Why doesn't a car moving on a curved path have a horizontal gravity force but it has a horizontal normal force?

Question

So I was wondering..
When we solve questions of an object sliding on an incline, we consider the force of gravity to have two components, a horizontal and a vertical, and the normal force to have only one component directing perpendicular to the surface.

But when we solve a question about a car moving on a curved path, we only consider the force of gravity to have one component directing downward, and the normal force to have two components, horizontal and vertical.
So why is that?

Aren't a car on a curved path and an object on an incline essentially the same?

Cort Ammon · Accepted Answer

In both cases, you break the force into two components, one in one direction and one at 90 degrees to it.  However, in some situations, you can easily prove that one of those components is zero.  In those cases, we skip a few steps and say "it just has one component."
In the case of the object sliding on an incline, you have quietly made the choice to break vectors up into a direction along the inline plane and one that is normal to the inclined plane.  Since gravity is not purely in one of those two directions, it gets broken up into two non-zero components.
In the case of a car on the banked turn, you have quietly made the choice to break up the vectors into horizonal and vertical.  Since gravity is purely in the vertical direction, it gets broken up into one non-zero component and one zero component (which we quietly ignore).
You could solve the inclined plane problem in horizontal and vertical (with gravity having one non-zero component).  You could solve the car on a banked turn problem with a normal component and a component along the banked turn (with gravity having two non-zero components).  The difference is not the physics of the problem, but rather than you choose one orientation or the other for your coordinate system to make the math easier.  The choice of directions for your coordinate system does not change the physics of what happens, but it can make the math easier.  In particular, when it comes to inclined planes, there's often a big advantage computationally in using normal/along-the-plane coordinates because it makes the handling of normal forces and friction simpler.  But it's just a choice.  The physics didn't change.

Jerry Schirmer · Answer

It's only a trick for the "car on an incline" problem.
One makes the incline problem simpler by rotating the coordinate system so that the incline is flat.  This has the consequence of changing "down" from being just simply pointing in the negative y direction to pointing at an angle.  But, gravity still points downward, our new coordinate system has just changed what direction that is.  Similarly, the normal force, which is shorthand for "the net force that keeps you from just falling into surfaces", is still just perpendicular to the surface (which hey, in our new coordinate system, is now just in the y-direction!).  All that has changed is the coordinate system.
if you have something like a loop, to make this trick work, you'd have to constantly change your coordinate system, rather than just simply rotate it once.  The end result would be that this new coordinate system would be non-inertial, and Newton's laws would no longer hold, and you'd have to introduce "inertial forces" to make your prediction work.  This is more complicated than just using "normal" coordinates, so that's what we do.

Why doesn't a car moving on a curved path have a horizontal gravity force but it has a horizontal normal force?

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