In mechanics, the normal force F N {\displaystyle F_{N}} is the component of a contact force that is perpendicular to the surface that an object contacts. In this instance, the word normal is used in the geometric sense and means perpendicular, as opposed to its common meaning of "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 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. 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 of the object), that is, F n = m g {\displaystyle F_{n}=mg} , where m is mass, and g is the gravitational field strength (about 9.81 N/kg 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 = m g cos ( θ ) {\displaystyle F_{n}=mg\cos(\theta )}
where F n {\displaystyle 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 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 Cauchy stress tensor describing the stress state of the surface. That is:
N = n N = n ( T ⋅ n ) = n ( n ⋅ τ ⋅ n ) . {\displaystyle \mathbf {N} =\mathbf {n} \,N=\mathbf {n} \,(\mathbf {T} \cdot \mathbf {n} )=\mathbf {n} \,(\mathbf {n} \cdot \mathbf {\tau } \cdot \mathbf {n} ).}
or, in indicial notation,
N i = n i N = n i T j n j = n i n k τ j k n j . {\displaystyle N_{i}=n_{i}N=n_{i}T_{j}n_{j}=n_{i}n_{k}\tau _{jk}n_{j}.}
The parallel shear component of the contact force is known as the frictional force ( F f r {\displaystyle F_{fr}} ). The static coefficient of friction for an object on an inclined plane can be calculated as follows:
μ s = tan ( θ ) {\displaystyle \mu _{s}=\tan(\theta )}
for an object on the point of sliding where θ {\displaystyle \theta } is the angle between the slope and the horizontal.
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