bullet

Banked Curves

 

Let us consider now the same car as before driving over a bunked curve of angle with respect to the horizontal . Notice that the radius of the circle extended from the position of the car to the center of the curve is a horizontal  line that draws a horizontal circle. That is, the radius of the circle is not parallel to the surface of the inclined (consider that the curve actually completes the circle and the result will be that the motion of the car describes a horizontal circle). For simplicity of the analysis, there will not be friction between the road and the car. Then, there are only two forces acting on the car, the weight of the car, and the normal force, . As always, the normal force is perpendicular to the surface.

In the present case, the acceleration and the weight are along the horizontal and vertical axis while the normal force is diagonal, the most convenient coordinate system to be used for solving the problem is one that coincide with the horizontal and vertical axis. Thus, the normal force must be breaking down into a horizontal and vertical component, . In order to do so, the dependence of these components on the angle must be understood. First, the triangle of the drawing determines the position of the complement of , . And second, from this new triangle , the location of the angle can be resolute . Finally, the components of the normal force can be calculated, .

Analyzing the vertical and horizontal forces of the diagram, the following equations are developed:

Net force in the vertical direction:

However, the car does not accelerates in this direction, .

Thus,

which is an equation for the normal force
implying that

 

In the case of the horizontal component,

Net force in the horizontal direction:

In this case, this net force is the car centripetal acceleration

   
Therefore,

Just as in the case of the unbanked curve, the maximum velocity for a given set of condition can be calculated by solving for the velocity in this equation.

That is,  

 

bullet
by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com