Inclined planes, Banked highways, and Normal force

AI Thread Summary
The discussion centers on the differences in normal force calculations for inclined planes and banked curves. For inclined planes, the normal force (FN) is expressed as FN = mgcos(∅), where it acts perpendicular to the surface, assisting in opposing gravity. In contrast, for banked curves, the equation FNcos(∅) = mg indicates that the normal force also contributes to the centripetal force required for circular motion. The key distinction lies in how the third force is resolved: it is perpendicular to the normal force on inclined planes and horizontal for banked curves. This clarification helps reconcile the apparent differences in notation and force dynamics between the two scenarios.
trogdor5
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Alright, I don't have a specific homework problem here, just a general question. I've attached two pages that I will be referencing.

Figure 1. Inclined Plane
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As can be seen here, the FN = mgcos(∅)

Figure 2. Banked Curve
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It seems as if the opposite is true here. FNcos(∅) = mg

The same holds true for the horizontal components. My question is, what is happening here? Why is there different notation all of a sudden. I completely understand the proof for the banked curve I just don't follow the inclined plane problem. It seems as if the two should be identical. What am I missing here?

Any effort would be extremely appreciated! Honestly, the work the guys who answer questions here is amazing and I really plan on contributing to the community when I feel I'm qualified to help!
 
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Look at the third 'force' in each case.
For the inclined plane, the third force is perpendicular to the normal force. You can think of it as assisting the normal force in opposing gravity. To eliminate the third force you resolve parallel to the normal force.
For the banked curve, the third force (the resultant in this case) is the one required to keep the car going around a vertical axis, so is horizontal. Some of the normal force goes into providing that, and only what's left over gets to oppose gravity. To eliminate the third force you resolve vertically.
 
Oh wow, thank you very much! I get it now
 
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