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When we talk about an object on an incline, the force of gravity has two components, one vertical and one parallel to the incline, as you know, while the normal force is matched in magnitude by the vertical component of gravity. When we talk about a banked road, however, the normal force becomes the force whose both components are considered, while gravity becomes the force that is matched by the vertical component of normal force. In a way, they almost reverse their roles. Why is that?


Perhaps the problem here isn't actually one of physics, but one of semantics. When physicists talk about an object on an incline, they describe the system with a "free body diagram" like is shown in the figure. Gravity always acts "down" (in other words, toward the center of the Earth). To assist with the math, the coordinates can be defined in any number of ways, but the physical system is the same. So in the figure, the free body diagram on the right is equivalent to the free body diagram on the left! To a physicist, a car on a banked road is the same as an object on an incline.

You do have to be careful to define your coordinate system ahead of time, and to stick to it throughout the problem you're trying to solve. Otherwise, you end up confusing what's meant by parallel and perpendicular, or horizontal and vertical. Is it vertical in relation to the flat surface, or the inclined surface? Is it parallel to gravity or to the incline? And so on. In that sense, it's meaningless to describe gravity and the normal force as "reversing roles," because you haven't defined your coordinate system. Even if engineers and physicists use different coordinate systems (which is often the case), like in the figure, the solution is always the same, so long as the work is self-consistent!

Answered by:

Kelly Chipps (AKA nuclear.kelly)
Postdoctoral Fellow
Department of Physics
Colorado School of Mines

Submitted by:

Berkan from his chair