## Hooke’s Law Calculator: Fs = -kx

Hooke’s Law states that the force needed to extend or compress a spring is directly proportional to the distance the spring is stretched or compressed. This means that there is a linear relationship between the force required to stretch a spring, and the length the spring is stretched. Hooke’s Law Equation F s = −k x

## Equation/Formula:

Hooke’s Law Equation F s = −k x

Missing Value(s): Hooke’s law calculator will find missing values in any of these given fields Enter one known value from either Force (N), Distance (m) or Tension (N), then press the ‘Calculate’ button.

Hooke’s law is an approximation that can be used to approximately determine stresses on objects under a wide variety of loads, when little information about their structure or inherent material characteristics are available. The equation shows that stress is directly proportional to strain. Hooke’s law can be used in the following two forms:

where formula_2 is stress, formula_3 strain or elongation, and k a proportionality constant. Hooke’s law is usually stated with Hooke’s Law F = -kx as opposed to Hooke’s Law E = -kx when the extension (or compression) of a spring is discussed. Hooke’s law may accurately describe an object’s deformations under many types of loads; however, it does not account for the effects of friction or non-constant coefficient of thermal expansion which are prevalent in many real world situations. It also requires that material properties remain constant with applied load, i.e., if YSI material tests are used to determine Hooke’s Law K values then Hooke’s Law must be reconsidered if the specimen is subsequently tested in tension or compression. Hooke’s law can also only approximate the true nature of a materials response because it does not account for other aspects of materials behavior, such as elastic memory or hysteresis (energy losses).

## Hooke’s law has been generalized to apply to many different types of systems and conditions:

There are some cases where Hooke’s law holds exactly, but these are rare. In most situations, Hooke’s law will give inaccurate answers and should not be used for precision work. There are two common inaccuracies that occur when Hooke’s law is used:

By definition, Hooke’s Law assumes that the stress within a spring is uniform. Hooke’s Law will still be applicable to situations where this is not the case, but significant deviations from Hooke’s Law can occur. Hooke’s Law does not account for frictional forces or the non-constant coefficient of thermal expansion. Both of these are very common sources of error when working with real springs.

Hooke’s law should only be applied to linear springs, which do not stretch more (or less) than proportionally when force is applied at both ends equally; see torsion for an example where Hooke’s law breaks down. Hooke did indeed investigate such a spring and found that Hooke’s law breaks down in this case. Hooke then proposed that the extension of a body depended upon its shape and size, as well as how it was stretched. Hooke specified four different laws of elasticity:

The last three laws were shown to follow mathematically from Hooke’s first law, but they are also experimentally verified. Hooke stated all his laws using “strict mathematical language” for his time; he had considerable difficulty devising the third law because he had no understanding of negative numbers (he actually reversed them). The experiments Hooke conducted to arrive at these laws involved placing small pieces of metal in between two powerful magnets and noting their deflection when attached to a balance wheel. When Hooke reported his results, Hooke’s Laws were not recognized for their importance (which was probably Hooke’s expectation given that he didn’t publish them as a separate paper). De Moivre published this work in 1727 and it is from him that Hooke’s laws acquired their modern name. Hooke considered the shape of the body, whereas de Moivre proposed Hooke’s third law but used various bodies, circles or spheres of different radii. Hooke had described such deformations in “Micrographia”.

In 1830 Augustin-Louis Cauchy derived mathematical expressions to describe Hooke’s first law of elasticity (that material stretches proportionally to its stress) and applied mathematics to further develop Hooke’s law. Cauchy stated Hooke’s third law using mathematical expressions in 1832 and Hooke’s second law was mathematically derived by Poisson in 1833.

Cauchy also gave Hooke credit for Hooke’s Law of Varying Force but failed to mention Hooke as its discoverer, which sparked a controversy between Hooke and Cauchy that lasted until Hooke died in 1703 (published posthumously). Hooke claimed that he had discovered a universal law with his experiments on the staves of wine barrels and published it under the name “Law IV” along with his three other laws, all of which were validated by experiment; however, Newton disagreed with this attribution and referred to Hooke’s law as “the swaggering paradox of Hooke.” Hooke also claimed that Newton stole his ideas and this controversy over Hooke’s Law is referred to as the Hooke–Newton debate.

The constant in Hooke’s law, formula_1 for a spring (or another elastic system) is often known as Hooke’s constant or Young modulus. It has units of force/distance² and depends on the stiffness of the material being stretched or compressed. Hooke’s law implies that it does not depend upon how much it is stretched or compressed; if formula_2 where x can be different values, then F = kx so that k would have to double if x was halved, and Hooke’s law would break down. Hooke’s constant is different for each type of material, but it has some properties that are universal to all materials, and Hooke’s law will hold regardless of the value of Hooke’s constant. Hooke’s Law is applicable to Hookean springs: that is, idealized objects which obey Hooke’s Law (i.e., the spring obeys Hooke’s Law if its spring constant k equals Hooke’s Constant).