Normal Force
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In
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, the normal force F_n is the component of a
contact force A contact force is any force that occurs as a result of two objects making contact with each other. Contact forces are ubiquitous and are responsible for most visible interactions between macroscopic collections of matter. Pushing a car or kicking ...
that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance ''
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
'' is used in the geometric sense and means
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
, as opposed to the common language use of ''normal'' meaning "ordinary" or "expected". A person standing still on a platform is acted upon by gravity, which would pull them down towards the
Earth's core The internal structure of Earth is the solid portion of the Earth, excluding its atmosphere and hydrosphere. The structure consists of an outer silicate solid crust, a highly viscous asthenosphere and solid mantle, a liquid outer core whose ...
unless there were a countervailing force from the resistance of the platform's molecules, a force which is named the "normal force". The normal force is one type of
ground reaction force In physics, and in particular in biomechanics, the ground reaction force (GRF) is the force exerted by the ground on a body in contact with it. For example, a person standing motionless on the ground exerts a contact force on it (equal to the perso ...
. If the person stands on a slope and does not sink into the ground or slide downhill, the total ground reaction force can be divided into two components: a normal force perpendicular to the ground and a frictional force parallel to the ground. In another common situation, if an object hits a surface with some speed, and the surface can withstand the impact, the normal force provides for a rapid deceleration, which will depend on the flexibility of the surface and the object.


Equations

In the case of an object resting upon a flat table (unlike on an incline as in Figures 1 and 2), the normal force on the object is equal but in opposite direction to the gravitational force applied on the object (or the
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
of the object), that is, F_n = mg, where ''m'' is mass, and ''g'' is the
gravitational field strength The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in th ...
(about 9.81 m/s2 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 be sturdy enough to deliver this normal force without breaking. However, it is easy to assume that the normal force and weight are action-reaction force pairs (a common mistake). In this case, the normal force and weight need to be equal in magnitude to explain why there is no upward acceleration of the object. For example, a ball that bounces upwards accelerates upwards because the normal force acting on the ball is larger in magnitude than the weight of the ball. Where an object rests on an incline as in Figures 1 and 2, 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: F_n = mg \cos(\theta) where F_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 the 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 is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
and the force of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
.


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 Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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 In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of the unit normal with the
Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
describing the stress state of the surface. That is: \mathbf = \mathbf\, N = \mathbf\, (\mathbf \cdot \mathbf) = \mathbf\, (\mathbf \cdot \mathbf \cdot \mathbf). or, in indicial notation, N_i = n_i N = n_i T_j n_j = n_i n_k \tau_ n_j. The parallel shear component of the contact force is known as the
frictional force Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
(F_). The static
coefficient of friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
for an object on an inclined plane can be calculated as follows: \mu_s=\tan(\theta) for an object on the point of sliding where \theta is the angle between the slope and the horizontal.


Physical origin

Normal force is directly a result of
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
and not a true force ''per se'': it is a result of the interactions of the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s at the surfaces of the objects. The atoms in the two surfaces cannot penetrate one another without a large investment of energy because there is no low energy state for which the electron
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s from the two surfaces overlap; thus no microscopic force is needed to prevent this penetration.Lieb, E. H. (1991). The stability of matter. In The Stability of Matter: From Atoms to Stars (pp. 483-499). Springer, Berlin, Heidelberg However these interactions are often modeled as
van der Waals force In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and th ...
, a force that grows very large very quickly as distance becomes smaller. On the more macroscopic level, such surfaces can be treated as a single object, and two bodies do not penetrate each other due to the stability of matter, which is again a consequence of Pauli exclusion principle, but also of the fundamental forces of nature: cracks in the bodies do not widen due to electromagnetic forces that create the
chemical bond A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of ...
s between the atoms; the atoms themselves do not disintegrate because of the electromagnetic forces between the electrons and the nuclei; and the nuclei do not disintegrate due to the nuclear forces.


Applications in real life

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). When we define upward to be the positive direction, constructing
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
and solving for the normal force on a passenger yields the following equation: N = m(g + a) In a
gravitron The Gravitron is an amusement ride, most commonly found as a portable ride at fairs and carnivals. The Gravitron first appeared at Morey's Piers in 1983 and quickly became a fixture at amusement parks in many countries. It is a modification of ...
amusement ride, the
static friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
caused by and perpendicular to the normal force acting on the passengers against the walls results in suspension of the passengers above the floor as the ride rotates. In such a scenario, the walls of the ride apply normal force to the passengers in the direction of the center, which is a result of the centripetal force applied to the passengers as the ride rotates. As a result of the normal force experienced by the passengers, the static friction between the passengers and the walls of the ride counteracts the pull of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
on the passengers, resulting in suspension above ground of the passengers throughout the duration of the ride. When we define the center of the ride to be the positive direction, solving for the normal force on a passenger that is suspended above ground yields the following equation: N = \frac where N is the normal force on the passenger, m is the mass of the passenger, v is the tangential velocity of the passenger and r is the distance of the passenger from the center of the ride. With the normal force known, we can solve for the static coefficient of friction needed to maintain a net force of zero in the vertical direction: \mu = \frac where \mu is the static coefficient of friction, and g is the gravitational field strength.


See also

*
Contact mechanics Contact mechanics is the study of the deformation of solids that touch each other at one or more points.Johnson, K. L, 1985, Contact mechanics, Cambridge University Press.Popov, Valentin L., 2010, ''Contact Mechanics and Friction. Physical P ...
*
Normal stress In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...


References

{{DEFAULTSORT:Normal Force Force Statics