Mixing (process engineering)
In industrial process engineering, mixing is a unit operation that involves manipulation of a heterogeneous physical system with the intent to make it more homogeneous. Familiar examples include pumping of the water in a swimming pool to homogenize the water temperature, the stirring of pancake batter to eliminate lumps. Mixing is performed to allow heat and/or mass transfer to occur between one or more streams, components or phases. Modern industrial processing always involves some form of mixing; some classes of chemical reactors are mixers. With the right equipment, it is possible to mix a solid, liquid or gas into another solid, liquid or gas. A biofuel fermenter may require the mixing of microbes and liquid medium for optimal yield; the opposite of mixing is segregation. A classical example of segregation is the brazil nut effect; the type of operation and equipment used during mixing depends on the state of materials being mixed and the miscibility of the materials being processed. In this context, the act of mixing may be synonymous with stirring kneading-processes.
Mixing of liquids occurs in process engineering. The nature of liquids to blend determines the equipment used. Single-phase blending tends to involve low-shear, high-flow mixers to cause liquid engulfment, while multi-phase mixing requires the use of high-shear, low-flow mixers to create droplets of one liquid in laminar, turbulent or transitional flow regimes, depending on the Reynolds number of the flow. Turbulent or transitional mixing is conducted with turbines or impellers. Mixing of liquids that are miscible or at least soluble in each other occurs in process engineering. An everyday example would be the addition of milk or cream to coffee. Since both liquids are water-based, they dissolve in one another; the momentum of the liquid being added is sometimes enough to cause enough turbulence to mix the two, since the viscosity of both liquids is low. If necessary, a spoon or paddle could be used to complete the mixing process. Blending in a more viscous liquid, such as honey, requires more mixing power per unit volume to achieve the same homogeneity in the same amount of time.
Blending powders is one of the oldest unit-operations in the solids handling industries. For many decades powder blending has been used just to homogenize bulk materials. Many different machines have been designed to handle materials with various bulk solids properties. On the basis of the practical experience gained with these different machines, engineering knowledge has been developed to construct reliable equipment and to predict scale-up and mixing behavior. Nowadays the same mixing technologies are used for many more applications: to improve product quality, to coat particles, to fuse materials, to wet, to disperse in liquid, to agglomerate, to alter functional material properties, etc; this wide range of applications of mixing equipment requires a high level of knowledge, long time experience and extended test facilities to come to the optimal selection of equipment and processes.. Solids-solids mixing can be performed either in batch mixers, the simpler form of mixing, or in certain cases in continuous dry-mix, more complex but which provide interesting advantages in terms of segregation and validation.
One example of a solid–solid mixing process is mulling foundry molding sand, where sand, bentonite clay, fine coal dust and water are mixed to a plastic and reusable mass, applied for molding and pouring molten metal to obtain sand castings that are metallic parts for automobile, machine building, construction or other industries. In powder two different dimensions in the mixing process can be determined: convective mixing and intensive mixing. In the case of convective mixing material in the mixer is transported from one location to another; this type of mixing leads to a less ordered state inside the mixer, the components that must be mixed are distributed over the other components. With progressing time the mixture becomes more randomly ordered. After a certain mixing time the ultimate random state is reached; this type of mixing is applied for free-flowing and coarse materials. Possible threats during macro mixing is the de-mixing of the components, since differences in size, shape or density of the different particles can lead to segregation.
When materials are cohesive, the case with e.g. fine particles and with wet material, convective mixing is no longer sufficient to obtain a randomly ordered mixture. The relative strong inter-particle forces form lumps, which are not broken up by the mild transportation forces in the convective mixer. To decrease the lump size additional forces are necessary; these additional forces can either be impact forces or shear forces. Liquid–solid mixing is done to suspend coarse free-flowing solids, or to break up lumps of fine agglomerated solids. An example of the former is the mixing granulated sugar into water. In the first case, the particles can be lifted into suspension by bulk motion of the fluid. One example of a solid–liquid mixing process in industry is concrete mixing, where cement, small stones or gravel and water are co
Navier–Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances. These balance equations arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term and a pressure term—hence describing viscous flow; the main difference between them and the simpler Euler equations for inviscid flow is that Navier–Stokes equations factor in the Froude limit and are not conservation equations, but rather a dissipative system, in the sense that they cannot be put into the quasilinear homogeneous form: y t + A y x = 0. Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest, they may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, many other things.
Coupled with Maxwell's equations, they can be used to study magnetohydrodynamics. The Navier–Stokes equations are of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether solutions always exist in three dimensions and, if they do exist, whether they are smooth – i.e. they are infinitely differentiable at all points in the domain. These are called the Navier–Stokes existence and smoothness problems; the Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample. The solution of the equations is a flow velocity, it is a field, since it is defined at every point in an interval of time. Once the velocity field is calculated other quantities of interest, such as pressure or temperature, may be found using additional equations and relations; this is different from what one sees in classical mechanics, where solutions are trajectories of position of a particle or deflection of a continuum.
Studying velocity instead of position makes more sense for a fluid. The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is D u D t = 1 ρ ∇ ⋅ σ + g By setting the Cauchy stress tensor σ to be the sum of a viscosity term τ and a pressure term − p I we arrive at where D D t is the material derivative, defined as D D t = d e f ∂ ∂ t + u ⋅ ∇, ρ is the density, u is the flow velocity, ∇ ⋅ is the divergence, p is the pressure, t is time, τ is the deviatoric stress tensor, which has order two, g represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electrostatic accelerations, so on,In this form, it is apparent that in the assumption of an inviscid fluid -no deviatoric stress- Cauchy equations reduce to the Euler equations. Assuming conservation of mass we can use the continuity equation, ∂ ρ ∂ t + ∇ ⋅ = 0 to arrive to the conservation form of the equations of motion; this is written: where ⊗ is the outer product: u ⊗ v = u v T.
The left side of the equation describes acceleration, may be composed of time-dependent and convective components. The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces. All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation. By expressing the deviatoric stress tensor in terms of viscosity and the fluid velocity gradient, assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below. A significant feature of the Cauchy equation and all other continuum equations is the presence
Plasma (physics)
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
Conservation of mass
The law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as system's mass cannot change, so quantity can neither be added nor be removed. Hence, the quantity of mass is conserved over time; the law implies that mass can neither be created nor destroyed, although it may be rearranged in space, or the entities associated with it may be changed in form. For example, in chemical reactions, the mass of the chemical components before the reaction is equal to the mass of the components after the reaction. Thus, during any chemical reaction and low-energy thermodynamic processes in an isolated system, the total mass of the reactants, or starting materials, must be equal to the mass of the products; the concept of mass conservation is used in many fields such as chemistry and fluid dynamics. Mass conservation was demonstrated in chemical reactions independently by Mikhail Lomonosov and rediscovered by Antoine Lavoisier in the late 18th century.
The formulation of this law was of crucial importance in the progress from alchemy to the modern natural science of chemistry. The conservation of mass only holds and is considered part of a series of assumptions coming from classical mechanics; the law has to be modified to comply with the laws of quantum mechanics and special relativity under the principle of mass-energy equivalence, which states that energy and mass form one conserved quantity. For energetic systems the conservation of mass-only is shown not to hold, as is the case in nuclear reactions and particle-antiparticle annihilation in particle physics. Mass is not conserved in open systems; such is the case when various forms of matter are allowed into, or out of, the system. However, unless radioactivity or nuclear reactions are involved, the amount of energy escaping such systems as heat, mechanical work, or electromagnetic radiation is too small to be measured as a decrease in the mass of the system. For systems where large gravitational fields are involved, general relativity has to be taken into account, where mass-energy conservation becomes a more complex concept, subject to different definitions, neither mass nor energy is as and conserved as is the case in special relativity.
The law of conservation of mass can only be formulated in classical mechanics when the energy scales associated to an isolated system are much smaller than m c 2, where m is the mass of a typical object in the system, measured in the frame of reference where the object is at rest, c is the speed of light. The law can be formulated mathematically in the fields of fluid mechanics and continuum mechanics, where the conservation of mass is expressed using the continuity equation, given in differential form as ∂ ρ ∂ t + ∇ ⋅ = 0, where ρ is the density, t is the time, ∇ ⋅ is the divergence, v is the flow velocity field; the interpretation of the continuity equation for mass is the following: For a given closed surface in the system, the change in time of the mass enclosed by the surface is equal to the mass that traverses the surface, positive if matter goes in and negative if matter goes out. For the whole isolated system, this condition implies that the total mass M, sum of the masses of all components in the system, does not change in time, i.e. d M d t = d d t ∫ ρ d V = 0,where d V is the differential that defines the integral over the whole volume of the system.
The continuity equation for the mass is part of Euler equations of fluid dynamics. Many other convection–diffusion equations describe the conservation and flow of mass and matter in a given system. In chemistry, the calculation of the amount of reactant and products in a chemical reaction, or stoichiometry, is founded on the principle of conservation of mass; the principle implies that during a chemical reaction the total mass of the reactants is equal to the total mass of the products. For example, in the following reaction CH4 + 2 O2 → CO2 + 2 H2O,where one molecule of methane and two oxygen molecules O2 are converted into one molecule of carbon dioxide and two of water; the number of molecules as result from the reaction can be derived from the principle of conservation of mass, as four hydrogen atoms, 4 oxygen atoms and one carbon atom are present the number water molecules produced must be two per molecule of carbon dioxide produced. Many engineering problems are solved by following the mass distribution in time of a given system, this practice is known as mass balance.
An important idea in ancient Greek philosophy was that "Nothing comes from nothing", so that what exists now has always existed: no new matter can come into existence where there was none before. An explicit statement of this, along
Archimedes' principle
Archimedes' principle states that the upward buoyant force, exerted on a body immersed in a fluid, whether or submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid. Archimedes' principle is a law of physics fundamental to fluid mechanics, it was formulated by Archimedes of Syracuse. In On Floating Bodies, Archimedes suggested that: Any object or immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. Archimedes' principle allows the buoyancy of an object or immersed in a fluid to be calculated; the downward force on the object is its weight. The upward, or buoyant, force on the object is. Thus, the net force on the object is the difference between the magnitudes of the buoyant force and its weight. If this net force is positive, the object rises. In simple words, Archimedes' principle states that, when a body is or immersed in a fluid, it experiences an apparent loss in weight, equal to the weight of the fluid displaced by the immersed part of the body.
Consider a cuboid immersed in a fluid, with one of its sides orthogonal to the direction of gravity. The fluid will exert a normal force on each face, but only the normal forces on top and bottom will contribute to buoyancy; the pressure difference between the bottom and the top face is directly proportional to the height. Multiplying the pressure difference by the area of a face gives a net force on the cuboid – the buoyancy, equaling in size the weight of the fluid displaced by the cuboid. By summing up sufficiently many arbitrarily small cuboids this reasoning may be extended to irregular shapes, so, whatever the shape of the submerged body, the buoyant force is equal to the weight of the displaced fluid. Weight of displaced fluid = weight of object in vacuum − weight of object in fluid The weight of the displaced fluid is directly proportional to the volume of the displaced fluid; the weight of the object in the fluid is reduced, because of the force acting on it, called upthrust. In simple terms, the principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravity or Fb = ρ x g x V. Thus, among submerged objects with equal masses, objects with greater volume have greater buoyancy.
Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting on it. Suppose that, when the rock is lowered into water, it displaces water of weight 3 newtons; the force it exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk to the sea floor, it is easier to lift an object up through the water than it is to pull it out of the water. For a submerged object, Archimedes' principle can be reformulated as follows: apparent immersed weight = weight of object − weight of displaced fluid inserted into the quotient of weights, expanded by the mutual volume density of object density of fluid = weight weight of displaced fluid yields the formula below; the density of the immersed object relative to the density of the fluid can be calculated without measuring any volume is density of object density of fluid = weight weight − apparent immersed weight.
Example: If you drop wood into water, buoyancy will keep it afloat. Example: A helium balloon in a moving car; when increasing speed or driving in a curve, the air moves in the opposite direction to the car's acceleration. However, due to buoyancy, the balloon is pushed "out of the way" by the air, will drift in the same direction as the car's acceleration; when an object is immersed in a liquid, the liquid exerts an upward force, known as the buoyant force, proportional to the weight of the displaced liquid. The sum force acting on the object is equal to the difference between the weight of the object and the weight of displaced liquid. Equilibrium, or neutral buoyancy, is achieved. Archimedes' principle does not consider the surface tension acting on the body. Moreover, Archimedes' principle has been found to break down in complex fluids. There is an exception to Archimedes' principle known as the bottom
Robert Boyle
Robert Boyle was an Anglo-Irish natural philosopher, chemist and inventor. Boyle is regarded today as the first modern chemist, therefore one of the founders of modern chemistry, one of the pioneers of modern experimental scientific method, he is best known for Boyle's law, which describes the inversely proportional relationship between the absolute pressure and volume of a gas, if the temperature is kept constant within a closed system. Among his works, The Sceptical Chymist is seen as a cornerstone book in the field of chemistry, he is noted for his writings in theology. Boyle was born at Lismore Castle, in County Waterford, the seventh son and fourteenth child of The 1st Earl of Cork and Catherine Fenton. Lord Cork known as Richard Boyle, had arrived in Dublin from England in 1588 during the Tudor plantations of Ireland and obtained an appointment as a deputy escheator, he had amassed enormous wealth and landholdings by the time Robert was born, had been created Earl of Cork in October 1620.
Catherine Fenton, Countess of Cork, was the daughter of Sir Geoffrey Fenton, the former Secretary of State for Ireland, born in Dublin in 1539, Alice Weston, the daughter of Robert Weston, born in Lismore in 1541. As a child, Boyle was fostered to a local family. Boyle received private tutoring in Latin and French and when he was eight years old, following the death of his mother, he was sent to Eton College in England, his father's friend, Sir Henry Wotton, was the provost of the college. During this time, his father hired a private tutor, Robert Carew, who had knowledge of Irish, to act as private tutor to his sons in Eton. However, "only Mr. Robert sometimes desires it and is a little entered in it", but despite the "many reasons" given by Carew to turn their attentions to it, "they practice the French and Latin but they affect not the Irish". After spending over three years at Eton, Robert travelled abroad with a French tutor, they visited Italy in 1641 and remained in Florence during the winter of that year studying the "paradoxes of the great star-gazer" Galileo Galilei, elderly but still living in 1641.
Robert returned to England from continental Europe in mid-1644 with a keen interest in scientific research. His father, Lord Cork, had died the previous year and had left him the manor of Stalbridge in Dorset as well as substantial estates in County Limerick in Ireland that he had acquired. Robert made his residence at Stalbridge House, between 1644 and 1652, conducted many experiments there. From that time, Robert devoted his life to scientific research and soon took a prominent place in the band of enquirers, known as the "Invisible College", who devoted themselves to the cultivation of the "new philosophy", they met in London at Gresham College, some of the members had meetings at Oxford. Having made several visits to his Irish estates beginning in 1647, Robert moved to Ireland in 1652 but became frustrated at his inability to make progress in his chemical work. In one letter, he described Ireland as "a barbarous country where chemical spirits were so misunderstood and chemical instruments so unprocurable that it was hard to have any Hermetic thoughts in it."In 1654, Boyle left Ireland for Oxford to pursue his work more successfully.
An inscription can be found on the wall of University College, the High Street at Oxford, marking the spot where Cross Hall stood until the early 19th century. It was here that Boyle rented rooms from the wealthy apothecary. Reading in 1657 of Otto von Guericke's air pump, he set himself with the assistance of Robert Hooke to devise improvements in its construction, with the result, the "machina Boyleana" or "Pneumatical Engine", finished in 1659, he began a series of experiments on the properties of air. An account of Boyle's work with the air pump was published in 1660 under the title New Experiments Physico-Mechanical, Touching the Spring of the Air, its Effects. Among the critics of the views put forward in this book was a Jesuit, Francis Line, it was while answering his objections that Boyle made his first mention of the law that the volume of a gas varies inversely to the pressure of the gas, which among English-speaking people is called Boyle's Law after his name; the person who formulated the hypothesis was Henry Power in 1661.
Boyle in 1662 included a reference to a paper written by Power, but mistakenly attributed it to Richard Towneley. In continental Europe the hypothesis is sometimes attributed to Edme Mariotte, although he did not publish it until 1676 and was aware of Boyle's work at the time. In 1663 the Invisible College became The Royal Society of London for Improving Natural Knowledge, the charter of incorporation granted by Charles II of England named Boyle a member of the council. In 1680 he declined the honour from a scruple about oaths, he made a "wish list" of 24 possible inventions which included "the prolongation of life", the "art of flying", "perpetual light", "making armour light and hard", "a ship to sail with all winds, a ship not to be sunk", "practicable and certain way of finding longitudes", "potent drugs to alter or exalt imagination, waking and other functions and appease pain, procure innocent sleep, harmless dreams, etc." They are extraordinary. It was during his time at Oxford; the Chevaliers are thought to have been established by royal order a few years before Boyle's time at Oxford.
The early part
Hooke's law
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