Hooke's law: the force is proportional to the extension
Bourdon tubes are based on Hooke's law. The force created by gas pressure inside the coiled metal tube above unwinds it by an amount proportional to the pressure.
The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period.
Hooke's law is a law of physics that states that the force (F) needed to extend or compress a spring by some distance x scales linearly with respect to that distance. That is: , where k is a constant factor characteristic of the spring: its stiffness, and 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 Latinanagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law already in 1660.
Hooke's equation 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, and 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. 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 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.
Plot of applied force F vs. elongation X for a helical spring according to Hooke's law (red line) and what the actual plot might look like (dashed line). At bottom, pictures of spring states corresponding to some points of the plot; the middle one is in the relaxed state (no force applied).
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 x 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
where k is a positive real number, characteristic of the spring. Moreover, the same formula holds when the spring is compressed, with and x both negative in that case. According to this formula, the graph of the applied force 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 often stated under the convention that is the restoring 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.
Hooke's spring law usually 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 and the sideways displacement of the plates x obey Hooke's law (for small enough deformations).
Hooke's law also 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 also applies when a stretched steel wire is twisted by pulling on a lever attached to one end. In this case the stress can be taken as the force applied to the lever, and x as the distance traveled by it along its circular path. Or, equivalently, one can let be the torque applied by the lever to the end of the wire, and x be the angle by which that end turns. In either case is proportional to x (although the constant k is different in each case.)
In the case of a helical spring that is stretched or compressed along its axis, the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if and x are defined as vectors, Hooke's equation still holds and says that the force vector is the elongation vector multiplied by a fixed scalar.
Some elastic bodies will deform in one direction when subjected to a force with a different direction. One example is a horizontal wood beam with non-square rectangular cross section that is bent by a transverse load that is neither vertical nor horizontal. In such cases, the magnitude of the displacement x will be proportional to the magnitude of the force , as long as the direction of the latter remains the same (and its value is not too large); so the scalar version of Hooke's law will hold. However, the force and displacement vectors will not be scalar multiples of each other, since they have different directions. Moreover, the ratio k between their magnitudes will depend on the direction of the vector .
Yet, in such cases there is often a fixed linear relation between the force and deformation vectors, as long as they are small enough. Namely, there is a functionκ from vectors to vectors, such that F = κ(X), and κ(αX1 + βX2) = ακ(X1) + βκ(X2) for any real numbers α, β and any displacement vectors X1, X2. Such a function is called a (second-order) tensor.
With respect to an arbitrary Cartesian coordinate system, the force and displacement vectors can be represented by 3 × 1 matrices of real numbers. Then the tensor κ connecting them can be represented by a 3 × 3 matrix κ of real coefficients, that, when multiplied by the displacement vector, gives the force vector:
for i = 1, 2, 3. Therefore, Hooke's law F = κX can be said to hold also when X and F are vectors with variable directions, except that the stiffness of the object is a tensor κ, rather than a single real number k.
(a) Schematic of a polymer nanospring. The coil radius, R, pitch, P, length of the spring, L, and the number of turns, N, are 2.5 μm, 2.0 μm, 13 μm, and 4, respectively. Electron micrographs of the nanospring, before loading (b-e), stretched (f), compressed (g), bent (h), and recovered (i). All scale bars are 2 μm. The spring followed a linear response against applied force, demonstrating the validity of Hooke's law at the nanoscale.
The stresses and strains of the material inside a continuous elastic material (such as a block of rubber, the wall of a boiler, or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke's spring law, and is often referred to by that name.
However, the strain state in a solid medium around some point cannot be described by a single vector. The same parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing.
In order to capture this complexity, the relevant state of the medium around a point must be represented by two-second-order tensors, the strain tensorε (in lieu of the displacement X) and the stress tensorσ (replacing the restoring force F). The analogue of Hooke's spring law for continuous media is then
where c is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the stiffness tensor or elasticity tensor. One may also write it as
where the tensor s, called the compliance tensor, represents the inverse of said linear map.
In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices
Being a linear mapping between the nine numbers σij and the nine numbers εkl, the stiffness tensor c is represented by a matrix of 3 × 3 × 3 × 3 = 81 real numbers cijkl. Hooke's law then says that
where i,j = 1,2,3.
All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor ε merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor σ specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor c, on the other hand, is a property of the material, and often depends on physical state variables such as temperature, pressure, and microstructure.
Due to the inherent symmetries of σ, ε, and c, only 21 elastic coefficients of the latter are independent. For isotropic media (which have the same physical properties in any direction), c can be reduced to only two independent numbers, the bulk modulusK and the shear modulusG, that quantify the material's resistance to changes in volume and to shearing deformations, respectively.
Since Hooke's law is a simple proportionality between two quantities, its formulas and consequences are mathematically similar to those of many other physical laws, such as those describing the motion of fluids, or the polarization of a dielectric by an electric field.
In particular, the tensor equation σ = cε relating elastic stresses to strains is entirely similar to the equation τ = με̇ relating the viscous stress tensorτ and the strain rate tensorε̇ in flows of viscous fluids; although the former pertains to static stresses (related to amount of deformation) while the latter pertains to dynamical stresses (related to the rate of deformation).
In SI units, displacements are measured in meters (m), and forces in newtons (N or kg·m/s2). Therefore, the spring constant k, and each element of the tensor κ, is measured in newtons per meter (N/m), or kilograms per second squared (kg/s2).
For continuous media, each element of the stress tensor σ is a force divided by an area; it is therefore measured in units of pressure, namely pascals (Pa, or N/m2, or kg/(m·s2). The elements of the strain tensor ε are dimensionless (displacements divided by distances). Therefore, the entries of cijkl are also expressed in units of pressure.
Stress–strain curve for low-carbon steel, showing the relationship between the stress (force per unit area) and strain (resulting compression/stretching, known as deformation). Hooke's law is only valid for the portion of the curve between the origin and the yield point (2).
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 (i.e., for stresses below the yield strength). 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 dependent and sensitive to temperature and loading rate.
The potential energy Uel(x) stored in a spring is given by
which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative.
This potential Uel can be visualized as a parabola on the Ux-plane such that Uel(x) = 1/2kx2. As the spring is stretched in the positive x-direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate:
Note that the change in the change in U is constant even when the displacement and acceleration are zero.
Relaxed force constants (generalized compliance constants)
Relaxed force constants (the inverse of generalized compliance constants) are uniquely defined for molecular systems, in contradistinction to the usual "rigid" force constants, and thus their use allows meaningful correlations to be made between force fields calculated for reactants, transition states, and products of a chemical reaction. Just as the potential energy can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed compliance constants. A direct method exists for calculating the compliance constant for any internal coordinate of a molecule, without the need to do the normal mode analysis. The suitability of relaxed force constants (inverse compliance constants) as covalent bond strength descriptors was demonstrated as early as 1980. Recently, the suitability as non-covalent bond strength descriptors was demonstrated too.
A mass suspended by a spring is the classical example of a harmonic oscillator
A mass m attached to the end of a spring is a classic example of a harmonic oscillator. By pulling slightly on the mass and then releasing it, the system will be set in sinusoidal oscillating motion about the equilibrium position. To the extent that the spring obeys Hooke's law, and that one can neglect friction and the mass of the spring, the amplitude of the oscillation will remain constant; and its frequencyf will be independent of its amplitude, determined only by the mass and the stiffness of the spring:
This phenomenon made possible the construction of accurate mechanical clocks and watches that could be carried on ships and people's pockets.
For an analogous development for viscous fluids, see Viscosity.
Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials must therefore be independent of the coordinate system chosen to represent them. The strain tensor is a symmetric tensor. Since the trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor.:Ch. 10 Thus in index notation:
The first term on the right is the constant tensor, also known as the volumetric strain tensor, and the second term is the traceless symmetric tensor, also known as the deviatoric strain tensor or shear tensor.
The most general form of Hooke's law for isotropic materials may now be written as a linear combination of these two tensors:
Using the relationships between the elastic moduli, these equations may also be expressed in various other ways. A common form of Hooke's law for isotropic materials, expressed in direct tensor notation, is
where λ = K − 2/3G = c1111 − 2c1212 and μ = G = c1212 are the Lamé constants, I is the second-rank identity tensor, and I is the symmetric part of the fourth-rank identity tensor. In index notation:
The symmetry of the Cauchy stress tensor (σij = σji and the generalized Hooke's laws (σij = cijklεkl) implies that cijkl = cjikl. Similarly, the symmetry of the infinitesimal strain tensor implies that cijkl = cijlk. These symmetries are called the minor symmetries of the stiffness tensor c. This reduces the number of elastic constants from 81 to 36.
If in addition, since the displacement gradient and the Cauchy stress are work conjugate, the stress–strain relation can be derived from a strain energy density functional (U), then
The arbitrariness of the order of differentiation implies that cijkl = cklij. These are called the major symmetries of the stiffness tensor. This reduces the number of elastic constants from 36 to 21. The major and minor symmetries indicate that the stiffness tensor has only 21 independent components.
It is often useful to express the anisotropic form of Hooke's law in matrix notation, also called Voigt notation. To do this we take advantage of the symmetry of the stress and strain tensors and express them as six-dimensional vectors in an orthonormal coordinate system (e1,e2,e3) as
Then the stiffness tensor (c) can be expressed as
and Hooke's law is written as
Similarly the compliance tensor (s) can be written as
If a linear elastic material is rotated from a reference configuration to another, then the material is symmetric with respect to the rotation if the components of the stiffness tensor in the rotated configuration are related to the components in the reference configuration by the relation
Linear deformations of elastic materials can be approximated as adiabatic. Under these conditions and for quasistatic processes the first law of thermodynamics for a deformed body can be expressed as
where δU is the increase in internal energy and δW is the work done by external forces. The work can be split into two terms
where δWs is the work done by surface forces while δWb is the work done by body forces. If δu is a variation of the displacement field u in the body, then the two external work terms can be expressed as
where t is the surface traction vector, b is the body force vector, Ω represents the body and ∂Ω represents its surface. Using the relation between the Cauchy stress and the surface traction, t = n · σ (where n is the unit outward normal to ∂Ω), we have
An elastic material is defined as one in which the total internal energy is equal to the potential energy of the internal forces (also called the elastic strain energy). Therefore, the internal energy density is a function of the strains, U0 = U0(ε) and the variation of the internal energy can be expressed as
Since the variation of strain is arbitrary, the stress–strain relation of an elastic material is given by
For a linear elastic material, the quantity ∂U0/∂ε is a linear function of ε, and can therefore be expressed as
where c is a fourth-rank tensor of material constants, also called the stiffness tensor. We can see why c must be a fourth-rank tensor by noting that, for a linear elastic material,
In index notation
The right-hand side constant requires four indices and is a fourth-rank quantity. We can also see that this quantity must be a tensor because it is a linear transformation that takes the strain tensor to the stress tensor. We can also show that the constant obeys the tensor transformation rules for fourth-rank tensors.
^The anagram was given in alphabetical order, ceiiinosssttuu, representing Ut tensio, sic vis – "As the extension, so the force": Petroski, Henry (1996). Invention by Design: How Engineers Get from Thought to Thing. Cambridge, MA: Harvard University Press. p. 11. ISBN978-0674463684.
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas.
There are two valid solutions.
The plus sign leads to .
A balance wheel, or balance, is the timekeeping device used in mechanical watches and some clocks, analogous to the pendulum in a pendulum clock. It is a wheel that rotates back and forth, being returned toward its center position by a spiral torsion spring. It is driven by the escapement, which transforms the motion of the watch gear train into impulses delivered to the balance wheel. Each swing of the wheel allows the train to advance a set amount. From its invention in the 14th century until tuning fork and quartz movements became available in the 1960s, virtually every portable timekeeping device used some form of balance wheel. Modern watch balance wheels are made of Glucydur, a low thermal expansion alloy of beryllium, copper and iron. The two alloys are matched so their residual temperature responses cancel out, resulting in lower temperature error. The wheels are smooth, to air friction, and the pivots are supported on precision jewel bearings. Older balance wheels used weight screws around the rim to adjust the poise, Balance wheels rotate about 1½ turns with each swing, that is, about 270° to each side of their center equilibrium position. The rate of the wheel is adjusted with the regulator. This holds the part of the spring behind the slit stationary, moving the lever slides the slit up and down the balance spring, changing its effective length, and thus the resonant vibration rate of the balance. Since the regulator interferes with the action, chronometers and some precision watches have ‘free sprung’ balances with no regulator. Their rate is adjusted by weight screws on the balance rim, a balances vibration rate is traditionally measured in beats per hour, or BPH, although beats per second and Hz are also used. The length of a beat is one swing of the balance wheel, balances in precision watches are designed with faster beats, because they are less affected by motions of the wrist. Alarm clocks and kitchen timers often have a rate of 4 beats per second, Watches made prior to the 1970s usually had a rate of 5 beats per second. Current watches have rates of 6,8 and a few have 10 beats per second, during WWII, Elgin produced a very precise stopwatch that ran at 40 beats per second, earning it the nickname Jitterbug. Audemars Piguet currently produces a movement that allows for a high balance vibration of 12 beats/s. The precision of the best balance wheel watches on the wrist is around a few seconds per day, the most accurate balance wheel timepieces made were marine chronometers, which by WWII had achieved accuracies of 0.1 second per day
Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. 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-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
A spring is an elastic object used to store mechanical energy. Springs are usually out of spring steel. There are a number of spring designs, in everyday usage the term often refers to coil springs. When a spring is compressed or stretched from its resting position, the rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. That is, it is the gradient of the force versus deflection curve, an extension or compression springs rate is expressed in units of force divided by distance, for example lbf/in or N/m. A torsion spring is a spring that works by twisting, when it is twisted about its axis by an angle, a torsion springs rate is in units of torque divided by angle, such as N·m/rad or ft·lbf/degree. The inverse of spring rate is compliance, that is, if a spring has a rate of 10 N/mm, the stiffness of springs in parallel is additive, as is the compliance of springs in series. Springs are made from a variety of materials, the most common being spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel, some non-ferrous metals are also used including phosphor bronze and titanium for parts requiring corrosion resistance and beryllium copper for springs carrying electrical current. Simple non-coiled springs were used throughout history, e. g. the bow. In the Bronze Age more sophisticated spring devices were used, as shown by the spread of tweezers in many cultures, coiled springs appeared early in the 15th century, in door locks. The first spring powered-clocks appeared in that century and evolved into the first large watches by the 16th century, in 1676 British physicist Robert Hooke discovered Hookes law which states that the force a spring exerts is proportional to its extension. Compression spring – is designed to operate with a compression load, flat spring – this type is made of a flat spring steel. Machined spring – this type of spring is manufactured by machining bar stock with a lathe and/or milling operation rather than a coiling operation, since it is machined, the spring may incorporate features in addition to the elastic element. Machined springs can be made in the load cases of compression/extension, torsion. Serpentine spring - a zig-zag of thick wire - often used in modern upholstery/furniture, the most common types of spring are, Cantilever spring – a spring which is fixed only at one end. Coil spring or helical spring – a spring is of two types, Tension or extension springs are designed to become longer under load and their turns are normally touching in the unloaded position, and they have a hook, eye or some other means of attachment at each end. Compression springs are designed to become shorter when loaded and their turns are not touching in the unloaded position, and they need no attachment points
Robert Hooke FRS was an English natural philosopher, architect and polymath. Allan Chapman has characterised him as Englands Leonardo, Robert Gunthers Early Science in Oxford, a history of science in Oxford during the Protectorate, Restoration and Age of Enlightenment, devotes five of its fourteen volumes to Hooke. Hooke studied at Wadham College, Oxford during the Protectorate where he became one of a tightly knit group of ardent Royalists led by John Wilkins. Here he was employed as an assistant to Thomas Willis and to Robert Boyle and he built some of the earliest Gregorian telescopes and observed the rotations of Mars and Jupiter. In 1665 he inspired the use of microscopes for scientific exploration with his book, based on his microscopic observations of fossils, Hooke was an early proponent of biological evolution. Much of Hookes scientific work was conducted in his capacity as curator of experiments of the Royal Society, much of what is known of Hookes early life comes from an autobiography that he commenced in 1696 but never completed. Richard Waller mentions it in his introduction to The Posthumous Works of Robert Hooke, the work of Waller, along with John Wards Lives of the Gresham Professors and John Aubreys Brief Lives, form the major near-contemporaneous biographical accounts of Hooke. Robert Hooke was born in 1635 in Freshwater on the Isle of Wight to John Hooke, Robert was the last of four children, two boys and two girls, and there was an age difference of seven years between him and the next youngest. Their father John was a Church of England priest, the curate of Freshwaters Church of All Saints, Robert Hooke was expected to succeed in his education and join the Church. John Hooke also was in charge of a school, and so was able to teach Robert. He was a Royalist and almost certainly a member of a group who went to pay their respects to Charles I when he escaped to the Isle of Wight, Robert, too, grew up to be a staunch monarchist. As a youth, Robert Hooke was fascinated by observation, mechanical works and he dismantled a brass clock and built a wooden replica that, by all accounts, worked well enough, and he learned to draw, making his own materials from coal, chalk and ruddle. Hooke quickly mastered Latin and Greek, made study of Hebrew. Here, too, he embarked on his study of mechanics. It appears that Hooke was one of a group of students whom Busby educated in parallel to the work of the school. Contemporary accounts say he was not much seen in the school, in 1653, Hooke secured a choristers place at Christ Church, Oxford. He was employed as an assistant to Dr Thomas Willis. There he met the natural philosopher Robert Boyle, and gained employment as his assistant from about 1655 to 1662, constructing, operating and he did not take his Master of Arts until 1662 or 1663
A string is the vibrating element that produces sound in string instruments such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a material that a musical instrument holds under tension so that they can vibrate freely. Wound strings have a core of one material, with an overwinding of other materials and this is to make the string vibrate at the desired pitch, while maintaining a low profile and sufficient flexibility for playability. This enabled stringed instruments to be made with less thick bass strings, on string instruments that the player plucks or bows directly, this enabled instrument makers to use thinner strings for the lowest-pitched strings, which made the lower-pitch strings easier to play. The end of the string that mounts to the tuning mechanism is usually plain. Depending on the instrument, the other, fixed end may have either a plain, loop. When a ball or loop is used with a guitar, this ensures that the string stays fixed in the bridge of the guitar, when a ball or loop is used with a violin-family instrument, this keeps the string end fixed in the tailpiece. Fender Bullet strings have a cylinder for more stable tuning on guitars equipped with synchronized tremolo systems. Strings for some instruments may be wrapped with silk at the ends to protect the string, the color and pattern of the silk often identifies attributes of the string, such as manufacturer, size, intended pitch, etc. There are several varieties of wound strings available, the simplest wound strings are roundwound—with round wire wrapped in a tight spiral around either a round or hexagonal core. Such strings are usually simple to manufacture and the least expensive and they have several drawbacks, however, Roundwound strings have a bumpy surface profile that produce friction on the players fingertips. This causes squeaking sounds when the fingers slide over the strings. Roundwound strings higher friction surface profile may hasten fingerboard and fret wear, when the core is round, the winding is less secure and may rotate freely around the core, especially if the winding is damaged after use. Flatwound strings also have either a round or hex core, however, the winding wire has a rounded square cross-section that has a shallower profile when tightly wound. This makes for more playing, and decreased wear for frets. Squeaking sounds due to fingers sliding along the strings are also decreased significantly, flatwound strings also have a longer playable life because of smaller grooves for dirt and oil to build up in. On the other hand, flatwound strings sound less bright than roundwounds, flatwounds also usually cost more than roundwounds because of less demand, less production, and higher overhead costs. Manufacturing is also difficult, as precise alignment of the flat sides of the winding must be maintained
A toy balloon or party balloon, is a small balloon mostly used for decoration, advertising and child toy. Toy balloons are made of rubber or aluminized plastic. They come in a variety of sizes and shapes, but are most commonly 10 to 30 centimetres in diameter. Toy balloons are not considered to include sky lanterns, although these too are or were used as toys in some parts of the world. The Consumer Products Safety Commission found that children had inhaled latex balloons whole or choked on fragments of broken balloons, parents a monthly magazine about raising children advised parents to buy Mylar balloons instead of latex balloons. Early balloons were made from pig bladders and animal intestines, the Aztecs created the first balloon sculptures using cat intestines, which were then presented to the gods as a sacrifice. There are references to balloons made of whale intestine in Swiss Family Robinson, the first rubber balloons were made by Professor Michael Faraday in 1824 for use in his experiments with hydrogen at the Royal Institution in London. The caoutchouc is exceedingly elastic, he wrote in the Quarterly Journal of Science the same year, Faraday made his balloons by cutting round two sheets of rubber laid together and pressing the edges together. The tacky rubber welded automatically, and the inside of the balloon was rubbed with flour to prevent the opposing surfaces joining together. Toy balloons were introduced by pioneer rubber manufacturer Thomas Hancock the following year in the form of a kit consisting of a bottle of rubber solution. Vulcanized toy balloons, which unlike the kind were unaffected by changes in temperature, were first manufactured by J. G. Ingram of London in 1847 and can be regarded as the prototype of modern toy balloons, in the 1920s Neil Tillotson designed and produced a latex balloon with a cats face and ears from a cardboard form which he cut by hand with a pair of scissors. He managed to make his first sale of these balloons with an order of 15 gross to be delivered for the annual Patriots Day Parade on April 19,1931, the first colored balloons were sold at the 1933-34 Chicago Worlds Fair. Inflatable foil balloons are made from plastics, such as aluminized PET film, foil balloons are not elastic like rubber balloons, so that detailed and colorful pictures printed on their surfaces are not distorted when the balloon is inflated. When no longer required, it is recommended to cut the balloon to release the helium, every toy balloon has an opening through which gases are blown into it, followed by a connecting tube known as the neck. Balloons are usually filled by using ones breath, a pump, the opening can then be permanently tied off or clamped temporarily. By filling a balloon with a gas lighter than air, such as helium, helium is the preferred gas for floating balloons, because it is inert and will not catch fire or cause toxic effects when inhaled. Small, light objects are placed in balloons along with helium and released into the air and, when the balloon eventually falls
Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure and display pressure in a unit are called pressure gauges or vacuum gauges. A manometer is an example as it uses a column of liquid to both measure and indicate pressure. Likewise the widely used Bourdon gauge is a device which both measures and indicates, and is probably the best known type of gauge. A vacuum gauge is a pressure gauge used to measure the pressures lower than the ambient atmospheric pressure. Other methods of pressure measurement involve sensors which can transmit the pressure reading to an indicator or control system. Everyday pressure measurements, such as for vehicle tire pressure, are made relative to ambient air pressure. In other cases measurements are made relative to a vacuum or to other specific reference. Gauge pressure is zero-referenced against ambient air pressure, so it is equal to absolute pressure minus atmospheric pressure, to distinguish a negative pressure, the value may be appended with the word vacuum or the gauge may be labeled a vacuum gauge. These are further divided into two subcategories, high and low vacuum, the applicable pressure ranges of many of the techniques used to measure vacuums have an overlap. Hence, by combining different types of gauge, it is possible to measure system pressure continuously from 10 mbar down to 10−11 mbar. Differential pressure is the difference in pressure between two points, the zero reference in use is usually implied by context, and these words are added only when clarification is needed. Tire pressure and blood pressure are gauge pressures by convention, while atmospheric pressures, deep vacuum pressures, for most working fluids where a fluid exists in a closed system, gauge pressure measurement prevails. Pressure instruments connected to the system will indicate pressures relative to the current atmospheric pressure, the situation changes when extreme vacuum pressures are measured, absolute pressures are typically used instead. Differential pressures are used in industrial process systems. Differential pressure gauges have two ports, each connected to one of the volumes whose pressure is to be monitored. Moderate vacuum pressure readings can be ambiguous without the proper context, thus a vacuum of 26 inHg gauge is equivalent to an absolute pressure of 30 inHg −26 inHg =4 inHg. Atmospheric pressure is typically about 100 kPa at sea level, but is variable with altitude, if the absolute pressure of a fluid stays constant, the gauge pressure of the same fluid will vary as atmospheric pressure changes
A clock is an instrument to measure, keep, and indicate time. The word clock is derived from the Celtic words clagan and clocca meaning bell, a silent instrument missing such a striking mechanism has traditionally been known as a timepiece. In general usage today a clock refers to any device for measuring and displaying the time, Watches and other timepieces that can be carried on ones person are often distinguished from clocks. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units, the day, the lunar month. Devices operating on several physical processes have been used over the millennia, a sundial shows the time by displaying the position of a shadow on a flat surface. There is a range of duration timers, an example being the hourglass. Water clocks, along with the sundials, are possibly the oldest time-measuring instruments, spring-driven clocks appeared during the 15th century. During the 15th and 16th centuries, clockmaking flourished, the next development in accuracy occurred after 1656 with the invention of the pendulum clock. A major stimulus to improving the accuracy and reliability of clocks was the importance of precise time-keeping for navigation, the electric clock was patented in 1840. The development of electronics in the 20th century led to clocks with no clockwork parts at all, the timekeeping element in every modern clock is a harmonic oscillator, a physical object that vibrates or oscillates at a particular frequency. This object can be a pendulum, a fork, a quartz crystal. Analog clocks usually indicate time using angles, Digital clocks display a numeric representation of time. Two numeric display formats are used on digital clocks, 24-hour notation. Most digital clocks use electronic mechanisms and LCD, LED, or VFD displays, for convenience, distance, telephony or blindness, auditory clocks present the time as sounds. There are also clocks for the blind that have displays that can be read by using the sense of touch, some of these are similar to normal analog displays, but are constructed so the hands can be felt without damaging them. The evolution of the technology of clocks continues today, the study of timekeeping is known as horology. The apparent position of the Sun in the sky moves over the course of a day, shadows cast by stationary objects move correspondingly, so their positions can be used to indicate the time of day. A sundial shows the time by displaying the position of a shadow on a flat surface, sundials can be horizontal, vertical, or in other orientations
For example, when a solid vertical bar is supporting a weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a container under pressure, each particle gets pushed against by all the surrounding particles. The container walls and the pressure-inducing surface push against them in reaction and these macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules. Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to the material or to its surface. Any strain of a material generates an internal elastic stress, analogous to the reaction force of a spring. In liquids and gases, only deformations that change the volume generate persistent elastic stress, however, if the deformation is gradually changing with time, even in fluids there will usually be some viscous stress, opposing that change. Elastic and viscous stresses are usually combined under the mechanical stress. Significant stress may exist even when deformation is negligible or non-existent, stress may exist in the absence of external forces, such built-in stress is important, for example, in prestressed concrete and tempered glass. Stress may also be imposed on a material without the application of net forces, for example by changes in temperature or chemical composition, stress that exceeds certain strength limits of the material will result in permanent deformation or even change its crystal structure and chemical composition. In some branches of engineering, the stress is occasionally used in a looser sense as a synonym of internal force. For example, in the analysis of trusses, it may refer to the total traction or compression force acting on a beam, since ancient times humans have been consciously aware of stress inside materials. Until the 17th century the understanding of stress was largely intuitive and empirical, with those tools, Augustin-Louis Cauchy was able to give the first rigorous and general mathematical model for stress in a homogeneous medium. Cauchy observed that the force across a surface was a linear function of its normal vector, and, moreover. The understanding of stress in liquids started with Newton, who provided a formula for friction forces in parallel laminar flow. Stress is defined as the force across a small boundary per unit area of that boundary, following the basic premises of continuum mechanics, stress is a macroscopic concept. In a fluid at rest the force is perpendicular to the surface, in a solid, or in a flow of viscous liquid, the force F may not be perpendicular to S, hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude depend on the orientation of S. Thus the stress state of the material must be described by a tensor, called the stress tensor, with respect to any chosen coordinate system, the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers
A boiler is a closed vessel in which water or other fluid is heated. The fluid does not necessarily boil, the heated or vaporized fluid exits the boiler for use in various processes or heating applications, including water heating, central heating, boiler-based power generation, cooking, and sanitation. The pressure vessel of a boiler is usually made of steel, stainless steel, especially of the austenitic types, is not used in wetted parts of boilers due to corrosion and stress corrosion cracking. In live steam models, copper or brass is used because it is more easily fabricated in smaller size boilers. For much of the Victorian age of steam, the material used for boilermaking was the highest grade of wrought iron. In the 20th century, design practice instead moved towards the use of steel, which is stronger and cheaper, with welded construction, which is quicker and requires less labour. It should be noted, however, that wrought iron boilers corrode far slower than their steel counterparts. This makes the longevity of older wrought-iron boilers far superior to those of welded steel boilers, cast iron may be used for the heating vessel of domestic water heaters. Although such heaters are usually termed boilers in some countries, their purpose is usually to produce hot water, not steam, the brittleness of cast iron makes it impractical for high-pressure steam boilers. The source of heat for a boiler is combustion of any of several fuels, such as wood, coal, oil, electric steam boilers use resistance- or immersion-type heating elements. Nuclear fission is used as a heat source for generating steam, either directly or, in most cases. Heat recovery steam generators use the heat rejected from other such as gas turbine. 18th century Haycock boilers generally produced and stored large volumes of very low-pressure steam and these could burn wood or most often, coal. Flued boiler with one or two large flues—an early type or forerunner of fire-tube boiler, fire-tube boiler, Here, water partially fills a boiler barrel with a small volume left above to accommodate the steam. This is the type of boiler used in nearly all steam locomotives, the heat source is inside a furnace or firebox that has to be kept permanently surrounded by the water in order to maintain the temperature of the heating surface below the boiling point. Fire-tube boilers usually have a low rate of steam production. Fire-tube boilers mostly burn solid fuels, but are adaptable to those of the liquid or gas variety. Water-tube boiler, In this type, tubes filled with water are arranged inside a furnace in a number of possible configurations and this type generally gives high steam production rates, but less storage capacity than the above
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of society with the most obvious being the audio. Hearing is one of the most crucial means of survival in the animal world, accordingly, the science of acoustics spreads across many facets of human society—music, medicine, architecture, industrial production, warfare and more. Likewise, animal species such as songbirds and frogs use sound, art, craft, science and technology have provoked one another to advance the whole, as in many other fields of knowledge. Robert Bruce Lindsays Wheel of Acoustics is a well accepted overview of the fields in acoustics. The word acoustic is derived from the Greek word ἀκουστικός, meaning of or for hearing, ready to hear and that from ἀκουστός, heard, audible, which in turn derives from the verb ἀκούω, I hear. The Latin synonym is sonic, after which the term used to be a synonym for acoustics. Frequencies above and below the range are called ultrasonic and infrasonic. If, for example, a string of a length would sound particularly harmonious with a string of twice the length. In modern parlance, if a string sounds the note C when plucked, a string twice as long will sound a C an octave lower. In one system of tuning, the tones in between are then given by 16,9 for D,8,5 for E,3,2 for F,4,3 for G,6,5 for A. Aristotle understood that sound consisted of compressions and rarefactions of air which falls upon, a very good expression of the nature of wave motion. The physical understanding of acoustical processes advanced rapidly during and after the Scientific Revolution, mainly Galileo Galilei but also Marin Mersenne, independently, discovered the complete laws of vibrating strings. Experimental measurements of the speed of sound in air were carried out successfully between 1630 and 1680 by a number of investigators, prominently Mersenne, meanwhile, Newton derived the relationship for wave velocity in solids, a cornerstone of physical acoustics. The eighteenth century saw advances in acoustics as mathematicians applied the new techniques of calculus to elaborate theories of sound wave propagation. Also in the 19th century, Wheatstone, Ohm, and Henry developed the analogy between electricity and acoustics, the twentieth century saw a burgeoning of technological applications of the large body of scientific knowledge that was by then in place. The first such application was Sabine’s groundbreaking work in architectural acoustics, Underwater acoustics was used for detecting submarines in the first World War
Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the relative to the ambient pressure. Various units are used to express pressure, Pressure may also be expressed in terms of standard atmospheric pressure, the atmosphere is equal to this pressure and the torr is defined as 1⁄760 of this. Manometric units such as the centimetre of water, millimetre of mercury, Pressure is the amount of force acting per unit area. The symbol for it is p or P, the IUPAC recommendation for pressure is a lower-case p. However, upper-case P is widely used. The usage of P vs p depends upon the field in one is working, on the nearby presence of other symbols for quantities such as power and momentum. Mathematically, p = F A where, p is the pressure, F is the normal force and it relates the vector surface element with the normal force acting on it. It is incorrect to say the pressure is directed in such or such direction, the pressure, as a scalar, has no direction. The force given by the relationship to the quantity has a direction. If we change the orientation of the element, the direction of the normal force changes accordingly. Pressure is distributed to solid boundaries or across arbitrary sections of normal to these boundaries or sections at every point. It is a parameter in thermodynamics, and it is conjugate to volume. The SI unit for pressure is the pascal, equal to one newton per square metre and this name for the unit was added in 1971, before that, pressure in SI was expressed simply in newtons per square metre. Other units of pressure, such as pounds per square inch, the CGS unit of pressure is the barye, equal to 1 dyn·cm−2 or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre, but using the names kilogram, gram, kilogram-force, or gram-force as units of force is expressly forbidden in SI. The technical atmosphere is 1 kgf/cm2, since a system under pressure has potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume. It is therefore related to density and may be expressed in units such as joules per cubic metre. Similar pressures are given in kilopascals in most other fields, where the prefix is rarely used