too slowly around the curve is not an issue. In theory, then, accounting for friction, there is a range of speeds at which you can negotiate a curve. With no friction, if you went faster than the design speed you would veer towards the outside of the curve, and if you went slower than the design speed you would veer towards the inside of the curve. These off-ramps often have the recommended speed posted even if there was no friction between your car tires and the road, if you went around the curve at this "design speed" you would be fine. More circular motion More circular motionĪ good example of uniform circular motion is a car going around a banked turn, such as on a highway off-ramp.
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