1.
System of measurement
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A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce, systems of measurement in modern use include the metric system, the imperial system, and United States customary units. The French Revolution gave rise to the system, and this has spread around the world. In most systems, length, mass, and time are base quantities, later science developments showed that either electric charge or electric current could be added to extend the set of base quantities by which many other metrological units could be easily defined. Other quantities, such as power and speed, are derived from the set, for example. Such arrangements were satisfactory in their own contexts, the preference for a more universal and consistent system only gradually spread with the growth of science. Changing a measurement system has substantial financial and cultural costs which must be offset against the advantages to be obtained using a more rational system. However pressure built up, including scientists and engineers for conversion to a more rational. The unifying characteristic is that there was some definition based on some standard, eventually cubits and strides gave way to customary units to met the needs of merchants and scientists. In the metric system and other recent systems, a basic unit is used for each base quantity. Often secondary units are derived from the units by multiplying by powers of ten. Thus the basic unit of length is the metre, a distance of 1.234 m is 1,234 millimetres. Metrication is complete or nearly complete in almost all countries, US customary units are heavily used in the United States and to some degree in Liberia. Traditional Burmese units of measurement are used in Burma, U. S. units are used in limited contexts in Canada due to the large volume of trade, there is also considerable use of Imperial weights and measures, despite de jure Canadian conversion to metric. In the United States, metric units are used almost universally in science, widely in the military, and partially in industry, but customary units predominate in household use. At retail stores, the liter is a used unit for volume, especially on bottles of beverages. Some other standard non-SI units are still in use, such as nautical miles and knots in aviation. Metric systems of units have evolved since the adoption of the first well-defined system in France in 1795, during this evolution the use of these systems has spread throughout the world, first to non-English-speaking countries, and then to English speaking countries
2.
SI derived unit
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The International System of Units specifies a set of seven base units from which all other SI units of measurement are derived. Each of these units is either dimensionless or can be expressed as a product of powers of one or more of the base units. For example, the SI derived unit of area is the metre. The degree Celsius has an unclear status, and is arguably an exception to this rule. The names of SI units are written in lowercase, the symbols for units named after persons, however, are always written with an uppercase initial letter. In addition to the two dimensionless derived units radian and steradian,20 other derived units have special names, some other units such as the hour, litre, tonne, bar and electronvolt are not SI units, but are widely used in conjunction with SI units. Until 1995, the SI classified the radian and the steradian as supplementary units, but this designation was abandoned, International System of Quantities International System of Units International Vocabulary of Metrology Metric prefix Metric system Non-SI units mentioned in the SI Planck units SI base unit I. Mills, Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC, Quantities, Units and Symbols in Physical Chemistry. CS1 maint, Multiple names, authors list
3.
Solid angle
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In geometry, a solid angle is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point, in the International System of Units, a solid angle is expressed in a dimensionless unit called a steradian. A small object nearby may subtend the same angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse, an objects solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the angles vertex, that the object covers. A solid angle in steradians equals the area of a segment of a sphere in the same way a planar angle in radians equals the length of an arc of a unit circle. Solid angles are used in physics, in particular astrophysics. The solid angle of an object that is far away is roughly proportional to the ratio of area to squared distance. Here area means the area of the object when projected along the viewing direction. The solid angle of a sphere measured from any point in its interior is 4π sr, Solid angles can also be measured in square degrees, in square minutes and square seconds, or in fractions of the sphere, also known as spat. In spherical coordinates there is a formula for the differential, d Ω = sin θ d θ d φ where θ is the colatitude, at the equator you see all of the celestial sphere, at either pole only one half. Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where a →, b →, c → are the positions of the vertices A, B and C. Define the vertex angle θa to be the angle BOC and define θb, let φab be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define φac, φbc correspondingly. When implementing the above equation care must be taken with the function to avoid negative or incorrect solid angles. One source of errors is that the scalar triple product can be negative if a, b, c have the wrong winding. Computing abs is a sufficient solution since no other portion of the equation depends on the winding, the other pitfall arises when the scalar triple product is positive but the divisor is negative. Indices are cycled, s0 = sn and s1 = sn +1, the solid angle of a latitude-longitude rectangle on a globe is s r, where φN and φS are north and south lines of latitude, and θE and θW are east and west lines of longitude. Mathematically, this represents an arc of angle φN − φS swept around a sphere by θE − θW radians, when longitude spans 2π radians and latitude spans π radians, the solid angle is that of a sphere
4.
International System of Units
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The International System of Units is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, the system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system was published in 1960 as the result of an initiative began in 1948. It is based on the system of units rather than any variant of the centimetre-gram-second system. The motivation for the development of the SI was the diversity of units that had sprung up within the CGS systems, the International System of Units has been adopted by most developed countries, however, the adoption has not been universal in all English-speaking countries. The metric system was first implemented during the French Revolution with just the metre and kilogram as standards of length, in the 1830s Carl Friedrich Gauss laid the foundations for a coherent system based on length, mass, and time. In the 1860s a group working under the auspices of the British Association for the Advancement of Science formulated the requirement for a coherent system of units with base units and derived units. Meanwhile, in 1875, the Treaty of the Metre passed responsibility for verification of the kilogram, in 1921, the Treaty was extended to include all physical quantities including electrical units originally defined in 1893. The units associated with these quantities were the metre, kilogram, second, ampere, kelvin, in 1971, a seventh base quantity, amount of substance represented by the mole, was added to the definition of SI. On 11 July 1792, the proposed the names metre, are, litre and grave for the units of length, area, capacity. The committee also proposed that multiples and submultiples of these units were to be denoted by decimal-based prefixes such as centi for a hundredth, on 10 December 1799, the law by which the metric system was to be definitively adopted in France was passed. Prior to this, the strength of the magnetic field had only been described in relative terms. The technique used by Gauss was to equate the torque induced on a magnet of known mass by the earth’s magnetic field with the torque induced on an equivalent system under gravity. The resultant calculations enabled him to assign dimensions based on mass, length, a French-inspired initiative for international cooperation in metrology led to the signing in 1875 of the Metre Convention. Initially the convention only covered standards for the metre and the kilogram, one of each was selected at random to become the International prototype metre and International prototype kilogram that replaced the mètre des Archives and kilogramme des Archives respectively. Each member state was entitled to one of each of the prototypes to serve as the national prototype for that country. Initially its prime purpose was a periodic recalibration of national prototype metres. The official language of the Metre Convention is French and the version of all official documents published by or on behalf of the CGPM is the French-language version
5.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
6.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
7.
Angle
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
8.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
9.
SI supplementary unit
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The International System of Units specifies a set of seven base units from which all other SI units of measurement are derived. Each of these units is either dimensionless or can be expressed as a product of powers of one or more of the base units. For example, the SI derived unit of area is the metre. The degree Celsius has an unclear status, and is arguably an exception to this rule. The names of SI units are written in lowercase, the symbols for units named after persons, however, are always written with an uppercase initial letter. In addition to the two dimensionless derived units radian and steradian,20 other derived units have special names, some other units such as the hour, litre, tonne, bar and electronvolt are not SI units, but are widely used in conjunction with SI units. Until 1995, the SI classified the radian and the steradian as supplementary units, but this designation was abandoned, International System of Quantities International System of Units International Vocabulary of Metrology Metric prefix Metric system Non-SI units mentioned in the SI Planck units SI base unit I. Mills, Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC, Quantities, Units and Symbols in Physical Chemistry. CS1 maint, Multiple names, authors list
10.
Unit sphere
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Usually a specific point has been distinguished as the origin of the space under study and it is understood that a unit sphere or unit ball is centered at that point. Therefore one speaks of the ball or the unit sphere. For example, a sphere is the surface of what is commonly called a circle, while such a circles interior. Similarly, a sphere is the surface of the Euclidean solid known colloquially as a sphere, while the interior. A unit sphere is simply a sphere of radius one, the importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of translation and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere. In Euclidean space of n dimensions, the sphere is the set of all points which satisfy the equation x 12 + x 22 + ⋯ + x n 2 =1. The volume of the ball in n dimensions, which we denote Vn. It is V n = π n /2 Γ = { π n /2 /, I f n ≥0 i s e v e n, π ⌊ n /2 ⌋2 ⌈ n /2 ⌉ / n. I f n ≥0 i s o d d, where n. is the double factorial, the surface areas and the volumes for some values of n are as follows, where the decimal expanded values for n ≥2 are rounded to the displayed precision. The An values satisfy the recursion, A0 =0 A1 =2 A2 =2 π A n =2 π n −2 A n −2 for n >2. The Vn values satisfy the recursion, V0 =1 V1 =2 V n =2 π n V n −2 for n >1. The surface area of a sphere with radius r is An rn−1. For instance, the area is A = 4π r 2 for the surface of the ball of radius r. The volume is V = 4π r 3 /3 for the ball of radius r. More precisely, the unit ball in a normed vector space V. It is the interior of the unit ball of. The latter is the disjoint union of the former and their common border, the shape of the unit ball is entirely dependent on the chosen norm, it may well have corners, and for example may look like n, in the case of the norm l∞ in Rn
11.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
12.
Surface area
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The surface area of a solid object is a measure of the total area that the surface of the object occupies. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces and this definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of area was sought by Henri Lebesgue. Their work led to the development of measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface, while the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function S ↦ A which assigns a real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the area is its additivity. More rigorously, if a surface S is a union of many pieces S1, …, Sr which do not overlap except at their boundaries. Surface areas of polygonal shapes must agree with their geometrically defined area. Since surface area is a notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface. This means that surface area is invariant under the group of Euclidean motions and these properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of many pieces that can be represented in the parametric form S D, r → = r →, ∈ D with a continuously differentiable function r →. The area of a piece is defined by the formula A = ∬ D | r → u × r → v | d u d v. Thus the area of SD is obtained by integrating the length of the vector r → u × r → v to the surface over the appropriate region D in the parametric uv plane. The area of the surface is then obtained by adding together the areas of the pieces. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f and surfaces of revolution. It was demonstrated by Hermann Schwarz that already for the cylinder, various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a natural notion of surface area, if a surface is very irregular, or rough
13.
Spherical cap
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In geometry, a spherical cap, spherical dome, or spherical segment of one base is a portion of a sphere cut off by a plane. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the red section of the illustration is also a spherical cap. The volume may also be found by integrating under a surface of rotation, using x = r cos , V = ∫ x r π d x = π = π3 r 32. If d < r1 + r2 is the distance between the two centers, elimination of the variables h1 and h2 leads to V = π12 d 2. The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the dome is circularly symmetric. For odd n =2 k +1, G n = ∑ i =0 k i q 2 i +12 i +1 and it is shown in that, if n → ∞ and q n = const. Then p n →1 − F where F is the integral of the normal distribution. Geometry of four hard fused spheres in a spatial configuration. Gibson, K. D. Scheraga, Harold A, volume of the intersection of three spheres of unequal size, a simplified formula. Gibson, K. D. Scheraga, Harold A, exact calculation of the volume and surface area of fused hard-sphere molecules with unequal atomic radii. On the analytical calculation of van der Waals surfaces and volumes, grant, J. A. Pickup, B. T. A Gaussian description of molecular shape, busa, Jan, Dzurina, Jozef, Hayryan, Edik, Hayryan, Shura. ARVO, A fortran package for computing the solvent accessible surface area, online calculator for spherical cap volume and area
14.
Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
15.
Spherical trigonometry
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Spherical trigonometry is of great importance for calculations in astronomy, geodesy and navigation. The origins of spherical trigonometry in Greek mathematics and the developments in Islamic mathematics are discussed fully in History of trigonometry. This book is now available on the web. The only significant developments since then have been the application of methods for the derivation of the theorems. A spherical polygon is a polygon on the surface of the sphere defined by a number of great-circle arcs, such polygons may have any number of sides. Two planes define a lune, also called a digon or bi-angle, the analogue of the triangle. Three planes define a triangle, the principal subject of this article. Four planes define a spherical quadrilateral, such a figure, and higher sided polygons, from this point the article will be restricted to spherical triangles, denoted simply as triangles. Both vertices and angles at the vertices are denoted by the upper case letters A, B and C. The angles of spherical triangles are less than π so that π < A + B + C < 3π. The sides are denoted by letters a, b, c. On the unit sphere their lengths are equal to the radian measure of the angles that the great circle arcs subtend at the centre. The sides of proper spherical triangles are less than π so that 0 < a + b + c < 3π, the radius of the sphere is taken as unity. For specific practical problems on a sphere of radius R the measured lengths of the sides must be divided by R before using the identities given below, likewise, after a calculation on the unit sphere the sides a, b, c must be multiplied by R. The polar triangle associated with a triangle ABC is defined as follows, consider the great circle that contains the side BC. This great circle is defined by the intersection of a plane with the surface. The points B and C are defined similarly, the triangle ABC is the polar triangle corresponding to triangle ABC. Therefore, if any identity is proved for the triangle ABC then we can derive a second identity by applying the first identity to the polar triangle by making the above substitutions
16.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
17.
Arc length
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Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves, the advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. A curve in the plane can be approximated by connecting a number of points on the curve using line segments to create a polygonal path. If the curve is not already a polygonal path, using a larger number of segments of smaller lengths will result in better approximations. For some curves there is a smallest number L that is a bound on the length of any polygonal approximation. These curves are called rectifiable and the number L is defined as the arc length, let f, → R n be a continuously differentiable function. The length of the curve defined by f can be defined as the limit of the sum of line segment lengths for a partition of as the number of segments approaches infinity. This means L = lim N → ∞ ∑ i =1 N | f − f | where t i = a + i / N = a + i Δ t for i =0,1, …, N. This means ∑ i =1 N | f − f Δ t | Δ t − ∑ i =1 N | f ′ | Δ t has absolute value less than ϵ for N > / δ. This means that in the limit N → ∞, the left term above equals the right term and this definition of arc length shows that the length of a curve f, → R n continuously differentiable on is always finite. In other words, the curve is always rectifiable and this definition is also valid if f is merely continuous, not differentiable. A curve can be parameterized in infinitely many ways, let φ, → be any continuously differentiable bijection. Then g = f ∘ φ −1, → R n is another continuously differentiable parameterization of the curve defined by f. Curves with closed-form solutions for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola, the lack of a closed form solution for the arc length of an elliptic arc led to the development of the elliptic integrals. In most cases, including even simple curves, there are no solutions for arc length. Numerical integration of the arc length integral is very efficient. For example, consider the problem of finding the length of a quarter of the circle by numerically integrating the arc length integral. The upper half of the circle can be parameterized as y =1 − x 2
18.
N-sphere
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In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. It is a manifold that can be embedded in Euclidean -space. Thus, the n-sphere would be defined by, S n =, an n-sphere embedded in an -dimensional Euclidean space is called a hypersphere. The n-sphere of unit radius is called the unit n-sphere, denoted Sn, the unit n-sphere is often referred to as the n-sphere. When embedded as described, an n-sphere is the surface or boundary of an -dimensional ball, for n ≥2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature. In particular, a 0-sphere is a pair of points, and is the boundary of a line segment, a 1-sphere is a circle of radius r centered at c, and is the boundary of a disk. A 2-sphere is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, a 3-sphere is a sphere in 4-dimensional Euclidean space. The set of points in -space, that define an n-sphere, is represented by the equation, where c is a center point, and r is the radius. The above n-sphere exists in -dimensional Euclidean space and is an example of an n-manifold, as a result, d r ∧ ω = d x 1 ∧ ⋯ ∧ d x n +1. The space enclosed by an n-sphere is called an -ball, an -ball is closed if it includes the n-sphere, and it is open if it does not include the n-sphere. Specifically, A 1-ball, a segment, is the interior of a 0-sphere. A 2-ball, a disk, is the interior of a circle, a 3-ball, an ordinary ball, is the interior of a sphere. A 4-ball is the interior of a 3-sphere, etc, topologically, an n-sphere can be constructed as a one-point compactification of n-dimensional Euclidean space. Briefly, the n-sphere can be described as S n = R n ∪, in particular, if a single point is removed from an n-sphere, it becomes homeomorphic to R n. This forms the basis for stereographic projection, in general, the volumes of the n-ball in n-dimensional Euclidean space, and the n-sphere in -dimensional Euclidean, of radius R, are proportional to the nth power of the radius, R. The 0-ball consists of a single point, the 0-dimensional Hausdorff measure is the number of points in a set, so V0 =1. The unit 1-ball is the interval of length 2, the 0-sphere consists of its two end-points. The unit 1-sphere is the circle in the Euclidean plane
19.
Point (geometry)
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In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, in particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a location in Euclidean space. Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects, Euclid originally defined the point as that which has no part. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by a triplet with the additional third number representing depth. Further generalizations are represented by an ordered tuplet of n terms, many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points, As an example, a line is a set of points of the form L =. Similar constructions exist that define the plane, line segment and other related concepts, a line segment consisting of only a single point is called a degenerate line segment. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, in spite of this, modern expansions of the system serve to remove these assumptions. There are several inequivalent definitions of dimension in mathematics, in all of the common definitions, a point is 0-dimensional. The dimension of a space is the maximum size of a linearly independent subset. In a vector space consisting of a point, there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero,1 ⋅0 =0, if no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a open set. The Hausdorff dimension of X is defined by dim H , = inf, a point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius. Although the notion of a point is considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry. More precisely, such structures generalize well-known spaces of functions in a way that the operation take a value at this point may not be defined
20.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
21.
SI base unit
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The International System of Units defines seven units of measure as a basic set from which all other SI units can be derived. The SI base units form a set of mutually independent dimensions as required by dimensional analysis commonly employed in science, thus, the kelvin, named after Lord Kelvin, has the symbol K and the ampere, named after André-Marie Ampère, has the symbol A. Many other units, such as the litre, are not part of the SI. The definitions of the units have been modified several times since the Metre Convention in 1875. Since the redefinition of the metre in 1960, the kilogram is the unit that is directly defined in terms of a physical artifact. However, the mole, the ampere, and the candela are linked through their definitions to the mass of the platinum–iridium cylinder stored in a vault near Paris. It has long been an objective in metrology to define the kilogram in terms of a fundamental constant, two possibilities have attracted particular attention, the Planck constant and the Avogadro constant. The 23rd CGPM decided to postpone any formal change until the next General Conference in 2011
22.
Ampere
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The ampere, often shortened to amp, is a unit of electric current. In the International System of Units the ampere is one of the seven SI base units and it is named after André-Marie Ampère, French mathematician and physicist, considered the father of electrodynamics. SI defines the ampere in terms of base units by measuring the electromagnetic force between electrical conductors carrying electric current. The ampere was then defined as one coulomb of charge per second, in SI, the unit of charge, the coulomb, is defined as the charge carried by one ampere during one second. In the future, the SI definition may shift back to charge as the base unit, ampères force law states that there is an attractive or repulsive force between two parallel wires carrying an electric current. This force is used in the definition of the ampere. The SI unit of charge, the coulomb, is the quantity of electricity carried in 1 second by a current of 1 ampere, conversely, a current of one ampere is one coulomb of charge going past a given point per second,1 A =1 C s. In general, charge Q is determined by steady current I flowing for a time t as Q = It, constant, instantaneous and average current are expressed in amperes and the charge accumulated, or passed through a circuit over a period of time is expressed in coulombs. The relation of the ampere to the coulomb is the same as that of the watt to the joule, the ampere was originally defined as one tenth of the unit of electric current in the centimetre–gram–second system of units. That unit, now known as the abampere, was defined as the amount of current that generates a force of two dynes per centimetre of length between two wires one centimetre apart. The size of the unit was chosen so that the derived from it in the MKSA system would be conveniently sized. The international ampere was a realization of the ampere, defined as the current that would deposit 0.001118 grams of silver per second from a silver nitrate solution. Later, more accurate measurements revealed that this current is 0.99985 A, at present, techniques to establish the realization of an ampere have a relative uncertainty of approximately a few parts in 107, and involve realizations of the watt, the ohm and the volt. Rather than a definition in terms of the force between two current-carrying wires, it has proposed that the ampere should be defined in terms of the rate of flow of elementary charges. Since a coulomb is equal to 6. 2415093×1018 elementary charges. The proposed change would define 1 A as being the current in the direction of flow of a number of elementary charges per second. In 2005, the International Committee for Weights and Measures agreed to study the proposed change, the new definition was discussed at the 25th General Conference on Weights and Measures in 2014 but for the time being was not adopted. The current drawn by typical constant-voltage energy distribution systems is usually dictated by the power consumed by the system, for this reason the examples given below are grouped by voltage level
23.
Candela
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The candela is the SI base unit of luminous intensity, that is, luminous power per unit solid angle emitted by a point light source in a particular direction. A common candle emits light with an intensity of roughly one candela. If emission in some directions is blocked by an opaque barrier, the word candela means candle in Latin. Like most other SI base units, the candela has an operational definition—it is defined by a description of a process that will produce one candela of luminous intensity. The definition describes how to produce a source that emits one candela. Such a source could then be used to calibrate instruments designed to measure luminous intensity, the candela is sometimes still called by the old name candle, such as in foot-candle and the modern definition of candlepower. The frequency chosen is in the spectrum near green, corresponding to a wavelength of about 555 nanometres. The human eye is most sensitive to frequency, when adapted for bright conditions. At other frequencies, more radiant intensity is required to achieve the same luminous intensity, if more than one wavelength is present, one must sum or integrate over the spectrum of wavelengths present to get the total luminous intensity. A common candle emits light with roughly 1 cd luminous intensity. A25 W compact fluorescent light bulb puts out around 1700 lumens, if light is radiated equally in all directions. Focused into a 20° beam, the light bulb would have an intensity of around 18,000 cd. The luminous intensity of light-emitting diodes is measured in millicandelas, or thousandths of a candela, indicator LEDs are typically in the 50 mcd range, ultra-bright LEDs can reach 15,000 mcd, or higher. Prior to 1948, various standards for luminous intensity were in use in a number of countries and these were typically based on the brightness of the flame from a standard candle of defined composition, or the brightness of an incandescent filament of specific design. One of the best-known of these was the English standard of candlepower, one candlepower was the light produced by a pure spermaceti candle weighing one sixth of a pound and burning at a rate of 120 grains per hour. Germany, Austria and Scandinavia used the Hefnerkerze, a based on the output of a Hefner lamp. It became clear that a unit was needed. Jules Violle had proposed a standard based on the emitted by 1 cm2 of platinum at its melting point
24.
Kelvin
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The kelvin is a unit of measure for temperature based upon an absolute scale. It is one of the seven units in the International System of Units and is assigned the unit symbol K. The kelvin is defined as the fraction 1⁄273.16 of the temperature of the triple point of water. In other words, it is defined such that the point of water is exactly 273.16 K. The Kelvin scale is named after the Belfast-born, Glasgow University engineer and physicist William Lord Kelvin, unlike the degree Fahrenheit and degree Celsius, the kelvin is not referred to or typeset as a degree. The kelvin is the unit of temperature measurement in the physical sciences, but is often used in conjunction with the Celsius degree. The definition implies that absolute zero is equivalent to −273.15 °C, Kelvin calculated that absolute zero was equivalent to −273 °C on the air thermometers of the time. This absolute scale is known today as the Kelvin thermodynamic temperature scale, when spelled out or spoken, the unit is pluralised using the same grammatical rules as for other SI units such as the volt or ohm. When reference is made to the Kelvin scale, the word kelvin—which is normally a noun—functions adjectivally to modify the noun scale and is capitalized, as with most other SI unit symbols there is a space between the numeric value and the kelvin symbol. Before the 13th CGPM in 1967–1968, the unit kelvin was called a degree and it was distinguished from the other scales with either the adjective suffix Kelvin or with absolute and its symbol was °K. The latter term, which was the official name from 1948 until 1954, was ambiguous since it could also be interpreted as referring to the Rankine scale. Before the 13th CGPM, the form was degrees absolute. The 13th CGPM changed the name to simply kelvin. Its measured value was 7002273160280000000♠0.01028 °C with an uncertainty of 60 µK, the use of SI prefixed forms of the degree Celsius to express a temperature interval has not been widely adopted. In 2005 the CIPM embarked on a program to redefine the kelvin using a more experimentally rigorous methodology, the current definition as of 2016 is unsatisfactory for temperatures below 20 K and above 7003130000000000000♠1300 K. In particular, the committee proposed redefining the kelvin such that Boltzmanns constant takes the exact value 6977138065049999999♠1. 3806505×10−23 J/K, from a scientific point of view, this will link temperature to the rest of SI and result in a stable definition that is independent of any particular substance. From a practical point of view, the redefinition will pass unnoticed, the kelvin is often used in the measure of the colour temperature of light sources. Colour temperature is based upon the principle that a black body radiator emits light whose colour depends on the temperature of the radiator, black bodies with temperatures below about 7003400000000000000♠4000 K appear reddish, whereas those above about 7003750000000000000♠7500 K appear bluish
25.
Kilogram
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The kilogram or kilogramme is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype of the Kilogram. The avoirdupois pound, used in both the imperial and US customary systems, is defined as exactly 0.45359237 kg, making one kilogram approximately equal to 2.2046 avoirdupois pounds. Other traditional units of weight and mass around the world are also defined in terms of the kilogram, the gram, 1/1000 of a kilogram, was provisionally defined in 1795 as the mass of one cubic centimeter of water at the melting point of ice. The final kilogram, manufactured as a prototype in 1799 and from which the IPK was derived in 1875, had an equal to the mass of 1 dm3 of water at its maximum density. The kilogram is the only SI base unit with an SI prefix as part of its name and it is also the only SI unit that is still directly defined by an artifact rather than a fundamental physical property that can be reproduced in different laboratories. Three other base units and 17 derived units in the SI system are defined relative to the kilogram, only 8 other units do not require the kilogram in their definition, temperature, time and frequency, length, and angle. At its 2011 meeting, the CGPM agreed in principle that the kilogram should be redefined in terms of the Planck constant, the decision was originally deferred until 2014, in 2014 it was deferred again until the next meeting. There are currently several different proposals for the redefinition, these are described in the Proposed Future Definitions section below, the International Prototype Kilogram is rarely used or handled. In the decree of 1795, the term gramme thus replaced gravet, the French spelling was adopted in the United Kingdom when the word was used for the first time in English in 1797, with the spelling kilogram being adopted in the United States. In the United Kingdom both spellings are used, with kilogram having become by far the more common, UK law regulating the units to be used when trading by weight or measure does not prevent the use of either spelling. In the 19th century the French word kilo, a shortening of kilogramme, was imported into the English language where it has used to mean both kilogram and kilometer. In 1935 this was adopted by the IEC as the Giorgi system, now known as MKS system. In 1948 the CGPM commissioned the CIPM to make recommendations for a practical system of units of measurement. This led to the launch of SI in 1960 and the subsequent publication of the SI Brochure, the kilogram is a unit of mass, a property which corresponds to the common perception of how heavy an object is. Mass is a property, that is, it is related to the tendency of an object at rest to remain at rest, or if in motion to remain in motion at a constant velocity. Accordingly, for astronauts in microgravity, no effort is required to hold objects off the cabin floor, they are weightless. However, since objects in microgravity still retain their mass and inertia, the ratio of the force of gravity on the two objects, measured by the scale, is equal to the ratio of their masses. On April 7,1795, the gram was decreed in France to be the weight of a volume of pure water equal to the cube of the hundredth part of the metre
26.
Metre
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The metre or meter, is the base unit of length in the International System of Units. The metre is defined as the length of the path travelled by light in a vacuum in 1/299792458 seconds, the metre was originally defined in 1793 as one ten-millionth of the distance from the equator to the North Pole. In 1799, it was redefined in terms of a metre bar. In 1960, the metre was redefined in terms of a number of wavelengths of a certain emission line of krypton-86. In 1983, the current definition was adopted, the imperial inch is defined as 0.0254 metres. One metre is about 3 3⁄8 inches longer than a yard, Metre is the standard spelling of the metric unit for length in nearly all English-speaking nations except the United States and the Philippines, which use meter. Measuring devices are spelled -meter in all variants of English, the suffix -meter has the same Greek origin as the unit of length. This range of uses is found in Latin, French, English. Thus calls for measurement and moderation. In 1668 the English cleric and philosopher John Wilkins proposed in an essay a decimal-based unit of length, as a result of the French Revolution, the French Academy of Sciences charged a commission with determining a single scale for all measures. In 1668, Wilkins proposed using Christopher Wrens suggestion of defining the metre using a pendulum with a length which produced a half-period of one second, christiaan Huygens had observed that length to be 38 Rijnland inches or 39.26 English inches. This is the equivalent of what is now known to be 997 mm, no official action was taken regarding this suggestion. In the 18th century, there were two approaches to the definition of the unit of length. One favoured Wilkins approach, to define the metre in terms of the length of a pendulum which produced a half-period of one second. The other approach was to define the metre as one ten-millionth of the length of a quadrant along the Earths meridian, that is, the distance from the Equator to the North Pole. This means that the quadrant would have defined as exactly 10000000 metres at that time. To establish a universally accepted foundation for the definition of the metre, more measurements of this meridian were needed. This portion of the meridian, assumed to be the length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator
27.
Mole (unit)
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The mole is the unit of measurement in the International System of Units for amount of substance. This number is expressed by the Avogadro constant, which has a value of 6. 022140857×1023 mol−1, the mole is one of the base units of the SI, and has the unit symbol mol. The mole is used in chemistry as a convenient way to express amounts of reactants and products of chemical reactions. For example, the chemical equation 2 H2 + O2 →2 H2O implies that 2 moles of dihydrogen and 1 mole of dioxygen react to form 2 moles of water. The mole may also be used to express the number of atoms, ions, the concentration of a solution is commonly expressed by its molarity, defined as the number of moles of the dissolved substance per litre of solution. For example, the relative molecular mass of natural water is about 18.015, therefore. The term gram-molecule was formerly used for essentially the same concept, the term gram-atom has been used for a related but distinct concept, namely a quantity of a substance that contains Avogadros number of atoms, whether isolated or combined in molecules. Thus, for example,1 mole of MgBr2 is 1 gram-molecule of MgBr2 but 3 gram-atoms of MgBr2, in honor of the unit, some chemists celebrate October 23, which is a reference to the 1023 scale of the Avogadro constant, as Mole Day. Some also do the same for February 6 and June 2, thus, by definition, one mole of pure 12C has a mass of exactly 12 g. It also follows from the definition that X moles of any substance will contain the number of molecules as X moles of any other substance. The mass per mole of a substance is called its molar mass, the number of elementary entities in a sample of a substance is technically called its amount. Therefore, the mole is a convenient unit for that physical quantity, one can determine the chemical amount of a known substance, in moles, by dividing the samples mass by the substances molar mass. Other methods include the use of the volume or the measurement of electric charge. The mass of one mole of a substance depends not only on its molecular formula, since the definition of the gram is not mathematically tied to that of the atomic mass unit, the number NA of molecules in a mole must be determined experimentally. The value adopted by CODATA in 2010 is NA =6. 02214129×1023 ±0. 00000027×1023, in 2011 the measurement was refined to 6. 02214078×1023 ±0. 00000018×1023. The number of moles of a sample is the sample mass divided by the mass of the material. The history of the mole is intertwined with that of mass, atomic mass unit, Avogadros number. The first table of atomic mass was published by John Dalton in 1805
28.
Second
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The second is the base unit of time in the International System of Units. It is qualitatively defined as the division of the hour by sixty. SI definition of second is the duration of 9192631770 periods of the corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. Seconds may be measured using a mechanical, electrical or an atomic clock, SI prefixes are combined with the word second to denote subdivisions of the second, e. g. the millisecond, the microsecond, and the nanosecond. Though SI prefixes may also be used to form multiples of the such as kilosecond. The second is also the unit of time in other systems of measurement, the centimetre–gram–second, metre–kilogram–second, metre–tonne–second. Absolute zero implies no movement, and therefore zero external radiation effects, the second thus defined is consistent with the ephemeris second, which was based on astronomical measurements. The realization of the second is described briefly in a special publication from the National Institute of Standards and Technology. 1 international second is equal to, 1⁄60 minute 1⁄3,600 hour 1⁄86,400 day 1⁄31,557,600 Julian year 1⁄, more generally, = 1⁄, the Hellenistic astronomers Hipparchus and Ptolemy subdivided the day into sixty parts. They also used an hour, simple fractions of an hour. No sexagesimal unit of the day was used as an independent unit of time. The modern second is subdivided using decimals - although the third remains in some languages. The earliest clocks to display seconds appeared during the last half of the 16th century, the second became accurately measurable with the development of mechanical clocks keeping mean time, as opposed to the apparent time displayed by sundials. The earliest spring-driven timepiece with a hand which marked seconds is an unsigned clock depicting Orpheus in the Fremersdorf collection. During the 3rd quarter of the 16th century, Taqi al-Din built a clock with marks every 1/5 minute, in 1579, Jost Bürgi built a clock for William of Hesse that marked seconds. In 1581, Tycho Brahe redesigned clocks that displayed minutes at his observatory so they also displayed seconds, however, they were not yet accurate enough for seconds. In 1587, Tycho complained that his four clocks disagreed by plus or minus four seconds, in 1670, London clockmaker William Clement added this seconds pendulum to the original pendulum clock of Christiaan Huygens. From 1670 to 1680, Clement made many improvements to his clock and this clock used an anchor escapement mechanism with a seconds pendulum to display seconds in a small subdial
29.
Becquerel
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The becquerel is the SI derived unit of radioactivity. One becquerel is defined as the activity of a quantity of material in which one nucleus decays per second. The becquerel is therefore equivalent to a second, s−1. The becquerel is named after Henri Becquerel, who shared a Nobel Prize in Physics with Pierre, as with every International System of Units unit named for a person, the first letter of its symbol is uppercase. 1 Bq =1 s−1 A special name was introduced for the second to represent radioactivity to avoid potentially dangerous mistakes with prefixes. For example,1 µs−1 could be taken to mean 106 disintegrations per second, other names considered were hertz, a special name already in use for the reciprocal second, and fourier. The hertz is now used for periodic phenomena. Whereas 1 Hz is 1 cycle per second,1 Bq is 1 aperiodic radioactivity event per second, the gray and the becquerel were introduced in 1975. Between 1953 and 1975, absorbed dose was often measured in rads, decay activity was measured in curies before 1946 and often in rutherfords between 1946 and 1975. Like any SI unit, Bq can be prefixed, commonly used multiples are kBq, MBq, GBq, TBq, for practical applications,1 Bq is a small unit, therefore, the prefixes are common. For example, the roughly 0.0169 g of potassium-40 present in a human body produces approximately 4,400 disintegrations per second or 4.4 kBq of activity. The global inventory of carbon-14 is estimated to be 8. 5×1018 Bq, the nuclear explosion in Hiroshima is estimated to have produced 8×1024 Bq. The becquerel succeeded the curie, an older, non-SI unit of radioactivity based on the activity of 1 gram of radium-226, the curie is defined as 3. 7·1010 s−1, or 37 GBq. The following table shows radiation quantities in SI and non-SI units
30.
Coulomb
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The coulomb is the International System of Units unit of electric charge. 242×1018 protons, and −1 C is equivalent to the charge of approximately 6. 242×1018 electrons. This SI unit is named after Charles-Augustin de Coulomb, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, the SI system defines the coulomb in terms of the ampere and second,1 C =1 A ×1 s. The second is defined in terms of a frequency emitted by caesium atoms. The ampere is defined using Ampères force law, the definition relies in part on the mass of the prototype kilogram. In practice, the balance is used to measure amperes with the highest possible accuracy. One coulomb is the magnitude of charge in 6. 24150934×10^18 protons or electrons. The inverse of this gives the elementary charge of 1. 6021766208×10−19 C. The magnitude of the charge of one mole of elementary charges is known as a faraday unit of charge. In terms of Avogadros number, one coulomb is equal to approximately 1.036 × NA×10−5 elementary charges, one ampere-hour =3600 C,1 mA⋅h =3.6 C. One statcoulomb, the obsolete CGS electrostatic unit of charge, is approximately 3. 3356×10−10 C or about one-third of a nanocoulomb, the elementary charge, the charge of a proton, is approximately 1. 6021766208×10−19 C. In SI, the charge in coulombs is an approximate value. However, in other systems, the elementary charge has an exact value by definition. Specifically, e90 = / C exactly, SI itself may someday change its definitions in a similar way. For example, one possible proposed redefinition is the ampere. is such that the value of the charge e is exactly 1. 602176487×10−19 coulombs. This proposal is not yet accepted as part of the SI, the charges in static electricity from rubbing materials together are typically a few microcoulombs. The amount of charge that travels through a lightning bolt is typically around 15 C, the amount of charge that travels through a typical alkaline AA battery from being fully charged to discharged is about 5 kC =5000 C ≈1400 mA⋅h. The hydraulic analogy uses everyday terms to illustrate movement of charge, the analogy equates charge to a volume of water, and voltage to pressure
31.
Celsius
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Celsius, also known as centigrade, is a metric scale and unit of measurement for temperature. As an SI derived unit, it is used by most countries in the world and it is named after the Swedish astronomer Anders Celsius, who developed a similar temperature scale. The degree Celsius can refer to a temperature on the Celsius scale as well as a unit to indicate a temperature interval. Before being renamed to honour Anders Celsius in 1948, the unit was called centigrade, from the Latin centum, which means 100, and gradus, which means steps. The scale is based on 0° for the point of water. This scale is widely taught in schools today, by international agreement the unit degree Celsius and the Celsius scale are currently defined by two different temperatures, absolute zero, and the triple point of VSMOW. This definition also precisely relates the Celsius scale to the Kelvin scale, absolute zero, the lowest temperature possible, is defined as being precisely 0 K and −273.15 °C. The temperature of the point of water is defined as precisely 273.16 K at 611.657 pascals pressure. This definition fixes the magnitude of both the degree Celsius and the kelvin as precisely 1 part in 273.16 of the difference between absolute zero and the point of water. Thus, it sets the magnitude of one degree Celsius and that of one kelvin as exactly the same, additionally, it establishes the difference between the two scales null points as being precisely 273.15 degrees. In his paper Observations of two persistent degrees on a thermometer, he recounted his experiments showing that the point of ice is essentially unaffected by pressure. He also determined with precision how the boiling point of water varied as a function of atmospheric pressure. He proposed that the point of his temperature scale, being the boiling point. This pressure is known as one standard atmosphere, the BIPMs 10th General Conference on Weights and Measures later defined one standard atmosphere to equal precisely 1013250dynes per square centimetre. On 19 May 1743 he published the design of a mercury thermometer, in 1744, coincident with the death of Anders Celsius, the Swedish botanist Carolus Linnaeus reversed Celsiuss scale. In it, Linnaeus recounted the temperatures inside the orangery at the University of Uppsala Botanical Garden, since the 19th century, the scientific and thermometry communities worldwide referred to this scale as the centigrade scale. Temperatures on the scale were often reported simply as degrees or. More properly, what was defined as centigrade then would now be hectograde.2 gradians, for scientific use, Celsius is the term usually used, with centigrade otherwise continuing to be in common but decreasing use, especially in informal contexts in English-speaking countries
32.
Farad
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The farad is the SI derived unit of electrical capacitance, the ability of a body to store an electrical charge. It is named after the English physicist Michael Faraday, one farad is defined as the capacitance across which, when charged with one coulomb, there is a potential difference of one volt. Equally, one farad can be described as the capacitance which stores a one-coulomb charge across a potential difference of one volt, the relationship between capacitance, charge and potential difference is linear. For example, if the difference across a capacitor is halved. For most applications, the farad is a large unit of capacitance. Most electrical and electronic applications are covered by the following SI prefixes,1 mF =1000 μF =1000000 nF1 μF =0.000001 F =1000 nF =1000000 pF1 nF =0. In 1881 at the International Congress of Electricians in Paris, the name farad was officially used for the unit of electrical capacitance, a capacitor consists of two conducting surfaces, frequently referred to as plates, separated by an insulating layer usually referred to as a dielectric. The original capacitor was the Leyden jar developed in the 18th century and it is the accumulation of electric charge on the plates that results in capacitance. Values of capacitors are specified in farads, microfarads, nanofarads and picofarads. The millifarad is rarely used in practice, while the nanofarad is uncommon in North America, the size of commercially available capacitors ranges from around 0.1 pF to 5000F supercapacitors. Capacitance values of 1 pF or lower can be achieved by twisting two short lengths of insulated wire together, the capacitance of the Earths ionosphere with respect to the ground is calculated to be about 1 F. The picofarad is sometimes pronounced as puff or pic, as in a ten-puff capacitor. Similarly, mic is sometimes used informally to signify microfarads, if the Greek letter μ is not available, the notation uF is often used as a substitute for μF in electronics literature. A micro-microfarad, an obsolete unit sometimes found in texts, is the equivalent of a picofarad. In texts prior to 1960, and on capacitor packages even more recently. Similarly, mmf or MMFD represented picofarads, the reciprocal of capacitance is called electrical elastance, the unit of which is the daraf. The abfarad is an obsolete CGS unit of equal to 109 farads. The statfarad is a rarely used CGS unit equivalent to the capacitance of a capacitor with a charge of 1 statcoulomb across a potential difference of 1 statvolt and it is 1/ farad, approximately 1.1126 picofarads
33.
Gray (unit)
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The gray is a derived unit of ionizing radiation dose in the International System of Units. It is defined as the absorption of one joule of energy per kilogram of matter. It is used as a measure of absorbed dose, specific energy and it is a physical quantity, and does not take into account any biological context. Unlike the pre-1971 non-SI roentgen unit of exposure, the gray when used for absorbed dose is defined independently of any target material. However, when measuring kerma the reference target material must be defined explicitly, usually as dry air at standard temperature and pressure. The corresponding cgs unit, the rad, remains common in the United States, though strongly discouraged in the guide for U. S. National Institute of Standards. The gray was named after British physicist Louis Harold Gray, a pioneer in the measurement of X-ray and radium radiation and it was adopted as part of the International System of Units in 1975. One gray is the absorption of one joule of energy, in the form of ionizing radiation, measuring and controlling the value of absorbed dose is vital to ensuring correct operation of these processes. Kerma is a measure of the energy of ionisation due to irradiation. Importantly, kerma dose is different from absorbed dose, depending on the energies involved. The measurement of absorbed dose is a problem, and so many different dosimeters are available for these measurements. These dosimeters cover measurements that can be done in 1-D, 2-D and 3-D, in radiation therapy, the amount of radiation applied varies depending on the type and stage of cancer being treated. For curative cases, the dose for a solid epithelial tumor ranges from 60 to 80 Gy. Preventive doses are typically around 45–60 Gy in 1. 8–2 Gy fractions, the absorbed dose also plays an important role in radiation protection, as it is the starting point for calculating the effect of low levels of radiation. In radiation protection dose assessment, the health risk is defined as the probability of cancer induction. This probability is related to the equivalent dose in sieverts, which has the dimensions as the gray. It is related to the gray by weighting factors described in the articles on equivalent dose, to avoid any risk of confusion between the absorbed dose and the equivalent dose, the absorbed dose is stated in grays and the equivalent dose is stated in sieverts. The accompanying diagrams show how absorbed dose is first obtained by computational techniques, radiation poisoning - The gray is conventionally used to express the severity of what are known as tissue effects from doses received in acute exposure to high levels of ionizing radiation
34.
Henry (unit)
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The henry is the SI derived unit of electrical inductance. The unit is named after Joseph Henry, the American scientist who discovered electromagnetic induction independently of, the magnetic permeability of vacuum is 4π × 10−7 H⋅m−1. The henry is a unit based on four of the seven base units of the International System of Units, kilogram, meter, second. The United States National Institute of Standards and Technology recommends English-speaking users of SI to write the plural as henries
35.
Hertz
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The hertz is the unit of frequency in the International System of Units and is defined as one cycle per second. It is named for Heinrich Rudolf Hertz, the first person to provide proof of the existence of electromagnetic waves. Hertz are commonly expressed in SI multiples kilohertz, megahertz, gigahertz, kilo means thousand, mega meaning million, giga meaning billion and tera for trillion. Some of the units most common uses are in the description of waves and musical tones, particularly those used in radio-. It is also used to describe the speeds at which computers, the hertz is equivalent to cycles per second, i. e. 1/second or s −1. In English, hertz is also used as the plural form, as an SI unit, Hz can be prefixed, commonly used multiples are kHz, MHz, GHz and THz. One hertz simply means one cycle per second,100 Hz means one hundred cycles per second, and so on. The unit may be applied to any periodic event—for example, a clock might be said to tick at 1 Hz, the rate of aperiodic or stochastic events occur is expressed in reciprocal second or inverse second in general or, the specific case of radioactive decay, becquerels. Whereas 1 Hz is 1 cycle per second,1 Bq is 1 aperiodic radionuclide event per second, the conversion between a frequency f measured in hertz and an angular velocity ω measured in radians per second is ω =2 π f and f = ω2 π. This SI unit is named after Heinrich Hertz, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, the hertz is named after the German physicist Heinrich Hertz, who made important scientific contributions to the study of electromagnetism. The name was established by the International Electrotechnical Commission in 1930, the term cycles per second was largely replaced by hertz by the 1970s. One hobby magazine, Electronics Illustrated, declared their intention to stick with the traditional kc. Mc. etc. units, sound is a traveling longitudinal wave which is an oscillation of pressure. Humans perceive frequency of waves as pitch. Each musical note corresponds to a frequency which can be measured in hertz. An infants ear is able to perceive frequencies ranging from 20 Hz to 20,000 Hz, the range of ultrasound, infrasound and other physical vibrations such as molecular and atomic vibrations extends from a few femtoHz into the terahertz range and beyond. Electromagnetic radiation is described by its frequency—the number of oscillations of the perpendicular electric and magnetic fields per second—expressed in hertz. Radio frequency radiation is measured in kilohertz, megahertz, or gigahertz
36.
Joule
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The joule, symbol J, is a derived unit of energy in the International System of Units. It is equal to the transferred to an object when a force of one newton acts on that object in the direction of its motion through a distance of one metre. It is also the energy dissipated as heat when a current of one ampere passes through a resistance of one ohm for one second. It is named after the English physicist James Prescott Joule, one joule can also be defined as, The work required to move an electric charge of one coulomb through an electrical potential difference of one volt, or one coulomb volt. This relationship can be used to define the volt, the work required to produce one watt of power for one second, or one watt second. This relationship can be used to define the watt and this SI unit is named after James Prescott Joule. As with every International System of Units unit named for a person, note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, section 5.2. The CGPM has given the unit of energy the name Joule, the use of newton metres for torque and joules for energy is helpful to avoid misunderstandings and miscommunications. The distinction may be also in the fact that energy is a scalar – the dot product of a vector force. By contrast, torque is a vector – the cross product of a distance vector, torque and energy are related to one another by the equation E = τ θ, where E is energy, τ is torque, and θ is the angle swept. Since radians are dimensionless, it follows that torque and energy have the same dimensions, one joule in everyday life represents approximately, The energy required to lift a medium-size tomato 1 m vertically from the surface of the Earth. The energy released when that same tomato falls back down to the ground, the energy required to accelerate a 1 kg mass at 1 m·s−2 through a 1 m distance in space. The heat required to raise the temperature of 1 g of water by 0.24 °C, the typical energy released as heat by a person at rest every 1/60 s. The kinetic energy of a 50 kg human moving very slowly, the kinetic energy of a 56 g tennis ball moving at 6 m/s. The kinetic energy of an object with mass 1 kg moving at √2 ≈1.4 m/s, the amount of electricity required to light a 1 W LED for 1 s. Since the joule is also a watt-second and the unit for electricity sales to homes is the kW·h. For additional examples, see, Orders of magnitude The zeptojoule is equal to one sextillionth of one joule,160 zeptojoules is equivalent to one electronvolt. The nanojoule is equal to one billionth of one joule, one nanojoule is about 1/160 of the kinetic energy of a flying mosquito
37.
Katal
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The katal is the SI unit of catalytic activity. It is a derived SI unit for quantifying the catalytic activity of enzymes and its use is recommended by the General Conference on Weights and Measures and other international organizations. It replaces the non-SI enzyme unit, enzyme units are, however, still more commonly used than the katal in practice at present, especially in biochemistry. The katal is not used to express the rate of a reaction, rather, it is used to express catalytic activity which is a property of the catalyst. The katal is invariant of the measurement procedure, but the quantity value is not. Therefore, in order to define the quantity of a catalyst, one katal of trypsin, for example, is that amount of trypsin which breaks a mole of peptide bonds per second under specified conditions. Kat = mol s The name katal has been used for decades, the name comes from the Ancient Greek κατάλυσις, meaning dissolution, which is the same origin as the word catalysis itself comes from. Unit katal for catalytic activity Pure Appl, the Tortuous Road to the Adoption of katal for the Expression of Catalytic Activity by the General Conference on Weights and Measures
38.
Lumen (unit)
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The lumen is the SI derived unit of luminous flux, a measure of the total quantity of visible light emitted by a source. Lumens are related to lux in that one lux is one lumen per square meter, the lumen is defined in relation to the candela as 1 lm =1 cd ⋅ sr. A full sphere has an angle of 4π steradians, so a light source that uniformly radiates one candela in all directions has a total luminous flux of 1 cd × 4π sr = 4π cd⋅sr ≈12.57 lumens. If a light source emits one candela of luminous intensity uniformly across a solid angle of one steradian, alternatively, an isotropic one-candela light-source emits a total luminous flux of exactly 4π lumens. If the source were partly covered by an ideal absorbing hemisphere, the luminous intensity would still be one candela in those directions that are not obscured. The lumen can be thought of casually as a measure of the amount of visible light in some defined beam or angle. The number of candelas or lumens from a source also depends on its spectrum, the difference between the units lumen and lux is that the lux takes into account the area over which the luminous flux is spread. A flux of 1000 lumens, concentrated into an area of one square metre, the same 1000 lumens, spread out over ten square metres, produces a dimmer illuminance of only 100 lux. Mathematically,1 lx =1 lm/m2, a source radiating a power of one watt of light in the color for which the eye is most efficient has luminous flux of 683 lumens. So a lumen represents at least 1/683 watts of light power. Lamps used for lighting are commonly labelled with their output in lumens. A23 W spiral compact fluorescent lamp emits about 1, 400–1,600 lm, many compact fluorescent lamps and other alternative light sources are labelled as being equivalent to an incandescent bulb with a specific wattage. Below is a table that shows typical luminous flux for common incandescent bulbs, on September 1,2010, European Union legislation came into force mandating that lighting equipment must be labelled primarily in terms of luminous flux, instead of electric power. This change is a result of the EUs Eco-design Directive for Energy-using Products, for example, according to the European Union standard, an energy-efficient bulb that claims to be the equivalent of a 60 W tungsten bulb must have a minimum light output of 700–750 lm. The light output of projectors is typically measured in lumens, a standardized procedure for testing projectors has been established by the American National Standards Institute, which involves averaging together several measurements taken at different positions. For marketing purposes, the flux of projectors that have been tested according to this procedure may be quoted in ANSI lumens. ANSI lumen measurements are in more accurate than the other measurement techniques used in the projector industry. This allows projectors to be easily compared on the basis of their brightness specifications
39.
Lux
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The lux is the SI unit of illuminance and luminous emittance, measuring luminous flux per unit area. It is equal to one lumen per square metre, in photometry, this is used as a measure of the intensity, as perceived by the human eye, of light that hits or passes through a surface. In English, lux is used as both the singular and plural form, illuminance is a measure of how much luminous flux is spread over a given area. One can think of flux as a measure of the total amount of visible light present. A given amount of light will illuminate a surface more dimly if it is spread over a larger area, however, the same 1000 lumens, spread out over ten square metres, produces a dimmer illuminance of only 100 lux. Achieving an illuminance of 500 lux might be possible in a kitchen with a single fluorescent light fixture with an output of 12000 lumens. To light a factory floor with dozens of times the area of the kitchen would require dozens of such fixtures, thus, lighting a larger area to the same level of lux requires a greater number of lumens. As with other SI units, SI prefixes can be used, for instance, a star of apparent magnitude 0 provides 2.08 microlux at the earths surface. A barely perceptible magnitude 6 star provides 8 nanolux, the unobscured sun provides an illumination of up to 100 kilolux on the Earths surface, the exact value depending on time of year and atmospheric conditions. This direct normal illuminance is related to the solar illuminance constant Esc, the illumination provided on a surface by a point source equals the number of lux just described times the cosine of the angle between a ray coming from the source and a normal to the surface. The number of lux falling on the surface equals this cosine times a number that characterizes the source from the point of view in question and it differs from the luminance, which does depend on the angular distribution of the emission. A perfectly white surface with one lux falling on it will emit one lux, like all photometric units, the lux has a corresponding radiometric unit. The weighting factor is known as the luminosity function, the lux is one lumen per square metre, and the corresponding radiometric unit, which measures irradiance, is the watt per square metre. The peak of the luminosity function is at 555 nm, the eyes image-forming visual system is sensitive to light of this wavelength than any other. Other wavelengths of visible light produce fewer lux per watt-per-meter-squared, the luminosity function falls to zero for wavelengths outside the visible spectrum. For a light source with mixed wavelengths, the number of lumens per watt can be calculated by means of the luminosity function and this means that white light sources produce far fewer lumens per watt than the theoretical maximum of 683.002 lm/W. The ratio between the number of lumens per watt and the theoretical maximum is expressed as a percentage known as the luminous efficiency. For example, an incandescent light bulb has a luminous efficiency of only about 2%
40.
Newton (unit)
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The newton is the International System of Units derived unit of force. It is named after Isaac Newton in recognition of his work on classical mechanics, see below for the conversion factors. One newton is the force needed to one kilogram of mass at the rate of one metre per second squared in direction of the applied force. In 1948, the 9th CGPM resolution 7 adopted the name newton for this force, the MKS system then became the blueprint for todays SI system of units. The newton thus became the unit of force in le Système International dUnités. This SI unit is named after Isaac Newton, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, section 5.2. Newtons second law of motion states that F = ma, where F is the applied, m is the mass of the object receiving the force. The newton is therefore, where the symbols are used for the units, N for newton, kg for kilogram, m for metre. In dimensional analysis, F = M L T2 where F is force, M is mass, L is length, at average gravity on earth, a kilogram mass exerts a force of about 9.8 newtons. An average-sized apple exerts about one newton of force, which we measure as the apples weight, for example, the tractive effort of a Class Y steam train and the thrust of an F100 fighter jet engine are both around 130 kN. One kilonewton,1 kN, is 102.0 kgf,1 kN =102 kg ×9.81 m/s2 So for example, a platform rated at 321 kilonewtons will safely support a 32,100 kilograms load. Specifications in kilonewtons are common in safety specifications for, the values of fasteners, Earth anchors. Working loads in tension and in shear, thrust of rocket engines and launch vehicles clamping forces of the various moulds in injection moulding machines used to manufacture plastic parts
41.
Ohm
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The ohm is the SI derived unit of electrical resistance, named after German physicist Georg Simon Ohm. The definition of the ohm was revised several times, today the definition of the ohm is expressed from the quantum Hall effect. In many cases the resistance of a conductor in ohms is approximately constant within a range of voltages, temperatures. In alternating current circuits, electrical impedance is also measured in ohms, the siemens is the SI derived unit of electric conductance and admittance, also known as the mho, it is the reciprocal of resistance in ohms. The power dissipated by a resistor may be calculated from its resistance, non-linear resistors have a value that may vary depending on the applied voltage. The rapid rise of electrotechnology in the last half of the 19th century created a demand for a rational, coherent, consistent, telegraphers and other early users of electricity in the 19th century needed a practical standard unit of measurement for resistance. Two different methods of establishing a system of units can be chosen. Various artifacts, such as a length of wire or a standard cell, could be specified as producing defined quantities for resistance, voltage. This latter method ensures coherence with the units of energy, defining a unit for resistance that is coherent with units of energy and time in effect also requires defining units for potential and current. Some early definitions of a unit of resistance, for example, the absolute-units system related magnetic and electrostatic quantities to metric base units of mass, time, and length. These units had the advantage of simplifying the equations used in the solution of electromagnetic problems. However, the CGS units turned out to have impractical sizes for practical measurements, various artifact standards were proposed as the definition of the unit of resistance. In 1860 Werner Siemens published a suggestion for a reproducible resistance standard in Poggendorffs Annalen der Physik und Chemie and he proposed a column of pure mercury, of one square millimetre cross section, one metre long, Siemens mercury unit. However, this unit was not coherent with other units, one proposal was to devise a unit based on a mercury column that would be coherent – in effect, adjusting the length to make the resistance one ohm. Not all users of units had the resources to carry out experiments to the required precision. The BAAS in 1861 appointed a committee including Maxwell and Thomson to report upon Standards of Electrical Resistance, in the third report of the committee,1864, the resistance unit is referred to as B. A. unit, or Ohmad. By 1867 the unit is referred to as simply Ohm, the B. A. ohm was intended to be 109 CGS units but owing to an error in calculations the definition was 1. 3% too small. The error was significant for preparation of working standards, on September 21,1881 the Congrès internationale délectriciens defined a practical unit of Ohm for the resistance, based on CGS units, using a mercury column at zero deg
42.
Pascal (unit)
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The pascal is the SI derived unit of pressure used to quantify internal pressure, stress, Youngs modulus and ultimate tensile strength. It is defined as one newton per square meter and it is named after the French polymath Blaise Pascal. Common multiple units of the pascal are the hectopascal which is equal to one millibar, the unit of measurement called standard atmosphere is defined as 101,325 Pa and approximates to the average pressure at sea-level at the latitude 45° N. Meteorological reports typically state atmospheric pressure in hectopascals, the unit is named after Blaise Pascal, noted for his contributions to hydrodynamics and hydrostatics, and experiments with a barometer. The name pascal was adopted for the SI unit newton per square metre by the 14th General Conference on Weights, one pascal is the pressure exerted by a force of magnitude one newton perpendicularly upon an area of one square metre. The unit of measurement called atmosphere or standard atmosphere is 101325 Pa and this value is often used as a reference pressure and specified as such in some national and international standards, such as ISO2787, ISO2533 and ISO5024. In contrast, IUPAC recommends the use of 100 kPa as a standard pressure when reporting the properties of substances, geophysicists use the gigapascal in measuring or calculating tectonic stresses and pressures within the Earth. Medical elastography measures tissue stiffness non-invasively with ultrasound or magnetic resonance imaging, in materials science and engineering, the pascal measures the stiffness, tensile strength and compressive strength of materials. In engineering use, because the pascal represents a small quantity. The pascal is also equivalent to the SI unit of energy density and this applies not only to the thermodynamics of pressurised gases, but also to the energy density of electric, magnetic, and gravitational fields. In measurements of sound pressure, or loudness of sound, one pascal is equal to 94 decibels SPL, the quietest sound a human can hear, known as the threshold of hearing, is 0 dB SPL, or 20 µPa. The airtightness of buildings is measured at 50 Pa, the units of atmospheric pressure commonly used in meteorology were formerly the bar, which was close to the average air pressure on Earth, and the millibar. Since the introduction of SI units, meteorologists generally measure pressures in hectopascals unit, exceptions include Canada and Portugal, which use kilopascals. In many other fields of science, the SI is preferred, many countries also use the millibar or hectopascal to give aviation altimeter settings. In practically all fields, the kilopascal is used instead. Centimetre of water Metric prefix Orders of magnitude Pascals law
43.
Siemens (unit)
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The siemens is the unit of electric conductance, electric susceptance and electric admittance in the International System of Units. The 14th General Conference on Weights and Measures approved the addition of the siemens as a unit in 1971. The unit is named after Ernst Werner von Siemens, as with every SI unit whose name is derived from the proper name of a person, the first letter of its symbol is upper case, the lower-case s is the symbol for the second. When an SI unit is spelled out in English, it should begin with a lower-case letter. In English, the same form siemens is used both for the singular and plural, the unit siemens for the conductance G is defined by S = Ω −1 = A V where Ω is the ohm, A is the ampere, and V is the volt. For a device with a conductance of one siemens, the current through the device will increase by one ampere for every increase of one volt of electric potential difference across the device. The conductance of a resistor with a resistance of five ohms, for example, is −1, mho /moʊ/ is an alternative name of the same unit, the reciprocal of one ohm. Mho is derived from spelling ohm backwards and is written with an upside-down capital Greek letter Omega, ℧, according to Maver the term mho was suggested by Sir William Thomson. The mho was officially renamed to the siemens, replacing the old meaning of the siemens unit, the term siemens, as it is an SI term, is used universally in science and often in electrical applications, while mho is still used primarily in electronic applications. Likewise, it is difficult to distinguish the symbol S from the lower-case s where second is meant, brochure The International System of Units issued by the BIPM Different units named after Siemens