Hooke's law is a principle of physics that states that the force needed to extend or compress a

spring by some distance is proportional to that distance. That is: where is a

Hooke's equation in fact holds (to some extent) in many other situations where an elastic body is
deformed, such as wind blowing on a tall building, a musician plucking a string of a guitar, or
the filling of a party balloon. An elastic body or material for which this equation can be assumed
is said to be linear-elastic or Hookean.

Hooke's law is only a first order linear approximation to the real response of springs and other
elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since
no material can be compressed beyond a certain minimum size, or stretched beyond a maximum
size, without some permanent deformation or change of state. In fact, many materials will
noticeably deviate from Hooke's law well before those elastic limits are reached.

On the other hand, Hooke's law is an accurate approximation for most solid bodies, as long as the
forces and deformations are small enough. For this reason, Hooke's law is extensively used in all
branches of science and engineering, and is the foundation of many disciplines such as
seismology, molecular mechanics and acoustics. It is also the fundamental principle behind the
spring scale, the manometer, and the balance wheel of the mechanical clock.

The modern theory of elasticity generalizes Hooke's law to say that the strain (deformation) of an
elastic object or material is proportional to the stress applied to it. However, since general
stresses and strains may have multiple independent components, the "proportionality factor" may
no longer be just a single real number, but rather a linear map (a tensor) that can be represented
by a matrix of real numbers.

In this general form, Hooke's law and Newton's laws of static equilibrium make it possible to
deduce the relation between strain and stress for complex objects in terms of intrinsic properties
of the materials it is made of. For example, one can deduce that a homogeneous rod with uniform
cross section will behave like a simple spring when stretched, with a stiffness directly
proportional to its cross-section area and inversely proportional to its length.

For linear springs

Consider a simple helical spring that has one end attached to some fixed object, while the free
end is being pulled by a force whose magnitude is . Suppose that the spring has reached a state
of equilibrium, where its length is not changing anymore. Let be the amount by which the free
end of the spring was displaced from its "relaxed" position (when it is not being stretched).
Hooke's law states that
or, equivalently,

where is a positive real number, characteristic of the spring. Moreover, the same formula holds
when the spring is compressed, with and both negative in that case. According to this
formula, the graph of the applied force as a function of the displacement will be a straight
line passing through the origin, whose slope is .

Hooke's law for a spring is often stated under the convention that is the restoring (reaction)
force exerted by the spring on whatever is pulling its free end. In that case the equation becomes

since the direction of the restoring force is opposite to that of the displacement.

Objects that quickly regain their original shape after being deformed by a force, with the
molecules or atoms of their material returning to the initial state of stable equilibrium, often obey
Hooke's law.

Hooke's law only holds for some materials under certain loading conditions. Steel exhibits linear-
elastic behavior in most engineering applications; Hooke's law is valid for it throughout its
elastic range. For some other materials, such as aluminium, Hooke's law is only valid for a
portion of the elastic range. For these materials a proportional limit stress is defined, below
which the errors associated with the linear approximation are negligible.

Rubber is generally regarded as a "non-hookean" material because its elasticity is stress

In physics, elasticity is the tendency of solid materials to return to their original shape after
being deformed. Solid objects will deform when forces are applied on them. If the material is
elastic, the object will return to its initial shape and size when these forces are removed.

The physical reasons for elastic behavior can be quite different for different materials. In metals,
the atomic lattice changes size and shape when forces are applied (energy is added to the
system). When forces are removed, the lattice goes back to the original lower energy state. For
rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces
are applied.

Perfect elasticity is an approximation of the real world, and few materials remain purely elastic
even after very small deformations. In engineering, the amount of elasticity of a material is
determined by two types of material parameter. The first type of material parameter is called a
modulus, which measures the amount of force per unit area (stress) needed to achieve a given
amount of deformation. The units of modulus are pascals (Pa) or pounds of force per square inch
(psi, also lbf/in2). A higher modulus typically indicates that the material is harder to deform. The
second type of parameter measures the elastic limit. The limit can be a stress beyond which the
material no longer behaves elastic and deformation of the material will take place. If the stress is
released, the material will elastically return to a permanent deformed shape instead of the
original shape.

When describing the relative elasticities of two materials, both the modulus and the elastic limit
have to be considered. Rubbers typically have a low modulus and tend to stretch a lot (that is,
they have a high elastic limit) and so appear more elastic than metals (high modulus and low
elastic limit) in everyday experience. Of two rubber materials with the same elastic limit, the one
with a lower modulus will appear to be more elastic.