Let us consider now the same car as before driving
over a bunked curve of angle
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
. In order to do
so, the dependence of these components on the angle
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
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
which is an equation for the normal force
In the case of the horizontal component,
Net force in the horizontal direction:
In this case, this net force is the car centripetal
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.