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Normal force

The normal force is defined as the net force compressing two parallel surfaces together, and its direction is perpendicular to the surfaces. In the simple case of a mass resting on a horizontal surface, the only component of the normal force is the force due to gravity, where N = m g {\displaystyle N=mg\,}. In this case, conditions of equilibrium tell us that the magnitude of the friction force is zero, F f = 0 {\displaystyle F_{f}=0}. In fact, the friction force always satisfies F f ≤ μ N {\displaystyle F_{f}\leq \mu N}, with equality reached only at a critical ramp angle (given by tan − 1 μ {\displaystyle \tan ^{-1}\mu }) that is steep enough to initiate sliding.

The friction coefficient is an empirical (experimentally measured) structural property that depends only on various aspects of the contacting materials, such as surface roughness. The coefficient of friction is not a function of mass or volume. For instance, a large aluminum block has the same coefficient of friction as a small aluminum block. However, the magnitude of the friction force itself depends on the normal force, and hence on the mass of the block.

Depending on the situation, the calculation of the normal force N {\displaystyle N} might include forces other than gravity. If an object is on a level surface and subjected to an external force P {\displaystyle P} tending to cause it to slide, then the normal force between the object and the surface is just N = m g + P y {\displaystyle N=mg+P_{y}}, where m g {\displaystyle mg} is the block's weight and P y {\displaystyle P_{y}} is the downward component of the external force. Prior to sliding, this friction force is F f = − P x {\displaystyle F_{f}=-P_{x}}, where P x {\displaystyle P_{x}} is the horizontal component of the external force. Thus, F f ≤ μ N {\displaystyle F_{f}\leq \mu N} in general. Sliding commences only after this frictional force reaches the value F f = μ N {\displaystyle F_{f}=\mu N}. Until then, friction is whatever it needs to be to provide equilibrium, so it can be treated as simply a reaction.

If the object is on a tilted surface such as an inclined plane, the normal force from gravity is smaller than m g {\displaystyle mg}, because less of the force of gravity is perpendicular to the face of the plane. The normal force and the frictional force are ultimately determined using vector analysis, usually via a free body diagram.

In general, process for solving any statics problem with friction is to treat contacting surfaces tentatively as immovable so that the corresponding tangential reaction force between them can be calculated. If this frictional reaction force satisfies F f ≤ μ N {\displaystyle F_{f}\leq \mu N}, then the tentative assumption was correct, and it is the actual frictional force. Otherwise, the friction force must be set equal to F f = μ N {\displaystyle F_{f}=\mu N}, and then the resulting force imbalance would then determine the acceleration associated with slipping.