Sir George Stokes, 1st Baronet
Sir George Gabriel Stokes, 1st Baronet, was an Anglo-Irish physicist and mathematician. Born in County Sligo, Stokes spent all of his career at the University of Cambridge, where he was the Lucasian Professor of Mathematics from 1849 until his death in 1903; as a physicist, Stokes made seminal contributions to fluid dynamics, including the Navier-Stokes equation, to physical optics, with notable works on polarization and fluorescence. As a mathematician, he popularised "Stokes' theorem" in vector calculus and contributed to the theory of asymptotic expansions. Stokes was made a baronet by the British monarch in 1889. In 1893 he received the Royal Society's Copley Medal the most prestigious scientific prize in the world, "for his researches and discoveries in physical science", he represented Cambridge University in the British House of Commons from 1887 to 1892, sitting as a Tory. Stokes served as president of the Royal Society from 1885 to 1890 and was the Master of Pembroke College, Cambridge.
George Stokes was the youngest son of the Reverend Gabriel Stokes, a clergyman in the Church of Ireland who served as rector of Skreen, in County Sligo. Stokes home life was influenced by his father's evangelical Protestantism. After attending schools in Skreen and Bristol, in 1837 Stokes matriculated at Pembroke College, Cambridge. Four years he graduated as senior wrangler and first Smith's prizeman, achievements that earned him election of a fellow of the college. In accordance with the college statutes, Stokes had to resign the fellowship when he married in 1857. Twelve years under new statutes, he was re-elected to the fellowship and he retained that place until 1902, when on the day before his 83rd birthday, he was elected as the college's Master. Stokes did not hold that position for long, for he died at Cambridge on 1 February the following year, was buried in the Mill Road cemetery. In 1849, Stokes was appointed to the Lucasian professorship of mathematics at Cambridge, a position he held until his death in 1903.
On 1 June 1899, the jubilee of this appointment was celebrated there in a ceremony, attended by numerous delegates from European and American universities. A commemorative gold medal was presented to Stokes by the chancellor of the university and marble busts of Stokes by Hamo Thornycroft were formally offered to Pembroke College and to the university by Lord Kelvin. Stokes, made a baronet in 1889, further served his university by representing it in parliament from 1887 to 1892 as one of the two members for the Cambridge University constituency. During a portion of this period he was president of the Royal Society, of which he had been one of the secretaries since 1854. Since he was Lucasian Professor at this time, Stokes was the first person to hold all three positions simultaneously. Stokes was the oldest of the trio of natural philosophers, James Clerk Maxwell and Lord Kelvin being the other two, who contributed to the fame of the Cambridge school of mathematical physics in the middle of the 19th century.
Stokes's original work began about 1840, from that date onwards the great extent of his output was only less remarkable than the brilliance of its quality. The Royal Society's catalogue of scientific papers gives the titles of over a hundred memoirs by him published down to 1883; some of these are only brief notes, others are short controversial or corrective statements, but many are long and elaborate treatises. In scope, his work covered a wide range of physical inquiry but, as Marie Alfred Cornu remarked in his Rede lecture of 1899, the greater part of it was concerned with waves and the transformations imposed on them during their passage through various media, his first published papers, which appeared in 1842 and 1843, were on the steady motion of incompressible fluids and some cases of fluid motion. These were followed in 1845 by one on the friction of fluids in motion and the equilibrium and motion of elastic solids, in 1850 by another on the effects of the internal friction of fluids on the motion of pendulums.
To the theory of sound he made several contributions, including a discussion of the effect of wind on the intensity of sound and an explanation of how the intensity is influenced by the nature of the gas in which the sound is produced. These inquiries together put the science of fluid dynamics on a new footing, provided a key not only to the explanation of many natural phenomena, such as the suspension of clouds in air, the subsidence of ripples and waves in water, but to the solution of practical problems, such as the flow of water in rivers and channels, the skin resistance of ships, his work on fluid motion and viscosity led to his calculating the terminal velocity for a sphere falling in a viscous medium. This became known as Stokes's law, he derived an expression for the frictional force exerted on spherical objects with small Reynolds numbers. His work is the basis of the falling sphere viscometer, in which the fluid is stationary in a vertical glass tube. A sphere of known size and density is allowed to descend through the liquid.
If selected, it reaches terminal velocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, the density of the liquid, Stokes's law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of different diameter is used in the classic experiment to improve the accuracy of the calculation; the school experiment uses glycerine as the fluid, t
Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture. In modern materials science, fracture mechanics is an important tool used to improve the performance of mechanical components, it applies the physics of stress and strain behavior of materials, in particular the theories of elasticity and plasticity, to the microscopic crystallographic defects found in real materials in order to predict the macroscopic mechanical behavior of those bodies. Fractography is used with fracture mechanics to understand the causes of failures and verify the theoretical failure predictions with real life failures; the prediction of crack growth is at the heart of the damage tolerance mechanical design discipline. There are three ways of applying a force to enable a crack to propagate: Mode I fracture – Opening mode, Mode II fracture – Sliding mode, Mode III fracture – Tearing mode.
The processes of material manufacture, processing and forming may introduce flaws in a finished mechanical component. Arising from the manufacturing process and surface flaws are found in all metal structures. Not all such flaws are unstable under service conditions. Fracture mechanics is the analysis of flaws to discover those that are safe and those that are liable to propagate as cracks and so cause failure of the flawed structure. Despite these inherent flaws, it is possible to achieve through damage tolerance analysis the safe operation of a structure. Fracture mechanics as a subject for critical study has been around for a century and thus is new. Fracture mechanics should attempt to provide quantitative answers to the following questions: What is the strength of the component as a function of crack size? What crack size can be tolerated under service loading, i.e. what is the maximum permissible crack size? How long does it take for a crack to grow from a certain initial size, for example the minimum detectable crack size, to the maximum permissible crack size?
What is the service life of a structure when a certain pre-existing flaw size is assumed to exist? During the period available for crack detection how should the structure be inspected for cracks? Fracture mechanics was developed during World War I by English aeronautical engineer A. A. Griffith – thus the term Griffith crack – to explain the failure of brittle materials. Griffith's work was motivated by two contradictory facts: The stress needed to fracture bulk glass is around 100 MPa; the theoretical stress needed for breaking atomic bonds of glass is 10,000 MPa. A theory was needed to reconcile these conflicting observations. Experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, used extensively to predict material failure before Griffith, could not be a specimen-independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material.
To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental glass specimens. The artificial flaw was in the form of a surface crack, much larger than other flaws in a specimen; the experiments showed that the product of the square root of the flaw length and the stress at fracture was nearly constant, expressed by the equation: σ f a ≈ C An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress at the tip of a sharp flaw in a linear elastic material is infinite. To avoid that problem, Griffith developed a thermodynamic approach to explain the relation that he observed; the growth of a crack, the extension of the surfaces on either side of the crack, requires an increase in the surface energy. Griffith found an expression for the constant C in terms of the surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate; the approach was: Compute the potential energy stored in a perfect specimen under a uniaxial tensile load.
Fix the boundary so that the applied load does no work and introduce a crack into the specimen. The crack hence reduces the elastic energy near the crack faces. On the other hand, the crack increases the total surface energy of the specimen. Compute the change in the free energy as a function of the crack length. Failure occurs when the free energy attains a peak value at a critical crack length, beyond which the free energy decreases as the crack length increases, i.e. by causing fracture. Using this procedure, Griffith found that C = 2 E γ π where E is the Young's modulus of the material and γ is the surface energy density of the material. Assuming E = 62 GPa and γ = 1 J/m2 gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass. Griffith's criterion
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a constant volume independent of pressure. As such, it is one of the four fundamental states of matter, is the only state with a definite volume but no fixed shape. A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Water is, by far, the most common liquid on Earth. Like a gas, a liquid is able to take the shape of a container. Most liquids resist compression. Unlike a gas, a liquid does not disperse to fill every space of a container, maintains a constant density. A distinctive property of the liquid state is surface tension; the density of a liquid is close to that of a solid, much higher than in a gas. Therefore and solid are both termed condensed matter. On the other hand, as liquids and gases share the ability to flow, they are both called fluids. Although liquid water is abundant on Earth, this state of matter is the least common in the known universe, because liquids require a narrow temperature/pressure range to exist.
Most known matter in the universe is in gaseous form as interstellar clouds or in plasma from within stars. Liquid is one of the four primary states of matter, with the others being solid and plasma. A liquid is a fluid. Unlike a solid, the molecules in a liquid have a much greater freedom to move; the forces that bind the molecules together in a solid are only temporary in a liquid, allowing a liquid to flow while a solid remains rigid. A liquid, like a gas, displays the properties of a fluid. A liquid can flow, assume the shape of a container, and, if placed in a sealed container, will distribute applied pressure evenly to every surface in the container. If liquid is placed in a bag, it can be squeezed into any shape. Unlike a gas, a liquid is nearly incompressible, meaning that it occupies nearly a constant volume over a wide range of pressures; these properties make a liquid suitable for applications such as hydraulics. Liquid particles are bound but not rigidly, they are able to move around one another resulting in a limited degree of particle mobility.
As the temperature increases, the increased vibrations of the molecules causes distances between the molecules to increase. When a liquid reaches its boiling point, the cohesive forces that bind the molecules together break, the liquid changes to its gaseous state. If the temperature is decreased, the distances between the molecules become smaller; when the liquid reaches its freezing point the molecules will lock into a specific order, called crystallizing, the bonds between them become more rigid, changing the liquid into its solid state. Only two elements are liquid at standard conditions for temperature and pressure: mercury and bromine. Four more elements have melting points above room temperature: francium, caesium and rubidium. Metal alloys that are liquid at room temperature include NaK, a sodium-potassium metal alloy, galinstan, a fusible alloy liquid, some amalgams. Pure substances that are liquid under normal conditions include water and many other organic solvents. Liquid water is of vital importance in biology.
Inorganic liquids include water, inorganic nonaqueous solvents and many acids. Important everyday liquids include aqueous solutions like household bleach, other mixtures of different substances such as mineral oil and gasoline, emulsions like vinaigrette or mayonnaise, suspensions like blood, colloids like paint and milk. Many gases can be liquefied by cooling, producing liquids such as liquid oxygen, liquid nitrogen, liquid hydrogen and liquid helium. Not all gases can be liquified at atmospheric pressure, however. Carbon dioxide, for example, can only be liquified at pressures above 5.1 atm. Some materials cannot be classified within the classical three states of matter. Examples include liquid crystals, used in LCD displays, biological membranes. Liquids have a variety of uses, as lubricants and coolants. In hydraulic systems, liquid is used to transmit power. In tribology, liquids are studied for their properties as lubricants. Lubricants such as oil are chosen for viscosity and flow characteristics that are suitable throughout the operating temperature range of the component.
Oils are used in engines, gear boxes and hydraulic systems for their good lubrication properties. Many liquids are used as solvents, to dissolve other solids. Solutions are found in a wide variety of applications, including paints and adhesives. Naphtha and acetone are used in industry to clean oil and tar from parts and machinery. Body fluids are water based solutions. Surfactants are found in soaps and detergents. Solvents like alcohol are used as antimicrobials, they are found in cosmetics and liquid dye lasers. They are used in processes such as the extraction of vegetable oil. Liquids tend to have better thermal conductivity than gases, the ability to flow makes a liquid suitable for removing excess heat from mechanical components; the heat can be removed by channeling the liquid through a heat exchanger, such as a radiator, or the heat can be removed with the liquid durin
Physics is the natural science that studies matter, its motion, behavior through space and time, that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, its main goal is to understand how the universe behaves. Physics is one of the oldest academic disciplines and, through its inclusion of astronomy the oldest. Over much of the past two millennia, chemistry and certain branches of mathematics, were a part of natural philosophy, but during the scientific revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, the boundaries of physics which are not rigidly defined. New ideas in physics explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics enable advances in new technologies.
For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have transformed modern-day society, such as television, domestic appliances, nuclear weapons. Astronomy is one of the oldest natural sciences. Early civilizations dating back to beyond 3000 BCE, such as the Sumerians, ancient Egyptians, the Indus Valley Civilization, had a predictive knowledge and a basic understanding of the motions of the Sun and stars; the stars and planets were worshipped, believed to represent gods. While the explanations for the observed positions of the stars were unscientific and lacking in evidence, these early observations laid the foundation for astronomy, as the stars were found to traverse great circles across the sky, which however did not explain the positions of the planets. According to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, all Western efforts in the exact sciences are descended from late Babylonian astronomy.
Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey. Natural philosophy has its origins in Greece during the Archaic period, when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause, they proposed ideas verified by reason and observation, many of their hypotheses proved successful in experiment. The Western Roman Empire fell in the fifth century, this resulted in a decline in intellectual pursuits in the western part of Europe. By contrast, the Eastern Roman Empire resisted the attacks from the barbarians, continued to advance various fields of learning, including physics. In the sixth century Isidore of Miletus created an important compilation of Archimedes' works that are copied in the Archimedes Palimpsest. In sixth century Europe John Philoponus, a Byzantine scholar, questioned Aristotle's teaching of physics and noting its flaws.
He introduced the theory of impetus. Aristotle's physics was not scrutinized until John Philoponus appeared, unlike Aristotle who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics John Philoponus wrote: “But this is erroneous, our view may be corroborated by actual observation more than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a small one, and so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other”John Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries during the Scientific Revolution.
Galileo cited Philoponus in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan, a teacher in the faculty of arts at the University of Paris, developed the concept of impetus, it was a step toward the modern ideas of momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during the Islamic Golden Age developed it further placing emphasis on observation and a priori reasoning, developing early forms of the scientific method; the most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn al-Haytham, in which he conclusively disproved the ancient Greek idea about vision, but came up with a new theory. In the book, he presented a study of the phenomenon of the camera obscura (his thousand-year-old
A rheometer is a laboratory device used to measure the way in which a liquid, suspension or slurry flows in response to applied forces. It is used for those fluids which cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case for a viscometer, it measures the rheology of the fluid. There are two distinctively different types of rheometers. Rheometers that control the applied shear stress or shear strain are called rotational or shear rheometers, whereas rheometers that apply extensional stress or extensional strain are extensional rheometers. Rotational or shear type rheometers are designed as either a native strain-controlled instrument or a native stress-controlled instrument; the word rheometer comes from the Greek, means a device for measuring main flow. In the 19th century it was used for devices to measure electric current, until the word was supplanted by galvanometer and ammeter, it was used for the measurement of flow of liquids, in medical practice and in civil engineering.
This latter use persisted to the second half of the 20th century in some areas. Following the coining of the term rheology the word came to be applied to instruments for measuring the character rather than quantity of flow, the other meanings are obsolete; the principle and working of rheometers is described in several texts. Four basic shearing planes can be defined according to their geometry, Couette drag plate flow Cylindrical flow Poiseuille flow in a tube and Plate-plate flowThe various types of shear rheometers use one or a combination of these geometries. One example of a linear shear rheometer is the Goodyer linear skin rheometer, used to test cosmetic cream formulations, for medical research purposes to quantify the elastic properties of tissue; the device works by attaching a linear probe to the surface of the tissue under test, a controlled cyclical force is applied, the resultant shear force measured using a load cell. Displacement is measured using an LVDT, thus the basic stress–strain parameters are captured and analysed to derive the dynamic spring rate of the tissue under tests.
Liquid is forced through a tube of constant cross-section and known dimensions under conditions of laminar flow. Either the flow-rate or the pressure drop are fixed and the other measured. Knowing the dimensions, the flow-rate can be converted into a value for the shear rate and the pressure drop into a value for the shear stress. Varying the pressure or flow allows a flow curve to be determined; when a small amount of fluid is available for rheometric characterization, a microfluidic rheometer with embedded pressure sensors can be used to measure pressure drop for a controlled flow rate. A dynamic shear rheometer known as DSR is used for research and development as well as for quality control in the manufacturing of a wide range of materials. Dynamic shear rheometers have been used since 1993 when Superpave was used for characterising and understanding high temperature rheological properties of asphalt binders in both the molten and solid state and is fundamental in order to formulate the chemistry and predict the end-use performance of these materials.
The liquid is placed within the annulus of one cylinder inside another. One of the cylinders is rotated at a set speed; this determines the shear rate inside the annulus. The liquid tends to drag the other cylinder round, the force it exerts on that cylinders is measured, which can be converted to a shear stress. One version of this is the Fann V-G Viscometer, which runs at two speeds, therefore only gives two points on the flow curve; this is sufficient to define a Bingham plastic model which used to be used in the oil industry for determining the flow character of drilling fluids. In recent years rheometers that spin at 600, 300, 200, 100, 6 & 3 RPM have been used; this allows for more complex fluids models such as Herschel–Bulkley to be used. Some models allow the speed to be continuously increased and decreased in a programmed fashion, which allows the measurement of time-dependent properties; the liquid is placed on a shallow cone placed into it. The angle between the surface of the cone and the plate is around 1–2 degrees but can vary depending on the types of tests being run.
The plate is rotated and the torque on the cone measured. A well-known version of this instrument is the Weissenberg rheogoniometer, in which the movement of the cone is resisted by a thin piece of metal which twists—known as a torsion bar; the known response of the torsion bar and the degree of twist give the shear stress, while the rotational speed and cone dimensions give the shear rate. In principle the Weissenberg rheogoniometer is an absolute method of measurement providing it is set up. Other instruments operating on this principle may be easier to use but require calibration with a known fluid. Cone and plate rheometers can be operated in an oscillating mode to measure elastic properties, or in combined rotational and oscillating modes; the development of extensional rheometers has proceeded more than shear rheometers, due to the challenges associated with generating a homogeneous extensional flow. Firstly, interactions of the test fluid or melt with solid interfaces will result in a component of shear flow, which will compromise the results.
Secondly, the strain history of all the material elements must be known. Thirdly, the strain rates and st
Plasma is one of the four fundamental states of matter, was first described by chemist Irving Langmuir in the 1920s. Plasma can be artificially generated by heating or subjecting a neutral gas to a strong electromagnetic field to the point where an ionized gaseous substance becomes electrically conductive, long-range electromagnetic fields dominate the behaviour of the matter. Plasma and ionized gases have properties and display behaviours unlike those of the other states, the transition between them is a matter of nomenclature and subject to interpretation. Based on the surrounding environmental temperature and density ionized or ionized forms of plasma may be produced. Neon signs and lightning are examples of ionized plasma; the Earth's ionosphere is a plasma and the magnetosphere contains plasma in the Earth's surrounding space environment. The interior of the Sun is an example of ionized plasma, along with the solar corona and stars. Positive charges in ions are achieved by stripping away electrons orbiting the atomic nuclei, where the total number of electrons removed is related to either increasing temperature or the local density of other ionized matter.
This can be accompanied by the dissociation of molecular bonds, though this process is distinctly different from chemical processes of ion interactions in liquids or the behaviour of shared ions in metals. The response of plasma to electromagnetic fields is used in many modern technological devices, such as plasma televisions or plasma etching. Plasma may be the most abundant form of ordinary matter in the universe, although this hypothesis is tentative based on the existence and unknown properties of dark matter. Plasma is associated with stars, extending to the rarefied intracluster medium and the intergalactic regions; the word plasma comes from Ancient Greek πλάσμα, meaning'moldable substance' or'jelly', describes the behaviour of the ionized atomic nuclei and the electrons within the surrounding region of the plasma. Each of these nuclei are suspended in a movable sea of electrons. Plasma was first identified in a Crookes tube, so described by Sir William Crookes in 1879; the nature of this "cathode ray" matter was subsequently identified by British physicist Sir J.
J. Thomson in 1897; the term "plasma" was coined by Irving Langmuir in 1928. Lewi Tonks and Harold Mott-Smith, both of whom worked with Irving Langmuir in the 1920s, recall that Langmuir first used the word "plasma" in analogy with blood. Mott-Smith recalls, in particular, that the transport of electrons from thermionic filaments reminded Langmuir of "the way blood plasma carries red and white corpuscles and germs."Langmuir described the plasma he observed as follows: "Except near the electrodes, where there are sheaths containing few electrons, the ionized gas contains ions and electrons in about equal numbers so that the resultant space charge is small. We shall use the name plasma to describe this region containing balanced charges of ions and electrons." Plasma is a state of matter in which an ionized gaseous substance becomes electrically conductive to the point that long-range electric and magnetic fields dominate the behaviour of the matter. The plasma state can be contrasted with the other states: solid and gas.
Plasma is an electrically neutral medium of unbound negative particles. Although these particles are unbound, they are not "free" in the sense of not experiencing forces. Moving charged particles generate an electric current within a magnetic field, any movement of a charged plasma particle affects and is affected by the fields created by the other charges. In turn this governs collective behaviour with many degrees of variation. Three factors define a plasma: The plasma approximation: The plasma approximation applies when the plasma parameter, Λ, representing the number of charge carriers within a sphere surrounding a given charged particle, is sufficiently high as to shield the electrostatic influence of the particle outside of the sphere. Bulk interactions: The Debye screening length is short compared to the physical size of the plasma; this criterion means that interactions in the bulk of the plasma are more important than those at its edges, where boundary effects may take place. When this criterion is satisfied, the plasma is quasineutral.
Plasma frequency: The electron plasma frequency is large compared to the electron-neutral collision frequency. When this condition is valid, electrostatic interactions dominate over the processes of ordinary gas kinetics. Plasma temperature is measured in kelvin or electronvolts and is, informally, a measure of the thermal kinetic energy per particle. High temperatures are needed to sustain ionisation, a defining feature of a plasma; the degree of plasma ionisation is determined by the electron temperature relative to the ionization energy, in a relationship called the Saha equation. At low temperatures and electrons tend to recombine into bound states—atoms—and the plasma will become a gas. In most cases the electrons are close enough to thermal equilibrium that their temperature is well-defined; because of the large difference in ma
Hooke's law is a law of physics that states that the force needed to extend or compress a spring by some distance x scales linearly with respect to that distance. That is: F s = k x, where k is a constant factor characteristic of the spring: its stiffness, x is small compared to the total possible deformation of the spring; the law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram, he published the solution of his anagram in 1678 as: sic vis. Hooke states in the 1678 work that he was aware of the law in 1660. Hooke's equation holds 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, 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 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. 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, is the foundation of many disciplines such as seismology, molecular mechanics and acoustics, it is the fundamental principle behind the spring scale, the manometer, the balance wheel of the mechanical clock. The modern theory of elasticity generalizes Hooke's law to say that the strain 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 that can be represented by a matrix of real numbers.
In this general form, Hooke's law makes 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 k directly proportional to its cross-section area and inversely proportional to its length. 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 F s. Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Let x be the amount by which the free end of the spring was displaced from its "relaxed" position. Hooke's law states that F s = k x or, equivalently, x = F s k where k is a positive real number, characteristic of the spring. Moreover, the same formula holds when the spring is compressed, with F s and x both negative in that case.
According to this formula, the graph of the applied force F s as a function of the displacement x will be a straight line passing through the origin, whose slope is k. Hooke's law for a spring is stated under the convention that F s is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F s = − k x since the direction of the restoring force is opposite to that of the displacement. Hooke's spring law applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative. For example, when a block of rubber attached to two parallel plates is deformed by shearing, rather than stretching or compression, the shearing force F s and the sideways displacement of the plates x obey Hooke's law. Hooke's law applies when a straight steel bar or concrete beam, supported at both ends, is bent by a weight F placed at some intermediate point.
The displacement x in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape. The law applies when a stretched steel wire is twisted by pulling on a lever attached to one end. In this case the stress F s can be taken as the force applied to the lever, x as the distance traveled by it along its circular path. Or, one can let F s be the torque applied by the lever to the end of the wire, x be the angle by which that end turns. In either case F s is proportional to x In the case of a helical spring, stretched or compressed along its axis, the applied force and the resulting elongation or compression have the same direction (which is the directi