Even in ancient times, the advantages of using systems of interrelated measures and units in comparison with separate, disparate measures and units of measurement were recognized.
The first systems that could reasonably be called systems of units were Gaussian (milligram, millimeter, second) and a number of CGS systems (centimeter, gram, second). Further development of such systems led to the development and adoption in 1960 by the XI General Conference on Weights and Measures of the International System of Units (Le Systeme international d'unites - abbreviated - SI).
The initial for SI, of course, is the metric system proposed in 1791. The next stage is the signing of the diplomatic document of the metric conference of 1875 by the seventeen leading industrial powers of the world.
In 1881, the CGS system appeared (development of the Gaussian system) and later, due to the need to use it to measure not only mechanical, but also electromagnetic quantities, its varieties (the most famous are CGSE and CGSM). Next milestone- the adoption in 1950 of the MKSA system - the Georgi system, in which the fourth basic unit appeared - the ampere. MKSA entered the SI as its component used for electrical and magnetic quantities. The need to include thermal and light quantities in the system led to the inclusion of two more basic units in the SI - kelvin and candela. In 1971, the mole was included in the basic units. Before proceeding to a detailed consideration of SI, it is necessary to dwell on the general principles for constructing systems of units of measurement.
Principles for constructing systems of units of measurement
The method of constructing systems of units, in its original form, was developed by F. Gauss. According to this method, the construction of systems of units of measurement begins with the choice of the minimum number of basic units through which all practically used units of measurement are expressed - called derivatives. It is interesting to note that there are no theoretically substantiated algorithms that make it possible to unambiguously determine the set (set) of the basic units necessary to build a system. The only criterion in choosing the basic units can only be the effectiveness and expediency of using this system. Different systems are based on different numbers of base units. As already mentioned, the metric system of 1791 was based on one basic unit - the meter, then on two - the meter and the kilogram. The Gaussian system and the CGS system - on three. GHS options - GHSέ0; CGSµ0; SGSF; SGSB - on four. The ISS system is again on three, its variants are MKSK, MKSA, MKSµ0; MKSKD and MKSLM - on four. SI includes 7 basic units. This is the maximum number for all known systems of units.
Initially, it was assumed that the basic units should be reproduced completely independently of each other. As will be shown below, in fact, significant deviations from this principle appeared in the systems of units.
The next stage in the development of the system is the assignment of letter symbols to the basic units of their dimensions. This is followed by the stage of including in the system a certain set of derived units, expressed in terms of the main ones and the dimensions assigned to them by substituting the symbols of the main units into the physical equations that define these units through the main ones.
Dimensionality of the measured quantities and units of measurements
Dimension is an expression in the form of a power monomial, composed of the products of symbols of basic units in various degrees and reflecting the connection of this derived unit with the main ones.
There are two interpretations of the concept of "dimension". One by one - dimensions are assigned to values, by the other - to units. Obviously, units, being particular realizations of quantities, have the same dimensions with them, therefore there is no fundamental contradiction between these points of view. In all physical, metrological literature and in this book, dimension is understood, first of all, only a generalized expression of the dependence of a unit of a given quantity on basic units.
Thus, the dimensions assigned to the basic and derived units are at the same time the dimensions of the corresponding quantities. It is necessary to warn against the thoughtless, automatic, use of the terms "basic and derived quantities". All quantities denote existing properties, among which there are neither basic nor derivative ones. All quantities are equal in this sense. Another thing is the units within the framework of the system that unites them. Forming a system of units, we have the right to subdivide them into basic and derivative.
It follows from the theory of measurement scales that only units of metric scales of differences and ratios have dimensions. Units of absolute scales are dimensionless in principle, even when included in any system of units. Name and order scales do not have units of measurement, therefore, the concept of “dimension” is not applicable to numbers, points and other signs characterizing these scales.
Recall that most of the classics of physics and metrology believed and still believe that "the dimension of a quantity is not a property associated with its essence, but is a kind of convention associated with the choice of a system of units" (M. Planck, P. Bridgman and others .). This opinion is confirmed by the dependence of the dimension of units on the chosen system, the coincidence of the dimensions of quantities that have a different physical nature, the dimensions of a number of quantities that are difficult to physically interpret (for example, electric capacitance), and the fact that quantities that are dimensional in one system may be dimensionless in another.
Here is what G. Hartley wrote about this in his monograph “Analysis of dimensions”: “There is no such thing as the absolute dimension of a physical quantity ... Dimensions ... are relative by definition. The formula for the dimension of a physical quantity is based on the definition of this quantity using the basic units of measurement, the choice of which (in certain limits) is arbitrary. It can be seen from the foregoing that the dimension symbols are specific logical operators, functionally defined only within the framework of the corresponding systems of units. Dimension symbols are not ordinary quantities, and the abstract algebra of operations with them differs from ordinary algebra. The use of these operators outside of systems of units is meaningless.
In practice, we are not interested in dimensions, as such, but in expressions that relate units of measurement to the basic units of the system and to each other. In structure, they are similar, but not identical: dimension symbols are abstracted from the specific dimensions of units of measurement. It is no coincidence that in the tables of the international document "Le Systeme international d´unites" there is no column "dimension", but only expressions of the relationship between different units of measurement are given.
The dimension of a quantity is simultaneously the dimension of its unit. Example: the dimension of the area (values) is L², the dimension of the area unit is m², and also - L². The dimension of the basic unit of the system coincides with its symbol to a degree equal to 1. The degrees of symbols of the basic units included in the monomial can be integer, fractional, positive, negative, they are called indicators of the dimension of derived units. The set of dimensions of the basic and derived units of this system forms a dimensional system. Its base is the dimensions of the basic units. Over the dimensions, you can perform the formal operations of multiplication, division, exponentiation, root extraction. Adding and subtracting dimensions does not make sense. The dimension of units (values) depends on the accepted system of units. A unit in the dimension of which at least one of the basic units is raised to a non-zero power is called dimensional, otherwise it is called dimensionless. Recall that a unit of a particular quantity, which is dimensionless in one system, can be dimensional in another, and vice versa.
International system of units - SI
SI is a corent system built according to the decimal principle: multiples and submultiples are formed by multiplying the original units by factors equal to ten to a positive or negative integer degree, and in equations linking the units of the system, the numerical coefficients are equal to one.
The adoption of SI made it possible to unify the units of measurement - for each quantity one and only one unit was adopted. SI covers most areas of the natural sciences and engineering. Its units, as a rule, have convenient practical application dimensions. The units of mass and force (weight) are clearly delineated. For all types of energy, one unit is set - the joule (thus, there is no need for various conversion factors). Simplified writing equations and formulas in various areas science and technology. But SI cannot be considered all-encompassing. It applies only to metric scales of scalar quantities. It must also be realized that, in fact, in the SI, dimensionless and counting units of absolute scales are used to form many derived units. We especially note the usual and unnoticed conditionality of the distribution of SI to vector quantities, such as speed, acceleration, angular velocity of rotation, force, moment of force, electric and magnetic field strengths, etc. In fact, the corresponding units of measurement (m/s, m/s², rad /s, N, Nm, V/m, A/m) can only correspond to the modules of these vectors - scalar quantities. For a complete description of vectors, including their direction, it is necessary to use a coordinate system - three-dimensional combined scales. Although specifications for non-metric scales are generally based on SI units, these scales cannot in principle be covered by the SI.
In the standard GOST 8.417 - 2002 "GSI. Units of quantities" there is an indication that this standard does not establish units of quantities evaluated on conditional scales, units of the quantity of products (for example, units of the International Sugar Scale, hardness scales, scales of photosensitivity of photographic materials, etc., as well as counting units). In terms of the theory of measurement scales, this indication is inaccurate, the units of any scales, except for absolute ones, are conditional, i.e. accepted by agreement. Therefore, it is more correct to write that SI and the above standards do not apply to quantities and properties described by non-metric scales. Also, outside the SI, there are many widely used counting units, such as "pair", "bag", "package", etc.
When constructing systems of units of physical quantities, two stages are distinguished: stage 1 - the choice of basic units; Stage 2 - the formation of derived units.
The sequence of arrangement of derived units must satisfy the following conditions:
the first should be a value that is expressed only in terms of basic quantities;
each subsequent one must be a value that is expressed only through the main and such derivatives that precede it. For example, such a sequence of units: area, volume, density.
The main principle in building a system of units is the convenience of using units in science, industry, and trade. At the same time, they are guided by a number of rules: the simplicity of the formation of derived units, the high accuracy of reproduction of basic and derived units, and the proximity of their sizes to the sizes of physical quantities most often encountered in practice. In addition, the number of basic units is always tried to be kept to a minimum.
Examples of systems of units of physical quantities
Gauss system. The millimeter, milligram, second are chosen as the basic units in it, and a system of magnetic quantities is built. The system is called absolute. In 1851, Weber extended it to the field of electrical quantities. Currently, it is only of historical interest, because. units are too small. However, the principle discovered by Gauss underlies the construction modern systems units is a division into basic and derived units.
The CGS system was adopted in 1881 with the basic units centimeter, gram, second. This system is suitable for physical research. Based on it, seven systems of electrical and magnetic quantities arose. Currently, the GHS system is used in theoretical sections physics and astronomy.
The natural system of units is based on physical constants. The first such system was proposed in 1906 by Planck. As the basic units were chosen: the speed of light in vacuum, the gravitational constant, the Boltzmann and Planck constants. The advantage of these systems is when building physical theories they give physical laws a simpler form and some formulas are freed from numerical coefficients. However, units of physical quantities in them have a size that is inconvenient for practice. For example, the unit of length in this system is 4.03 10-35 m. In addition, such an accuracy in measuring the selected universal constants has not yet been achieved so that all derived units can be established.
Relative and logarithmic quantities and units
Relative and logarithmic quantities are widely used in science and technology, because they characterize the composition and properties of materials, the ratio of energy quantities, for example, relative density, relative permittivity, amplification and attenuation of power.
A relative quantity is a dimensionless ratio of a physical quantity to the physical quantity of the same name, taken as the initial one. For example, atomic and molecular weights chemical elements in relation to 1/12 of the mass of the carbon-12 atom. Relative values can be expressed in dimensionless units, in percent, ppm (the ratio is 10-3), in parts per million.
The logarithmic quantity is the logarithm of the dimensionless ratio of two physical quantities of the same name. They are used, for example, to express the level sound pressure, amplification, attenuation, etc.
The unit of the logarithmic value is bel (B): 1 B \u003d lg (P2 / P1) at P2 \u003d 10P1, where P2 and P1 are the same names for power, energy, etc. For the ratio of two quantities of the same name associated with force (voltage, pressure, etc.), bel is determined by the formula:
1B = 2 lg (F2/F1) at F2 = 100.5 F1.
The submultiple of the bela is the decibel, which is equal to 0.1 B.
International System of Units (SI)
The development of science and technology more and more insistently demanded the unification of units of measurement. A unified system of units was required, convenient for practical use and covering various areas of measurement. In addition, it had to be coherent. Since the metric system of measures has been widely used in Europe since the beginning of the 19th century, it was taken as the basis for the transition to a single international system of units.
In 1960, the XI General Conference on Weights and Measures approved the International System of Units of Physical Quantities (Russian designation SI, international SI) based on six basic units. The decision was made:
- - assign the name "International System of Units" to the system based on six basic units;
- - establish an international abbreviation for the name of the SI system;
- - enter a table of prefixes for the formation of multiples and submultiples;
- - form 27 derived units, indicating that other derived units may be added.
In 1971, the seventh basic unit of quantity of a substance (mol) was added to the SI.
When constructing the SI, we proceeded from the following basic principles:
- - the system is based on basic units that are independent of each other;
- - derived units are formed according to the simplest connection equations and only one SI unit is established for each type of quantity;
- - the system is coherent;
- - along with SI units, non-system units widely used in practice are allowed;
- - the system includes decimal multiples and submultiples.
Advantages of SI:
- - versatility, because it covers all areas of measurement;
- - unification of units for all types of measurements - the use of one unit for a given physical quantity, for example, for pressure, work, energy;
- - SI units in their size are convenient for practical use;
- - switching to it increases the level of measurement accuracy, because the basic units of this system can be reproduced more accurately than the units of other systems;
- - this is a single international system and its units are common.
In the USSR, the International System (SI) was put into effect by GOST 8.417-81. As further development The SI class of additional units was excluded from it, a new definition of the meter was introduced, and a number of other changes were introduced. Currently, the Russian Federation has an interstate standard GOST 8.417-2002, which establishes the units of physical quantities used in the country. The standard states that SI units, as well as decimal multiples and submultiples of these units, are subject to mandatory use.
SI derived units are formed according to the rules for the formation of coherent derived units (see above for an example). Examples of such units and derived units with special names and designations are given. 21 derived units were given names and designations by the names of scientists, for example, hertz, newton, pascal, becquerel.
In a separate section of the standard, units that are not included in the SI are given. These include:
- 1. Non-systemic units allowed for use along with the SI because of their practical importance. They are divided into areas of application. For example, in all areas the units used are ton, hour, minute, day, liter; in optics, diopter, in electron-volt physics, and so on.
- 2. Some relative and logarithmic quantities and their units. For example, percentage, ppm, white.
- 3. Off-system units temporarily allowed for use. For example, nautical mile, carat (0.2 g), knot, bar.
In a separate section, the rules for writing unit designations, using unit designations in column headings of tables, etc. are given.
The appendices to the standard give rules for the formation of coherent derived SI units, a table of correlations of some non-systemic units with SI units, and recommendations for choosing decimal multiples and submultiples.
Units whose names include the names of the main units. Examples: area unit - square meter, dimension L2, designation of the unit m2; the unit of the ionizing particle flux is a second to the minus one degree, the dimension is T-1, the designation of the unit is s-1.
Units with special names. Examples:
force, weight - newton, dimension LMT-2, unit designation H (international N); energy, work, amount of heat - joule, dimension L2MT-2, designation J (J).
Units whose names are formed using special names. Examples:
moment of force - name newton meter, dimension L2MT-2, designation Nm (Nm); specific energy - the name of the joule per kilogram, the dimension L2T-2, the designation J / kg (J / kg).
Decimal multiples and submultiples are formed using multipliers and prefixes, from 1024 (yotta) to 10-24 (yokto).
Attaching two or more prefixes in a row to the name is not allowed, for example, not a kilogram, but a ton, which is an off-system unit allowed along with the SI.
Due to the fact that the name of the basic unit of mass contains the prefix kilo, for the formation of submultiple and multiple units of mass, the submultiple unit of gram is used and the prefixes are attached to the word "gram" - milligram, microgram.
The choice of a multiple or fractional unit of the SI unit is dictated primarily by the convenience of its application, and the numerical values of the obtained quantities should be acceptable in practice. It is believed that the numerical values of quantities are most easily perceived in the range from 0.1 to 1000.
In some areas of activity, the same submultiple or multiple unit is always used, for example, in mechanical engineering drawings, dimensions are always expressed in millimeters.
To reduce the likelihood of errors in calculations, it is recommended to substitute decimal and multiple submultiples only in the final result, and in the process of calculations, all quantities should be expressed in SI units, replacing prefixes with powers of 10.
GOST 8.417-2002 contains the rules for writing the designation of units, the main of which are as follows.
Designations of units by letters or signs should be used, and two types of letter designations are established: international and Russian. International designations are written in relations with foreign countries (contracts, product deliveries and documentation). When used on the territory of the Russian Federation, Russian designations are used. At the same time, only international designations are used on plates, scales and shields of measuring instruments.
Unit names are lowercase unless they appear at the beginning of a sentence. The exception is Celsius.
In the designations of units, a dot is not put as an abbreviation sign; they are printed in a direct font. Exceptions are abbreviations of words that are included in the name of the unit, but are not the names of the units themselves. For example, mm Hg. Art.
Unit designations are used after numerical values and placed in a line with them (without transfer to the next line). A space should be left between the last digit and the designation, except for the sign raised above the line.
When specifying the values of quantities with limit deviations, the numerical values should be enclosed in brackets and the unit designations should be placed after the brackets or put down after the numerical value of the quantity and after its maximum deviation.
The literal designations of the units included in the product should be separated by dots on the middle line, as multiplication signs. It is allowed to separate letters with spaces, if this does not lead to misunderstanding. Geometric dimensions are indicated by the "x" sign.
In the alphabetic notation of the ratio of units, only one stroke should be used as a division sign: oblique or horizontal. It is allowed to use unit designations in the form of a product of unit designations raised to a power.
When using a slash, the designations of units in the numerator and denominator should be placed on the same line, the product of the designations in the denominator should be enclosed in brackets.
When specifying a derived unit consisting of two or more units, it is not allowed to combine letter designations and unit names, i.e. for some designations, for others - names.
Designations of units, the names of which are formed by the names of scientists, are written with a capital (capital) letter.
It is allowed to use the notation of units in the explanations of the notation of quantities to formulas. The placement of unit designations in the same line with formulas expressing dependencies between quantities and their numerical values presented in alphabetical form is not allowed.
The standard singles out units by fields of knowledge in physics and indicates the recommended multiple and submultiple units. There are 9 areas of use of units:
- 1. space and time;
- 2. periodic and related phenomena;
- 3. mechanics;
- 4. warmth;
- 5. electricity and magnetism;
- 6. light and associated electromagnetic radiation;
- 7. acoustics;
- 8. physical chemistry and molecular physics;
- 9. ionizing radiation.
The first system of units of physical quantities, although it was not yet a system of units in the modern sense, was adopted by the National Assembly of France in 1791. It included units of length, area, volume, capacity and mass, the main of which were two units: meter and kilogram.
The system of units as a set of basic and derived units was first proposed in 1832 by the German scientist K. Gauss. He built a system of units, where he took the units of length (millimeter), mass (milligram) and time (second) as the basis, and called it the absolute system.
With the development of physics and technology, other systems of units of physical quantities appeared, based on a metric basis. All of them were built according to the principle developed by Gauss. These systems have found application in various branches of science and technology. The measuring instruments developed at that time are graduated in the corresponding units and are still used today.
The variety of units of measurement of physical quantities and systems of units complicated their application. The same equations between quantities had different coefficients of proportionality. The properties of materials, processes were expressed by various numerical values. The International Committee on Weights and Measures separated from its membership a commission to develop a unified International System of Units. The Commission developed a draft of the International System of Units, which was approved by the XI General Conference on Weights and Measures in 1960. The adopted system was called the International System of Units, abbreviated as SI (SI - the initial letters of the name System International).
Given the need to cover all areas of science and technology with the International System of Units, seven units are chosen as the main ones. In mechanics, these are units of length, mass and time; in electricity, a unit of electric current strength is added; in heat, a unit of thermodynamic temperature; in optics, a unit of light intensity; in molecular physics, thermodynamics and chemistry, a unit of the amount of matter. These seven units, respectively: meter, kilogram, second, ampere, Kelvin, candela, and mole, are chosen as the basic SI units (Table 2.1).
The unit of length (meter) is the length of the path traveled by light in a vacuum in 1/299,792,458 of a second.
The unit of mass (kilogram) is the mass equal to the mass of the international prototype of the kilogram.
The unit of time (second) is the duration of 9192631770 periods of radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom.
The unit of electric current strength (ampere) is the strength of an unchanging current, which, passing through two normal straight conductors of infinite length and an negligibly small circular cross-sectional area, located at a distance of I m from one another in a vacuum, causes an interaction force between the conductors equal to 2- 10~7N per meter of length.
2.1. Basic SI units
The unit of thermodynamic temperature (Kelvin) is 1/273.16 of the thermodynamic temperature of the triple point of water. You can also use the Celsius scale.
The unit of luminous intensity (candela) is the luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540 1012 Hz, the energy intensity of which in this direction is 1/683 W / sr.
A unit of the amount of a substance (mol) is the amount of substances in a system containing as many structural elements as there are atoms in carbon-12 weighing 0.012 kg.
The basic units of the International System have sizes that are convenient for practical purposes and are widely used in the relevant areas of measurement.
The international system of units also contains two additional units: for a flat angle - a radian and for a solid angle - a steradian (Table 2.1).
Radian (rad) - a unit of a plane angle, equal to the angle between two radii of a circle, the length of the arc between which is equal to the radius. In degree terms, I rad \u003d 57
Steradian (sr) - a unit equal to the solid angle with a vertex in the center of the sphere, cutting out on the surface of the sphere an area equal to the area of a square with a side equal to the radius of the sphere. The solid angle £) is measured indirectly - by measuring the flat angle a at the top of the cone, followed by calculation by the formula
The solid angle in I cf corresponds to a flat angle equal to 65
Angular units cannot be included among the basic ones, since this would cause difficulty in interpreting the dimensions of quantities associated with rotation (arcs of a circle, areas of a circle, work of a pair of forces, etc.). At the same time, angular units cannot be considered derivatives, since they do not depend on the choice of basic units. Indeed, for any unit of length, the dimensions of the radian and steradian remain unchanged.
From the seven basic units and two additional units, units are derived as derivatives for measuring physical quantities in all areas of science and technology.
In the decisions of the XI and XII General Conferences on Weights and Measures, 33 derived SI units are given. Examples of derived units with their own names are given in Table. 2.2.
An important principle observed in the International System of Units is its coherence (consistency). Thus, the choice of the basic units of the system ensured the complete consistency of mechanical and electrical units. For example, a watt is a unit of mechanical power (equal to a joule per second) equals the power generated by an electric current with a force of I amperes at a voltage of I volts.
In SI, like other coherent systems of units, the coefficients of proportionality in the physical equations that define derived units are equal to the dimensionless unit.
Coherent derived units of the International System are formed using the simplest equations of connection between quantities (defining equations), in which the quantities are taken equal to units SI.
For example, the unit of speed is formed using an equation that determines the speed of a rectilinearly and uniformly moving point V = y, where V is the speed; / - the length of the path traveled; / - time. Substituting /, / and K for their SI units gives [ V = [/] / M = I m / s.
2.2. SI derived units with their own name
Therefore, the SI unit of speed is meters per second. It is equal to the speed of a rectilinearly and uniformly moving point, at which this point moves over a distance of 1 m in time / I s.
For example, to form a unit of energy, the equation T \u003d mU is used where T is the kinetic energy; m - body weight; V is the speed of the point, then the coherent SI unit of energy is formed as follows:
That is, the SI unit of energy is the joule (equal to a newton meter). It is equal to the kinetic energy of a body with a mass of 2 kg moving at a speed of I m/s.
In the International System of Units, as in other systems of units of physical quantities, dimension plays an important role.
Dimension is a symbolic (literal) designation of the dependence of derived quantities (or units) on the main ones.
For example, if any physical quantity is expressed in terms of length L, mass M and time G (which are the main quantities in the system of units of the LMT type) by the formula X \u003d f (L, M, 7), then it can be shown that the measurement results will be independent of the choice of units in the event that the function / will be a homogeneous function of length, mass and time. Let X \u003d LpM "Tr. The dimension of the quantity A is expressed by the formula 6mX \u003d 11МЯТ where dim is an abbreviation for the word dimension - dimension.
This formula shows how the derived quantity is related to the basic quantities, and is called the dimension formula.
Since any quantity can be represented as the product of its numerical value (L) per unit X X \u003d (SCH, it can be represented as
The equality of values in this formula breaks down into two equalities: the equality of numerical values
The dimension serves as a qualitative characteristic of the quantity and is expressed as the product of the powers of the basic quantities, through which it can be determined.
The dimension does not fully reflect all the qualitative features of the quantities. There are different quantities that have the same dimension. For example, work and moment of force, current strength and magnetomotive force, etc.
Dimension plays an important role in checking the correctness of complex calculation formulas in the theory of similarity and the theory of dimensions.
2.4. Benefits of the International System of Units
The main advantages of the International System of Units are:
Unification of units of physical quantities based on SI. For each physical quantity, one unit is established and a system for the formation of multiple and submultiple units from it using multipliers (Table 2.3);
The SI system is a universal system. It covers all areas of science, technology and economic sectors;
The basic and most derived SI units have sizes that are convenient for practical use. The system distinguishes between units of mass (kilogram) and force (newton);
The writing of equations and formulas in various fields of science and technology is simplified. In SI, for all types of energy (mechanical, thermal, electrical, etc.), one common unit is established - the joule.
2.3. Multipliers and prefixes for the formation of decimal multiples and submultiples and their designation
The first system of units of physical quantities, although it was not yet a system of units in the modern sense, was adopted by the National Assembly of France in 1791. It included units of length, area, volume, capacity and mass, the main of which were two units: meter and kilogram.
The system of units as a set of basic and derived units was first proposed in 1832 by the German scientist K. Gauss. He built a system of units, where he took the units of length (millimeter), mass (milligram) and time (second) as the basis, and called it the absolute system.
With the development of physics and technology, other systems of units of physical quantities appeared, based on a metric basis. All of them were built according to the principle developed by Gauss. These systems have found application in various branches of science and technology. The measuring instruments developed at that time are graduated in the corresponding units and are still used today.
The variety of units of measurement of physical quantities and systems of units complicated their application. The same equations between quantities had different coefficients of proportionality. The properties of materials, processes were expressed by various numerical values. The International Committee on Weights and Measures separated from its membership a commission to develop a unified International System of Units. The Commission developed a draft of the International System of Units, which was approved by the XI General Conference on Weights and Measures in 1960. The adopted system was called the International System of Units, abbreviated as SI (SI - the initial letters of the name System International).
Given the need to cover all areas of science and technology with the International System of Units, seven units are chosen as the main ones. In mechanics, these are units of length, mass and time; in electricity, a unit of electric current strength is added; in heat, a unit of thermodynamic temperature; in optics, a unit of light intensity; in molecular physics, thermodynamics and chemistry, a unit of the amount of matter. These seven units, respectively: meter, kilogram, second, ampere, Kelvin, candela, and mole, are chosen as the basic SI units (Table 2.1).
The unit of length (meter) is the length of the path traveled by light in a vacuum in 1/299,792,458 of a second.
The unit of mass (kilogram) is the mass equal to the mass of the international prototype of the kilogram.
The unit of time (second) is the duration of 9192631770 periods of radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom.
The unit of electric current strength (ampere) is the strength of an unchanging current, which, passing through two normal straight conductors of infinite length and an negligibly small circular cross-sectional area, located at a distance of I m from one another in a vacuum, causes an interaction force between the conductors equal to 2- 10~7N per meter of length.
The unit of thermodynamic temperature (Kelvin) is 1/273.16 of the thermodynamic temperature of the triple point of water. You can also use the Celsius scale.
The unit of luminous intensity (candela) is the luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540 1012 Hz, the energy intensity of which in this direction is 1/683 W / sr.
A unit of the amount of a substance (mol) is the amount of substances in a system containing as many structural elements as there are atoms in carbon-12 weighing 0.012 kg.
The basic units of the International System have sizes that are convenient for practical purposes and are widely used in the relevant areas of measurement.
The international system of units also contains two additional units: for a flat angle - a radian and for a solid angle - a steradian (Table 2.1).
Radian (rad) - a unit of a plane angle, equal to the angle between two radii of a circle, the length of the arc between which is equal to the radius. In degrees, I rad \u003d 57 ° 1744.8 ".
Steradian (sr) - a unit equal to the solid angle with a vertex in the center of the sphere, cutting out on the surface of the sphere an area equal to the area of a square with a side equal to the radius of the sphere. The solid angle £) is measured indirectly - by measuring the flat angle a at the top of the cone, followed by calculation by the formula
The solid angle in I cf corresponds to a flat angle equal to 65 ° 32 ", the angle l-sr - a flat angle of 120 °, the angle 2sr - a flat angle of 180 °. Additional units are used only for theoretical calculations and the formation of derived units, for example, angular velocity, angular Acceleration Angles are measured in degrees, minutes, and seconds, and there are no instruments for measuring angles in radians.
Angular units cannot be included among the basic ones, since this would cause difficulty in interpreting the dimensions of quantities associated with rotation (arcs of a circle, areas of a circle, work of a pair of forces, etc.). At the same time, angular units cannot be considered derivatives, since they do not depend on the choice of basic units. Indeed, for any unit of length, the dimensions of the radian and steradian remain unchanged.
From the seven basic units and two additional units, units are derived as derivatives for measuring physical quantities in all areas of science and technology.
In the decisions of the XI and XII General Conferences on Weights and Measures, 33 derived SI units are given. Examples of derived units with their own names are given in Table. 2.2.
An important principle observed in the International System of Units is its coherence (consistency). Thus, the choice of the basic units of the system ensured the complete consistency of mechanical and electrical units. For example, a watt is a unit of mechanical power (equal to a joule per second) equals the power generated by an electric current with a force of I amperes at a voltage of I volts.
In SI, like other coherent systems of units, the coefficients of proportionality in the physical equations that define derived units are equal to the dimensionless unit.
Coherent derived units of the International System are formed using the simplest equations of connection between quantities (defining equations), in which the quantities are taken equal to SI units.
For example, the unit of speed is formed using an equation that determines the speed of a rectilinearly and uniformly moving point V = y, where V is the speed; / - the length of the path traveled; / - time. Substituting /, / and K for their SI units gives [ V = [/] / M = I m / s.
Therefore, the SI unit of speed is meters per second. It is equal to the speed of a rectilinearly and uniformly moving point, at which this point moves over a distance of 1 m in time / I s.
For example, to form a unit of energy, the equation T \u003d mU is used where T is the kinetic energy; m - body weight; V is the speed of the point, then the coherent SI unit of energy is formed as follows:
That is, the SI unit of energy is the joule (equal to a newton meter). It is equal to the kinetic energy of a body with a mass of 2 kg moving at a speed of I m/s.
In the International System of Units, as in other systems of units of physical quantities, dimension plays an important role.
Dimension is a symbolic (literal) designation of the dependence of derived quantities (or units) on the main ones.
For example, if any physical quantity is expressed in terms of length L, mass M and time G (which are the main quantities in the system of units of the LMT type) by the formula X \u003d f (L, M, 7), then it can be shown that the measurement results will be independent of the choice of units in the event that the function / will be a homogeneous function of length, mass and time. Let X \u003d LpM "Tr. The dimension of the quantity A is expressed by the formula 6mX \u003d 11МЯТ where dim is an abbreviation for the word dimension - dimension.
This formula shows how the derived quantity is related to the basic quantities, and is called the dimension formula.
Since any quantity can be represented as the product of its numerical value (L) per unit X X \u003d (SCH, it can be represented as (XX \u003d SHR (M) R (T) G1LRMYATG.
The equality of values in this formula breaks down into two equalities: the equality of numerical values
The dimension serves as a qualitative characteristic of the quantity and is expressed as the product of the powers of the basic quantities, through which it can be determined.
The dimension does not fully reflect all the qualitative features of the quantities. There are different quantities that have the same dimension. For example, work and moment of force, current strength and magnetomotive force, etc.
Dimension plays an important role in checking the correctness of complex calculation formulas in the theory of similarity and the theory of dimensions.
Benefits of the International System of Units
The main advantages of the International System of Units are:
- - unification of units of physical quantities based on SI. For each physical quantity, one unit is established and a system for the formation of multiple and submultiple units from it using multipliers (Table 2.3);
- - the SI system is a universal system. It covers all areas of science, technology and economic sectors;
- - the basic and most derived SI units have dimensions that are convenient for practical use. The system distinguishes between units of mass (kilogram) and force (newton);
- - simplifies the recording of equations and formulas in various fields of science and technology. In SI, for all types of energy (mechanical, thermal, electrical, etc.), one common unit is established - the joule.
5. Principles of constructing systems of units fv.
The formation of a system of units is based on objective regular relationships between physical quantities and on the arbitrary, but reasonable will of people and their agreements, the final of which is adopted at the General Conference on Weights and Measures.
When constructing or introducing a new system of units, scientists are guided by only one single principle - practical expediency, i.e. ease of use of units in human activities. This principle is based on the following basic criteria:
Ease of formation of PV derivatives and their units, i.e. equating to unity the coefficients of proportionality in the equations of communication;
High accuracy of materialization of basic and derived units and transfer of their size to lower standards;
Indestructibility of standards of basic units, i.e. the possibility of their reconstruction in case of loss;
Continuity of units, preservation of their sizes and names with the introduction of a new system of units, which is associated with the exclusion of material and psychological costs;
The proximity of the sizes of basic and derived units to the sizes of PV, most often encountered in practice;
Long-term storage of basic and derived units by their standards;
The choice as the main minimum number of PV, reflecting the most general properties of matter.
The above criteria are in conflict, therefore, by agreement, the most advantageous option for practice is chosen. .
6. International system of units (si). Basic and additional units of the si system.
The unified international system of units (SI system) was adopted by the XI General Conference on Weights and Measures in 1960. On the territory of our country, the SI system of units has been in force since January 1, 1982 in accordance with GOST 8.417-81 "GSI. Units of physical quantities" .
The SI system is the only system of PV units, which is accepted and used in most countries of the world. This is due to its advantages and advantages over other systems of units, which include:
Versatility, i.e. coverage of all areas of science and technology;
Unification of all areas and types of measurements;
Coherence of quantities;
Ability to reproduce units with high accuracy in accordance with their definition;
Simplification of writing formulas in physics, chemistry, and also in technical sciences due to the lack of conversion factors;
Reducing the number of allowed units;
A unified system for the formation of multiple and submultiple units that have their own names;
Facilitation of the pedagogical process in secondary and higher schools, since there is no need to study many systems of units and non-systemic units;
Better mutual understanding in the development of scientific, technical and economic ties between different countries.
Basic units of the SI system:
Meter- unit meas. length
Second- units of time
Kilogram– unit of mass
Kelvin– unit temp.
Ampere- unit current strength
Candella- unit of illuminance
mole- unit of measure qty in-va
Additional units:
Radian is the unit of flat angle
Steradian is the unit of the solid angle
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