In mechanics

Mechanics

Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, the normal force (occasionally N) is the component, perpendicular to the surface of contact, of the contact force

Contact force

In physics, a contact force is a force that acts at the point of contact between two objects, in contrast to body forces. Contact forces are described by Newton's laws of motion, as with all other forces in dynamics....

 exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.

The normal force is one of the components of the ground reaction force

Ground reaction force

In physics, and in particular in biomechanics, the ground reaction force is the force exerted by the ground on a body in contact with it.For example, a person standing on the ground exerts a contact force on it and at the same time an equal and opposite ground reaction force is exerted by the...

 and may coincide with it, for example considering a person standing still on the ground, in which case the ground reaction force reduces to the normal force. In another common situation, if an object hits a surface with some speed, and the surface can withstand it, the normal force provides for a rapid deceleration, which will depend on the flexibility of the surface.

Equations

In a simple case such as an object resting upon a table, the normal force on the object is equal but in opposite direction to the gravitational force applied on the object (or the weight

Weight

In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...

 of the object), that is, , where m is mass, and g is the gravitational field strength

Standard gravity

Standard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...

 (about 9.81 Newtons/kilogram on Earth). The normal force here represents the force applied by the table against the object that prevents it from sinking through the table, and requires that the table is sturdy enough to deliver this normal force without breaking.

Where an object rests on an incline, the normal force is perpendicular to the plane the object rests on. Still, the normal force will be as large as necessary to prevent sinking through the surface, presuming the surface is sturdy enough. The strength of the force can be calculated as:


where N is the normal force, m is the mass of the object, g is the gravitational field strength, and θ is the angle of the inclined surface measured from the horizontal.

The normal force is one of several forces which act on the object. In the simple situations so far considered, the most important other forces acting on it are friction

Friction

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

 and the force of gravity.

Using vectors

In general, the magnitude of the normal force, N, is the projection of the net surface interaction force, T, in the normal direction, n, and so the normal force vector can be found by scaling the normal direction by the net surface interaction force. The surface interaction force, in turn, is equal to the dot product of the unit normal with the stress tensor

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

 describing the stress state of the surface. That is,
Or, in indicial notation,

The parallel shear component of the contact force is known as the frictional force

Friction

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

 ().

The static coefficient of friction for an object on an inclined plane can be calculated as follows:

Real-world applications

For a person standing in an elevator either stationary or moving at constant velocity, the normal force on the person's feet balances the person's weight. In an elevator that is accelerating upward, the normal force is greater than the person's ground weight and so the person's perceived weight increases (making the person feel heavier). In an elevator that is accelerating downward, the normal force is less than the person's ground weight and so a passenger's perceived weight decreases. If a passenger were to stand on a "weighing scale", such as a conventional bathroom scale, while riding the elevator, the scale will be reading the normal force it delivers to the passenger's feet, and will be different than the person's ground weight if the elevator cab is accelerating up or down. The weighing scale measures normal force (which varies as the elevator cab accelerates), not gravitational force (which does not vary as the cab accelerates). It is impossible to measure true gravitational force without knowledge of the motion of one's immediate environment.

When we define upward to be the positive direction, constructing Newton's second law and solving for the normal force on a passenger yields the following equation:

Lagrangian Mechanics

In Lagrangian Mechanics

Lagrangian mechanics

Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....

, the normal force plays an important part in its formalization. Suppose a smooth surface, , is defined by the explicit equation . Furthermore, suppose a particle must move along this surface. The equation of motion would be given by



Where is the acceleration of the particle,