Friction is the force resisting the relative motion of solid surfaces, fluid layers, material elements sliding against each other. There are several types of friction: Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact. Dry friction is subdivided into static friction between non-moving surfaces, kinetic friction between moving surfaces. With the exception of atomic or molecular friction, dry friction arises from the interaction of surface features, known as asperities Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces. Skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. Internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation; when surfaces in contact move relative to each other, the friction between the two surfaces converts kinetic energy into thermal energy.
This property can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to thermal energy whenever motion with friction occurs, for example when a viscous fluid is stirred. Another important consequence of many types of friction can be wear, which may lead to performance degradation or damage to components. Friction is a component of the science of tribology. Friction is important in supplying traction to facilitate motion on land. Most land vehicles rely on friction for acceleration and changing direction. Sudden reductions in traction can cause loss of control and accidents. Friction is not itself a fundamental force. Dry friction arises from a combination of inter-surface adhesion, surface roughness, surface deformation, surface contamination; the complexity of these interactions makes the calculation of friction from first principles impractical and necessitates the use of empirical methods for analysis and the development of theory.
Friction is a non-conservative force - work done against friction is path dependent. In the presence of friction, some energy is always lost in the form of heat, thus mechanical energy is not conserved. The Greeks, including Aristotle and Pliny the Elder, were interested in the cause and mitigation of friction, they were aware of differences between static and kinetic friction with Themistius stating in 350 A. D. that "it is easier to further the motion of a moving body than to move a body at rest". The classic laws of sliding friction were discovered by Leonardo da Vinci in 1493, a pioneer in tribology, but the laws documented in his notebooks, were not published and remained unknown; these laws were rediscovered by Guillaume Amontons in 1699 and became known as Amonton's three laws of dry friction. Amontons presented the nature of friction in terms of surface irregularities and the force required to raise the weight pressing the surfaces together; this view was further elaborated by Bernard Forest de Bélidor and Leonhard Euler, who derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction.
John Theophilus Desaguliers first recognized the role of adhesion in friction. Microscopic forces cause surfaces to stick together; the understanding of friction was further developed by Charles-Augustin de Coulomb. Coulomb investigated the influence of four main factors on friction: the nature of the materials in contact and their surface coatings. Coulomb further considered the influence of sliding velocity and humidity, in order to decide between the different explanations on the nature of friction, proposed; the distinction between static and dynamic friction is made in Coulomb's friction law, although this distinction was drawn by Johann Andreas von Segner in 1758. The effect of the time of repose was explained by Pieter van Musschenbroek by considering the surfaces of fibrous materials, with fibers meshing together, which takes a finite time in which the friction increases. John Leslie noted a weakness in the views of Amontons and Coulomb: If friction arises from a weight being drawn up the inclined plane of successive asperities, why isn't it balanced through descending the opposite slope?
Leslie was skeptical about the role of adhesion proposed by Desaguliers, which should on the whole have the same tendency to accelerate as to retard the motion. In Leslie's view, friction should be seen as a time-dependent process of flattening, pressing down asperities, which creates new obstacles in what were cavities before. Arthur Jules Morin developed the concept of sliding versus rolling friction. Osborne Reynolds derived the equation of viscous flow; this completed the classic empirical model of friction used today in engineering. In 1877, Fleeming Jenkin and J. A. Ewing investigated the continuity between static and kinetic friction; the focus of research during the 20th century has been to understand the physical mechanisms behind friction. Frank Philip Bowden and David Tabor showed that, at a microscopic level, the actual area of contact between surfaces is a small fraction of the apparent area; this actual area of contact, caused by asperities increases with pressure. The development of the atomic force microscope
Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars and light—are brought toward one another. On Earth, gravity gives weight to physical objects, the Moon's gravity causes the ocean tides; the gravitational attraction of the original gaseous matter present in the Universe caused it to begin coalescing, forming stars – and for the stars to group together into galaxies – so gravity is responsible for many of the large-scale structures in the Universe. Gravity has an infinite range, although its effects become weaker on farther objects. Gravity is most described by the general theory of relativity which describes gravity not as a force, but as a consequence of the curvature of spacetime caused by the uneven distribution of mass; the most extreme example of this curvature of spacetime is a black hole, from which nothing—not light—can escape once past the black hole's event horizon. However, for most applications, gravity is well approximated by Newton's law of universal gravitation, which describes gravity as a force which causes any two bodies to be attracted to each other, with the force proportional to the product of their masses and inversely proportional to the square of the distance between them.
Gravity is the weakest of the four fundamental forces of physics 1038 times weaker than the strong force, 1036 times weaker than the electromagnetic force and 1029 times weaker than the weak force. As a consequence, it has no significant influence at the level of subatomic particles. In contrast, it is the dominant force at the macroscopic scale, is the cause of the formation and trajectory of astronomical bodies. For example, gravity causes the Earth and the other planets to orbit the Sun, it causes the Moon to orbit the Earth, causes the formation of tides, the formation and evolution of the Solar System and galaxies; the earliest instance of gravity in the Universe in the form of quantum gravity, supergravity or a gravitational singularity, along with ordinary space and time, developed during the Planck epoch from a primeval state, such as a false vacuum, quantum vacuum or virtual particle, in a unknown manner. Attempts to develop a theory of gravity consistent with quantum mechanics, a quantum gravity theory, which would allow gravity to be united in a common mathematical framework with the other three forces of physics, are a current area of research.
Archimedes discovered the center of gravity of a triangle. He postulated that if the centers of gravity of two equal weights wasn't the same, it would be located in the middle of the line that joins them; the Roman architect and engineer Vitruvius in De Architectura postulated that gravity of an object didn't depend on weight but its "nature". Aryabhata first identified the force to explain why objects are not thrown out when the earth rotates. Brahmagupta described gravity as an attractive force and used the term "gruhtvaakarshan" for gravity. Modern work on gravitational theory began with the work of Galileo Galilei in the late 16th and early 17th centuries. In his famous experiment dropping balls from the Tower of Pisa, with careful measurements of balls rolling down inclines, Galileo showed that gravitational acceleration is the same for all objects; this was a major departure from Aristotle's belief that heavier objects have a higher gravitational acceleration. Galileo postulated air resistance as the reason that objects with less mass fall more in an atmosphere.
Galileo's work set the stage for the formulation of Newton's theory of gravity. In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. In his own words, "I deduced that the forces which keep the planets in their orbs must reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth; the equation is the following: F = G m 1 m 2 r 2 Where F is the force, m1 and m2 are the masses of the objects interacting, r is the distance between the centers of the masses and G is the gravitational constant. Newton's theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted for by the actions of the other planets. Calculations by both John Couch Adams and Urbain Le Verrier predicted the general position of the planet, Le Verrier's calculations are what led Johann Gottfried Galle to the discovery of Neptune.
A discrepancy in Mercury's orbit pointed out flaws in Newton's theory. By the end of the 19th century, it was known that its orbit showed slight perturbations that could not be accounted for under Newton's theory, but all searches for another perturbing body had been fruitless; the issue was resolved in 1915 by Albert Einstein's new theory of general relativity, which accounted for the small discrepancy in Mercury's orbit. This discrepancy was the advance in the perihelion of Mercury of 42.98 arcseconds per century. Although Newton's theory has been superseded by Einstein's general relativity, most modern non-relativistic gravitational calculations are still made using Newton
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function; this article considers linear operators, which are the most common type. However, non-linear differential operators, such as the Schwarzian derivative exist. Assume that there is a map A from a function space F 1 to another function space F 2 and a function f ∈ F 2 so that f is the image of u ∈ F 1 i.e. F = A. A differential operator is represented as a linear combination, finitely generated by u and its derivatives containing higher degree such as P = ∑ | α | ≤ m a α D α, where the set of non-negative integers, α =, is called a multi-index, | α | = α 1 + α 2 + ⋯ + α n called length, a α are functions on some open domain in n-dimensional space and D α = D 1 α 1 D 2 α 2 ⋯ D n α n; the derivative above is one as functions or, distributions or hyperfunctions and D j = − i ∂ ∂ x j or sometimes, D j = ∂ ∂ x j.
The most common differential operator is the action of taking derivative. Common notations for taking the first derivative with respect to a variable x include: d d x, D, D x, ∂ x; when taking higher, nth order derivatives, the operator may be written: d n d x n, D n, D x n, or ∂ x n. The derivative of a function f of an argument x is sometimes given as either of the following: ′ f ′; the D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form ∑ k = 0 n c k D k in his study of differential equations. One of the most seen differential operators is the Laplacian operator, defined by Δ = ∇ 2 = ∑ k = 1 n ∂ 2 ∂ x k 2. Another differential operator is the Θ operator, or theta operator, defined by Θ = z d d z; this is sometimes called the homogeneity operator, because its eigenfunctions are the monomials in z: Θ = k z k, k = 0, 1, 2, … In n variables the homogeneity operator is given by Θ = ∑ k = 1 n x k ∂ ∂ x k. As in one variable
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions represent physical quantities, the derivatives represent their rates of change, the equation defines a relationship between the two; because such relations are common, differential equations play a prominent role in many disciplines including engineering, physics and biology. In pure mathematics, differential equations are studied from several different perspectives concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas. If a closed-form expression for the solution is not available, the solution may be numerically approximated using computers; the theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. Differential equations first came into existence with the invention of calculus by Newton and Leibniz.
In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Isaac Newton listed three kinds of differential equations: d y d x = f d y d x = f x 1 ∂ y ∂ x 1 + x 2 ∂ y ∂ x 2 = y He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. Jacob Bernoulli proposed the Bernoulli differential equation in 1695; this is an ordinary differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. The problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, within ten years Euler discovered the three-dimensional wave equation; the Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur, in which he based his reasoning on Newton's law of cooling, that the flow of heat between two adjacent molecules is proportional to the small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat; this partial differential equation is now taught to every student of mathematical physics. For example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance.
The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, air resistance may be modeled as proportional to the ball's velocity; this means that the ball's acceleration, a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation and verifying its validity. Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, Homogeneous/Inhomogeneous; this list is far from exhaustive. An ordinary differential equation is an equation containing an unknown function of one real or complex variable x, its derivatives, some
Autonomous system (mathematics)
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are called time-invariant systems. Many laws in physics, where the independent variable is assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future. Autonomous systems are related to dynamical systems. Any autonomous system can be transformed into a dynamical system and, using weak assumptions, a dynamical system can be transformed into an autonomous system. An autonomous system is a system of ordinary differential equations of the form d d t x = f where x takes values in n-dimensional Euclidean space and t is time, it is distinguished from systems of differential equations of the form d d t x = g in which the law governing the rate of motion of a particle depends not only on the particle's location, but on time.
Let x 1 be a unique solution of the initial value problem for an autonomous system d d t x = f, x = x 0. X 2 = x 1 solves d d t x = f, x = x 0. Indeed, denoting s = t − t 0 we have x 1 = x 2 and d s = d t, thus d d t x 2 = d d t x 1 = d d s x 1 = f = f. For the initial condition, the verification is trivial, x 2 = x 1 = x 1 = x 0; the equation y ′ = y is autonomous, since the independent variable, let us call it x, does not explicitly appear in the equation. To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB One can observe from the plot that the function y is x -invariant, so is the shape of the solution, i.e. y = y for any shift x 0. Solving the equation symbolically in MATLAB, by running we obtain two equilibrium solutions, y = 0 and y = 2, a third solution involving an unknown constant C 3, Picking up some specific values for the initial condition, we can add the plot of several solutions Autonomous systems can be analyzed qualitatively using the phase space.
The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order n is equivalent to an n -dimensional first-order system, but not vice versa; the first-order autonomous equation d x d t = f
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