The force arrow for centripetal force points in the opposite direction of my calculation? I need someone to clarify a conceptual problem I can't seem to surpass. Image there is a rollercoaster loop and a rollerbcoaster car enters the loop at high speed.
Once the car completes the full loop and is then at the last point at the bottom of the loop, I am trying to understanding the forces at play.
I can see that there is gravitational force:  $F_g$ = $mg$ and centripetal force: $F_c = \frac{mv^2}{r}$, and then there is the normal force $F_N$.
Let's say $F_g = 10$ ,and $F_c = 25$, wouldn't $F_N = 35$?
And if so, I am very confused about the force diagram. We have $F_N$ pointing up and $F_g$ pointing down, but what about $F_c$? My calculation has it pointing down, but in reality I thought centripetal force always points to the center of a circle?
Thanks for the help!
 A: The centripetal force isn't a separate force. It is the vector sum of the forces that are present. In the case of a roller coaster, the normal force and the force of gravity will add vectorically to produce a net force, and that net force must apply the centripetal force to keep you moving in a circle. Because the force of gravity is constant, the normal force will change (both in magnitude and direction) to provide the remainder of the centripetal force. 
When you are at the top of the loop, both the normal force and gravity point in the same direction. When you are at the bottom of the loop, the normal force points up, the opposite direction as gravity (down). So the normal force will have to be larger in magnitude when you are at the bottom of the loop than it is when you are at the top of the loop.
A: The centripetal force is always going to be supplied by some other force in the problem. You shouldn’t have a force labeled as the centripetal force but one of the forces in a problem will be serving as the centripetal force. What you should do is draw all the forces that are acting on the cart and determine which force is pointing towards the center of the circular path that the cart is taking. 
In this case at the bottom of the loop you only have the gravitational force and the normal force acting on the car. The normal force is the centripetal force because it is pointing towards the center of the circle. For example at the top of the loop the gravitational force would serve as the centripetal force and there would be no normal force. Anyway at the bottom of the loop since the normal force is serving as your centripetal force you should have:  
$F_N = m\frac{V^2}{r}$
Hopefully that helps to clear up your calculations.
A: The word "centripetal" comes from the latin, and means "seeking the
center". To seek or desire is one of the meanings of the latin verb
"petere". The word "petition" has the same origin. Hence that force is always towards the center, upward
in your case.
The Normal force is the (total) reaction of the rails on the car (in the absence of friction or lateral forces).
The simpler way to analyze it is to first ignore gravity. Your
rollercoaster is lost in space. It goes on its rails in a straight
line, at constant speed, just by inertia (no engine, no friction).
At some point, there is this loop on the rails. You pass the loop on
your inertia and reach the end of it.  Analyse what happens.
What are the forces (no gravity).
This analysis is simpler because you have uniform motion speed (not uniform velocity), no gravity and no falling cars.
Then you bring it back to earth. Horizontal linear motion works the
same (forget the air and other friction).  Almost the same analysis,
except that you have all the time an extra force produced by gravity,
which must be balanced too. That is all.
I am saying almost the same because your motion speed through the loop is no
longer uniform. But you can forget it, because the speed that you
loose going up, you get back going down, so that there is no change
in the case you analyze. And you are not asked about that anyway.
More importantly for you, you analyze at a point where the angular acceleration caused by gravity is null. Forget what I just wrote, if you do not get it. The effect of gravity is null because all it can do at the bottom is make the car go through the rails, which is necessarily compensated by a normal force (unless the rails are allowed to break).
( Fun question : assuming the loop is in the middle of a straight horizontal stretch, where are the rails most likely to break ?  i.e., where do they get the greater strain ? )
The conceptual difficulty may be that the centripetal force here is the
reaction of the rails balancing the inertial force of the car in circular motion (it may be only a part of that reaction of the rail, if other forces are to be balanced). That is the
part which I do not know how it is taught. As long as the motion stays
circular, this inertial force looks like an outward force (centrifugal)
that balances the centripetal force.  But go to your books for that
part.
This inertial force results from the circular motion. If the car stops
(or slows too much) at the top of the loop, then the inertial force
disappear, and so does the centripetal force. The car falls because of
gravity, not because of the lost centripetal force, now (close to) null. It is the
inertial force that keep the car on the rails. And that is why I say
that the centripetal force is a reaction in this case. (in the case of the moon it is earth gravity that causes it)
Now, in this problem, the normal force is the force that keeps the car from going
through the rails, i.e. the global reaction of the rails which may balance several other forces.
Concepts and Misconceptions :
The difficulty is that forces can be combined in different way and
take different names according to the role you are considering in the
system. But they must balance.
A centripetal force is not any force going through the center. It is
defined with respect to some motions around a center. Uniform
circular motion is one of them (actually the only one if you apply strictly what I say at the end of this paragraph). A body may well be submitted to other
forces going through the center (rocket push for example), but that
does not necessarily make them centripetal forces. Differentiation allows to
consider it for other non linear motions. It is exactly balanced by
the part of the inertial force orthogonal to the motion.
The Normal force is a reaction from a surface preventing a body from
entering that surface, and is orthogonal to that surface
("orthogonal"= "normal"="perpendicular" in this context). It balances
the other forces that would tend to make the body enter the surface.
Since There may be several such forces, the normal force can be
decomposed into a sum of colinear (i.e., parallel) forces
corresponding to the reaction to each of these other forces.
The reaction from a surface is not always orthogonal to that surface.
It can then be decomposed into a force orthogonal to the surface (the
normal force) and a force tangential to the surface ( the friction
force in general ... but it is actually a bit different when you talk of rails).
A centripetal force is any force (possibly resulting from a sum of
forces) that maintain the motion around the center as said above. It
may be gravity, the pull of a string, the push of a rocket engine (not
obvious), the reaction from an outer surface (normal force, or
friction, or both), or even combinations of these.  It may be only a
part of the Normal force (or of the combination of other forces) as explained above.
I hope I am not too much at odds with the way these things are taught.
