How can you show that for a given pair of surfaces and given normal force, the force of static friction increases with the applied force?

Frictional resistance forces are typically proportional to the force which presses the surfaces together. This force which will affect frictional resistance is the component of applied force which acts perpendicular or "normal" to the surfaces which are in contact and is typically referred to as the normal force. In many common situations, the normal force is just the weight of the object which is sitting on some surface, but if an object is on an incline or has components of applied forceperpendicular to the surface, then it is not equal to the weight.