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When describing an object sliding down a slope, we say that the normal force is less than the weight of the object, and that the vector weight minus the normal force equals the force with which the object is pushed down the slope. (bottom left in image)

However, when describing a car making a banking turn on an angled road, the normal force seems to be greater than the weight of the object, such that the normal force minus the weight is a horizontal centripetal force. (bottom right in image)

I don't understand how to reconcile these models and was hoping someone could share their insight about it.

diagram

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The normal force is one of a class of peculiar forces that vary depending on the situation. For example, the normal force of a table on a heavy book is greater than the normal force on a lightweight book. In this case you can't find the normal force unless you know the motion of the book (stationary). The normal force of the floor of an elevator on a book is different for a stationary elevator compared to the same book in an accelerating elevator. Forces of this type are called constraint forces.

So to reconcile your two pictures, you have to know the motion of the car. In one case it slides straight down the ramp, accelerating. In the other the it executes uniform circular motion. In both of your cases, the direction of the net force, which is constrained by the known motion, is enough information to completely solve the situation.

Bottom line: you can't find the normal force on an object without first knowing the motion of the object.

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