Relatedness of weight, normal, and centripetal force I'm taking a physics course. Since the beginning of the semester I've been troubled by this question: How do you determine how weight, normal and centripetal forces are related?
I've seen plenty of box-on-flat-surface examples where the Normal and Weight forces do not effect the Net force, but in centripetal problems, the solutions almost always say something along the lines of
FNet = W + N
I understand the net force is centripetal force, but not how it's related to W or N.
Here's a picture I drew of two situations I know are true from practice. 

I've been tasked with a problem like the third, and I'm not sure where to start, as I don't understand why the first two are true. You can assume this is all uniform circular motion.
Thanks for any input
 A: Forget for a moment about all these names of forces and just consider this situation from a physics point of view or from experience (in a roller-coaster for instance). Think of what forces are exerted on the cart. Think of what forces you would feel sitting in the car... 
1st image
Obviously gravity/weight ($W$) will always affect the cart. This force is always pointed downwards.
Now, if you move/drive along a curved (or circular) track, you feel being pushed to the outside, away from the center of the circle. This force is called ''centrifugal force'' ($C$) and equal in magnitude (opposite in direction) to the ''centripetal force'', which is the force that makes you move along the curved track.
In this 1st image the centrifugal force is pointing up, i.e. in the opposite direction of the weight. So the car is pressed onto the track by gravity (which is pointing down) and is lifted/pushed away from the track by the centrifugal force (which is pointing up).
The total (net) force is the vector sum of these forces. And since they are antiparallel, the magnitude of the net force that pushes the cart onto the track is just the difference of weight and centrifugal force. 
If the car is pushed on the track by some force, by Newton's third law there is an equal force in opposite direction exerted by the track on the car. This force is the ''normal force'' ($N$) here. 
So you have for the magnitude of the normal force the relation:
$$N=||W|-|C||$$
which tells you that the normal force due to weight is reduced by the centrifugal force. You can say the track does not need to push the car as much up anymore because the centrifugal force helps it.
2nd image
With the same reasoning as above, here you have gravity and centrifugal force both pointing down, i.e. both press the car onto the track. The track needs to not only keep the weight of the cart but also compensate for the centrifugal force which is pushing on it. The normal force is therefore the sum of these two forces:
$$N=|W|+|C|$$
3rd image
This should be easy now. Find out what the total force is that is pushing the figure onto the track. This should be equal in magnitude to the normal force.
PS: Note that all of the above discussion is only valid for the situation depicted. As the car moves around the circle, the forces will add differently depending on the position along the circle. 
A: The equation of motion should be the net force on the object is equal to the mass of the object times the centripetal acceleration of the object.
I would avoid using the term "centripetal force" as this can often cause confusion.

In terms of direction the situation in middle diagram will always have the forces in the directions as shown in your diagram so with up as positive one can write $N-W=ma_{\rm centripetal}$.

The third picture has three possibilities depending on the speed of the object.  
If the object is moving fast enough so that the weight is not sufficient to provide the necessary centripetal acceleration then the normal reaction force on the object must be in the same direction as the weight.
With down as positive the equation of motion is then $W+N=ma_{\rm centripetal}$
If on the other hand the object is moving slow enough so that the weight provides a force which is to large to provide the necessary centripetal acceleration then the track must provide an upward force on the object opposite in direction to the direction of the weight.
With down as positive the equation of motion is then $W-N=ma_{\rm centripetal}$
For a particular speed of the object the weight provides a force on the object which produces the necessary centripetal acceleration and there is no force on the object due the track.
With down as positive the equation of motion is then $W=ma_{\rm centripetal}$
To solve such problems all you need to do is draw a free body diagram and guess the direction of the force on the object due to the track.
After doing you calculations if your value for the force on the object due to the track turns out to be positive then the direction that you guessed was correct.
On the other hand, if your value of the force turns out to be negative then you know the actual direction of the force is opposite to the direction that you originally guessed.
