homework and exercises - Car on a frictionless banked curve

A car is on a banked curve, following a path which is part of a circle with radius $R$. The curve is banked at angle $\theta$ with the horizontal, and is a frictionless surface. What is the speed the car must go to accomplish this?

What I don't understand about this problem is why we assume there is only the normal force and the gravitational force on the vehicle. From that point onwards, I have no trouble following the solution.

The way I see it, if we're considering only that there exists a normal force and a gravitational force, something the car is doing (accelerating?) must be "adding" to the normal force. Or is it possible that the car could just be coasting, and it could keep a constant height along the banked curve?

Answer

You are right in thinking that the car's acceleration is what keeps it in place, but it is important to remember that an object moving at a constant speed in a circle is accelerating (despite not speeding up). The reason for this is that acceleration is defined as a "change in velocity," and velocity is a vector quantity (i.e. it has magnitude and direction). Therefore, the car's centripetal acceleration must have a force corresponding to it through Newton's Second Law (that force is the radial component of the normal force). And by requiring the position of the car to behave in certain ways, we can calculate that force exactly.

So yes, the acceleration is "adding" to the normal force. The sloped track must apply a greater force in the radial direction than it would if the car were not moving because in that situation the car would begin moving along the radial direction (sliding down the ramp). Likewise, it is not moving in the vertical direction (sliding down the ramp) so the normal force in that direction must be greater as well (to exactly cancel its weight). Since the normal force is the vector sum of its radial and vertical components, you can determine the increased normal force.

The truth is that by stating the problem this way, we have required certain constraints on the motion of the car from which we can deduce the forces acting on it by using Newton's Second Law.