From PER wiki
Contact forces are forces that we don't think much about because they seem so natural  if you bump into a wall, it will exert enough force on you to stop your motion toward the wall. This is because solid objects cannot pass through each other at normal speeds of encounter. Thus, each will modify the other's motion (yes, the wall will bend back a little). This fact implies that the objects interact. The resulting forces are called contact forces. The component of the contact force that prevents objects from passing through one another is called the normal force. The name normal force is meant to remind you that this component of the contact force is perpendicular (or normal) to the surface of contact. It is important to remember that both the objects in contact experience opposite normal forces as a result of Newton's Third Law.
Learning Goals
After completing this module, you should be able to:
 Describe the direction of the normal force acting on an object in contact with a surface.
 Use Newton's Second Law plus the constraint of equal accelerations perpendicular to the surface to solve for the normal force.
 Be able to draw normal forces on two touching objects that satisfy Newton's Third Law.
Notation for Normal Forces
In this course, we will give normal forces a special symbol. Instead of writing F with two subscripts, we will generally write N with two subscripts. The subscripts will still indicate the target and the source of the force, respectively. We will discuss the reason for this special notation when we introduce the full form of the contact force.
Illustrative Examples
Horizontal Surface (Floor)

Consider a box at rest on a horizontal surface, the normal force exerted on the box by the surface, N_{BF}, will be vertical.

Vertical Surface (Wall)

A book pushed against a wall or other vertical surface, the normal force will be horizontal as shown in the free body diagram below.

Inclined Surface (Ramp)

On a ramp, the normal force must remain perpendicular to the surface.

ObjectObject Interface

When two objects are in contact, they exert equal and opposite normal forces on each other, as required by Newton's Third Law. Note that the surface under the two objects will still exert a separate normal force, N_{1F}, on the lower object.

Normal Force as Constraint
The normal force is a constraint force, as is tension. It takes on whatever value is necessary to ensure that the objects in contact will not move into each other. This implies that the normal force constrains the motions of these objects so that they have the same velocity component perpendicular to the surface and will therefore remain in contact without getting closer or farther from each other. Thus when objects are in contact, the contact force will generate a force to cause the appropriate acceleration to change their velocities so they don't interpenetrate. Thus there is no "force law" for normal forces  the normal force must be found using Newton's second law and determining the acceleration of the objects from their (constrained) motion. To clarify this point, consider the following examples:
Stationary Horizontal Surface (Floor)

An object in contact with a stationary horizontal surface like a floor is constrained to have no acceleration in the vertical direction (or it will break contact), but may have a nonzero horizontal acceleration. Thus, the floor and the object need not have the same vector acceleration, but will have the same vertical component (zero in this case).
Pushing a box

A possible situation corresponding to this example is a box (mass m) being pushed along a horizontal surface with a horizontal force of magnitude F as shown in the figure. The box interacts with three objects, the person pushing the box, the surface and the Earth. As a result of these three interactions there are three forces acting on the box as shown in the free body diagram:
 is the force exerted on the box by the person.
 is the gravitational force on the box by the Earth of magnitude mg and pointing down
 is the normal force exerted on the box by the surface of unknown magnitude N_{bS} and pointing up.
To obtain the normal force we need to consider the forces only along the vertical direction. For that purpose we will write the vertical component of Newton's second law consistent with the coordinate system indicated in the figure, +y axis upwards.
Newton's second law, , implies that along the y  axis
Because there is no acceleration along the vertical direction, a_{y} = 0, we have
Realize that the value of the normal force is obtained as a consequence of Newton's second law (the object's motion) and the constraint imposed by the contact between both rigid surfaces.


Vertically Accelerating Horizontal Surface (Elevator)

An object in steady contact with a vertically moving horizontal surface like the floor of an accelerating elevator is constrained to have the same vertical velocity and acceleration as the elevator, but may have a nonzero horizontal acceleration. The elevator and the object will not have the same total acceleration, but will have the same vertical component.
Box on an elevator accelerating upwards

The box of mass m is at rest on the floor of an elevator which is accelerating upwards with acceleration a. The box interacts with two objects, the elevator's floor and the Earth. As a result of these two interactions there are two forces acting on the box as shown in the free body diagram:
 is the gravitational force on the box by the Earth of magnitude mg and pointing down
 is the normal force exerted on the box by the floor of unknown magnitude N_{bf} and pointing up.
We write the vertical component of Newton's second law consistent with the coordinate system indicated in the figure, +y axis upwards.
Newton's second law, , implies that along the y  axis
therefore
Realize that the value of the normal force is obtained as a consequence of Newton's second law (the object's motion) and it will adjust its value depending on the acceleration of the elevator to make the box and elevator move together.


Horizontally Moving Horizontal Surface (Truck Bed)

An object in contact with a horizontal surface that is moving horizontally like the bed of a moving pickup truck (assuming the road is smooth) is constrained to have have zero vertical acceleration, but might not have the same horizontal acceleration as the truck (the object could be sliding relative to the truck). The truck and the object are not required to have the same total acceleration, but will have the same vertical component.
