**Abstract**

It is common knowledge to everyone that when one applies a force on a spring or a rubber, it will stretch and be displaced. Questions then arise on the relationship between the applied force and the amount of stretch. Such a question was answered by the works of a scientist named Robert Hooke. His findings are famously known as Hooke’s law.

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**Introduction**

Hooke’s law is a physical principle that states, the force F that is required to stretch or compress a spring over a given distance X is usually directly proportional to that distance. The formula is expressed as F = kX with k being a factor that is constant in a spring. Constant factors can either be stiffness or brittleness.

Hooke’s equation applies in situations where an elastic body is deformed or destroyed. The elastic material or object to which Hooke’s law is applied is usually called a linier-elastic material or a hooked material. The law is regarded as a linier approximation of the first order to a real response of a spring and any other object that is compressible beyond a set limit. The law may fail to apply in the event that the limit set, usually referred to as the elastic limit, is exceeded at which point the object or material is deformed.

In the case of a solid object, it is considered to be an accurate approximation and applies as long as the applied force and deformities are minimal. Hooke’s law is applied in studies related to all branches of science.

Modern theory of elasticity sums up Hooke’s law as the strain or deformation of an elastic object or material that is proportional to the applied force on the same material or object. Forces applied may not be the same or uniform in terms of their components hence dictating that the proportionality factor in the form of a linier map is represented by a matrix of numbers. To obtain the relationship between strain and stress on complex objects characterized by intrinsic properties, Hooke’s law and Newton’s law can be applied. When applied, it obtains that a homogenous rod with uniform cross-section will certainly behave in the manner a simple spring does when it is stretched with stiffness (k) directly proportional to its cross-sectional area and inversely proportional to its height.

**Discussion**

This is a graph of force (F) as applied on an object against the elongation (X) for an object that is helical in nature. The dashed line represents what it would actually look like in nature. The red arrows represent extension while the blue arrows represent compression.

From the graph, when a force (F) is applied on a string from the loose end and pulled down, the spring reaches an equilibrium where its length changes no more with X being the distance the free end is displaced.

Hooke’s law states that F = kX with k being a positive real number and a characteristic of the object. The formula applies when the spring is equally compressed with F and X both being negative. This is show as plotted on the graph. It results into a straight line graph. It s application relies on the convention that F is the reaction force exerted by the spring on the object pulling it down. This changes the equation to F = -kX as the direction of reaction will be opposite to that of displacement.

In general scalar springs that are elastic in nature with arbitrary complexities, the deformation and stress is expressed in a single number which can either be positive or negative. When applied on a block of rubber that is attached to two parallel plates and deformed through shearing instead of stretching or compressing it. The shearing force F and the sideway displacement of plates X do obey Hooke’s law. Either, it can be applied in vector formulation in helical springs stretched or compressed along its axis where by the applied force and the resultant elongation or compression occur in the same direction. In this case, if F and X are defined as vectors, Hooke’s equation will hold and dictate that the force’s vector is the elongation vector as multiplied by a fixed scalar.

In the general tensor form of elastic bodies that are deformed or displaced in only one direction when force is usually exerted that is of a different direction, the magnitude of displacement X will tend to be proportional to the magnitude of the force F as long as the direction of the force remains the same.

**Conclusion**

Hooke’s law is measured in internationally accepted standard units. The SI unit for displacement is meters (m) and the unit for force is Newtons (N). The spring constant k is measured in newtons per meter (N/m).

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