Systems of physical quantities and units. Construction of systems of units of physical quantities
basic units of physical quantities) and in the choice of their size. For this reason, by defining the basic quantities and their units, very different systems of units of physical quantities can be constructed. It should be added to this that derived units of physical quantities can also be defined differently. This means that a lot of unit systems can be built. Let us dwell on the general features of all systems.
The main common feature is a clear definition of the essence and physical meaning of the basic physical units and quantities of the system. It is desirable, but as stated in the previous section, not necessary, that the underlying physical quantity can be reproduced with high accuracy and can be transmitted by the measuring instrument with minimal loss of accuracy.
The next important step in building a system is to establish the size of the main units, that is, agree and legislate the procedure for reproducing the main unit.
Since all physical phenomena are interconnected by laws written in the form of equations expressing the relationship between physical quantities, when establishing derived units, it is necessary to select a constitutive relation for the derived quantity. Then, in such an expression, the coefficient of proportionality included in the defining relation should be equated to one or another constant number. Thus, a derived unit is formed, which can be given the following definition: “ Derived unit of physical quantity- a unit, the size of which is associated with the sizes of the basic units by relationships expressing physical laws, or definitions of the corresponding quantities.”
When constructing a system of units consisting of basic and derived units, two most important points should be emphasized:
First, the division of units of physical quantities into basic and derivatives does not mean that the former have any advantage or are more important than the latter. In different systems, the basic units may be different, and the number of basic units in the system may also be different.
Secondly, one should distinguish between equations of connection between quantities and equations of connection between their numerical values and values. Communication equations are relationships in general form that do not depend on units. Equations for the relationship between numerical values can have different forms depending on the selected units for each of the quantities. For example, if you choose the meter, kilogram of mass and second as the basic units, then the relationships between mechanical derivative units, such as force, work, energy, speed, etc., will differ from those if the basic units are chosen centimeter, gram, second or meter, ton, second.
Characterizing various systems of units of physical quantities, remember that the first step in building systems was associated with an attempt to relate basic units to quantities found in nature. So, during the era of the Great French Revolution in 1790-1791. It was proposed that the unit of length should be considered one forty-millionth of the earth's meridian. In 1799, this unit was legalized in the form of a prototype meter - a special platinum-iridium ruler with divisions. At the same time, the kilogram was defined as the weight of one cubic decimeter of water at 4°C. To store the kilogram, a model weight was made - a prototype of the kilogram. As a unit of time, 1/86400 of the average solar day was legalized.
Subsequently, the natural reproduction of these values had to be abandoned, since the reproduction process is associated with large errors. These units were established by law according to the characteristics of their prototypes, namely:
This basis of all modern systems of units of physical quantities has been preserved to this day. Thermal (Kelvin), electrical (Ampere), optical (candela), chemical (mole) units were added to the mechanical basic units, but the basics have been preserved to this day. It should be added that the development of measuring technology and in particular the discovery and implementation of lasers in measurements made it possible to find and legitimize new, very accurate ways of reproducing the basic units of physical quantities. We will dwell on such points in the following sections devoted to individual types of measurements.
Here we will briefly list the most commonly used systems of units in the natural sciences of the 20th century, some of which still exist in the form of non-systemic or slang units.
In Europe over the past decades, three systems of units have been widely used: CGS (centimeter, gram, second), MKGSS (meter, kilogram-force, second) and the SI system, which is the main international system and preferred in the territory of the former USSR “in all fields of science , technology and national economy, as well as in teaching.”
The last quotation, taken in quotation marks, is from the USSR state standard GOST 9867-61 “International System of Units”, which came into force on January 1, 1963. We will discuss this system in more detail in the next paragraph. Here we just point out that the main mechanical units in the SI system are the meter, kilogram-mass and second.
The GHS system has been in existence for over a hundred years and is very useful in some scientific and engineering fields. The main advantage of the GHS system is the logic and consistency of its construction. When describing electromagnetic phenomena, there is only one constant - the speed of light. This system was developed between 1861 and 1870. British Electrical Standards Committee. The GHS system was based on the system of units of the German mathematician Gauss, who proposed a method for constructing a system based on three basic units - length, mass and time. Gauss's system used the millimeter, milligram and second.
For electrical and magnetic quantities, two different versions of the SGS system have been proposed - the absolute electrostatic system SGSE and the absolute electromagnetic system SGSM. In total, in the development of the GHS system, there were seven different systems, which had the centimeter, gram and second as their main units.
At the end of the last century, the MKGSS system appeared, the main units of which were the meter, kilogram-force and second. This system has become widespread in applied mechanics, heat engineering and related fields. This system has many shortcomings, starting with confusion in the names of the basic unit, the kilogram, which meant kilogram-force as opposed to the widely used kilogram-mass. There was not even a name for the unit of mass in the MKGSS system and it was designated as i.e. m (technical unit of mass). Nevertheless, the MKGSS system is still partially used, at least in determining engine power in horsepower.
- power equal to 75 kgf m/s - is still used in technology as a slang unit.
In 1901, the Italian physicist P. Giorgi proposed a system of mechanical units built on three mechanical basic units - the meter, kilogram of mass and second. The advantage of this system was that it was easy to relate to the absolute practical system of electrical and magnetic units, since the units of work (joule) and power (watt) in these systems were the same. Thus, the opportunity was found to take advantage of the comprehensive and convenient GHS system with the desire to “seam” electrical and magnetic units with mechanical units.
This was achieved by introducing two constants - the electrical permeability (ε 0) of the vacuum and the magnetic permeability of the vacuum (μ 0). There is some inconvenience in writing formulas that describe the forces of interaction between stationary and moving electric charges and, accordingly, in determining the physical meaning of these constants. However, these shortcomings are largely compensated by such conveniences as the unity of expression of energy when describing both mechanical and electromagnetic phenomena, because
1 joule = 1 newton, meter = 1 volt, coulomb = 1 ampere, weber.
As a result of the search for the optimal version of the international system of units in 1948 IX General Conference on Weights and Measures, based on a survey of member countries of the Metric Convention, adopted an option that proposed taking the meter, kilogram of mass and second as the basic units. It was proposed to exclude the kilogram-force and related derivative units from consideration. The final decision, based on the results of a survey of 21 countries, was formulated at the Tenth General Conference on Weights and Measures in 1954.
The resolution read:
“As the basic units of a practical system for international relations, accept:
Later, at the insistence of chemists, the international system was supplemented by the seventh basic unit of quantity of a substance - the mole.
Subsequently, the international SI system or in the English transcription Sl (System International) was somewhat clarified, for example, the temperature unit was called Kelvin instead of “degree Kelvin”, the system of standards of electrical units was reoriented from Ampere to Volt, since a standard of potential difference was created based on the quantum effect - the Josephson effect, which made it possible to reduce the error in reproducing the unit of potential difference - the Volt - by more than an order of magnitude. In 1983, at the XVIII General Conference on Weights and Measures, a new definition of the meter was adopted. According to the new definition, a meter is the distance traveled by light in 1/2997925 of a second. Such a definition, or rather a redefinition, was needed in connection with the introduction of lasers into the reference technology. It should immediately be noted that the size of the unit, in this case the meter, does not change. Only the methods and means of its reproduction change, characterized by less error (greater accuracy).
5. Principles for constructing systems of fv units.
The formation of a system of units is based on objective, natural connections between physical quantities and on the arbitrary but reasonable will of people and their agreements, the final of which is that 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 activity. This principle is based on the following basic criteria:
Ease of formation of PV derivatives and their units, i.e. equating proportionality coefficients in communication equations to unity;
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 when introducing a new system of units, which is associated with the elimination of material and psychological costs;
The proximity of the sizes of basic and derived units to the sizes of the PVs most often encountered in practice;
Long-term storage of basic and derived units by their standards;
Selection of the minimum number of PVs as the main ones, reflecting the most general properties of matter.
The above criteria conflict, so the most beneficial option for practice is selected by agreement .
6. International system of units (SI). Basic and additional units of the C system.
The Unified International System of Units (SI system) was adopted by the XI General Conference on Weights and Measures in 1960. In our country, the SI system of units has been in effect 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 that 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, as well as in technical sciences due to the lack of conversion factors;
Reducing the number of units allowed;
A unified system for the formation of multiple and submultiple units with their own names;
Facilitation of the pedagogical process in secondary and higher schools, since there is no need to study multiple systems of units and non-systemic units;
Better mutual understanding in the development of scientific, technical and economic ties between different countries.
Basic SI units:
Meter- unit length measurement
Second- units of time
Kilogram– units of mass
Kelvin– unit temperature change
Ampere- unit current strength
Candella- unit of luminous intensity
Mole- unit change quantity in quantity
Additional units:
Radian- this is the unit of plane angle
Steradian- this is the unit of solid angle
Principles for constructing a system of units of quantities Ø System of physical quantities - A set of physical quantities formed in accordance with accepted principles, when some quantities are taken as independent (basic quantities), and others (derived quantities) are determined as functions of independent quantities. Ø The systems of physical quantities that existed at different times and in different states had many differences: Ø they used different measures, Ø they had different multiplicities of units used, Ø they had different numbers of basic and derived units. 2
Systems of units that were used before the introduction of the international system Gaussian system (LMT - millimeter, milligram, second); 2. The GHS system (LMT+QJ – centimeter, gram, second + kelvin, candela) applies to the area of thermal and optical quantities; 3. The ISS system (LMT+QJ – meter, kilogram, second + kelvin, candela) extends to the area of thermal and light quantities; 4. MTS system (LMT – meter, ton, second); 5. MKGSS system (LFT – meter, kilogram-force, second). Area of distribution: mechanics, heat engineering. Kilogram-force is a force equal to the weight of a body weighing 1 kg with normal acceleration of gravity g 0 = 9.80665 m/s2 1 kgf = 9.80655 N 1. 3
Systems of units of electromagnetic quantities Electrostatic system of units (SGSE system) When constructing this system, the first derivative of the electrical unit is introduced by the unit of electric charge using Coulomb's law as the governing equation. In this case, the absolute dielectric constant is considered a dimensionless electrical quantity. As a consequence of this, in some equations relating electromagnetic quantities, the square root of the speed of light in vacuum appears explicitly. n Electromagnetic system of units (SGSM system). In constructing this system, the first derivative of the electrical unit is the unit of current, using Ampere's law as the governing equation. In this case, absolute magnetic permeability is considered a dimensionless electrical quantity. In this regard, in some equations relating electromagnetic quantities, the square root of the speed of light in vacuum appears explicitly. n 4
Symmetrical system of units (SGS system). This system is a combination of the SGSE and SGSM systems. In the SGS system, units of the SGSE system are used as units of electrical quantities, and units of the SGSM system are used as units of magnetic quantities. As a result of the combination of the two systems, in some equations connecting electrical and magnetic quantities, the square root of the speed of light in vacuum appears explicitly. n 5
Principles for constructing a system of units of quantities Ø With all these differences, the existing systems of physical quantities had common features: Ø the presence of generally recognized (legalized for a given state) measures for reproducing units of physical quantities, Ø the presence of connections between individual measures for the formation of derived units, Ø the presence of a system for transferring sizes units of physical quantities. ØTransfer of unit size - reduction of the unit size of a physical quantity stored by a measuring instrument to the unit size reproduced or stored by a standard 6
Principles for constructing a system of units of quantities The relationships of physical quantities in the system are reflected using such an important concept as dimension - (from dimension). The dimension of a quantity is an expression in the form of a power polynomial, revealing the relationship of the physical quantity Q with basic physical quantities. For example, in the LMT system adopted in mechanics, in which length L, mass M, time T are used as the basic units, the dimension has the form: Indicators a, b, g are called dimensional indices. In particular, the dimension of speed and the dimension of force, 7
Principles for constructing a system of units of quantities You can perform operations on dimensions: multiplication, division, exponentiation and root extraction. The concept of dimension is widely used: Ø to convert units from one system to another; Ø to check the correctness of the calculation formulas obtained as a result of theoretical derivation; Ø when clarifying the relationship between them; Ø in the theory of physical similarity. 8
Principles for constructing a system of units of quantities The dimension of a derived quantity is the simplest communication equation that determines a quantity, with a proportionality coefficient equal to one. However, the dimension does not reflect the physical nature of the quantity. In particular, for a number of quantities that are different in nature, the dimensions turn out to be the same. For example, work and moment of force have the same dimension: In addition, the dimension does not reveal how the quantity is measured, except in the simplest cases when the relation equation coincides with the expression of the dimension, which, for example, is typical for the area of a square. 9
Principles for constructing a system of units of quantities 1. Equations of connection between quantities, in which alphabetic symbols are understood as physical quantities: X=f (X 1, X 2, …Xm) (1) X 1, X 2, …Xm – quantities associated with measured by the quantity X by some equation of communication. 2. Equations of connection between numerical values of quantities, in which alphabetic symbols are understood as numerical values of physical quantities: n X = q [X]; X 1 = q 1 ; X 2 = q 2 ; X m = q m [ X m] Where q, q 1, …qm – numerical values; [X], , …, – units of quantities The equation of connection between numerical values can be reduced to an equation of dimension. 10
Principles for constructing a system of units of quantities Dependencies between units of measurement, manifested in physical laws, make it possible to obtain derived units of the system, the concept of which was first introduced by K. Gauss. The names and designations of derived quantities can be obtained: Ø from the names and designations of the basic units; Ø using special names and designations; Ø from the names and designations of basic and special names and designations of derived units; Ø using multiple and submultiple prefixes and multipliers. eleven
Principles for constructing a system of units of quantities Derived units are: coherent and incoherent. A derivative unit is called coherent if it is related to other units of the system by an equation in which the numerical factor is taken equal to one. For example, a unit of speed is formed using an equation that determines the speed of rectilinear uniform motion of a point: v = L/ t, where L is the length of the path traveled; t – time of movement. Substituting their units for L and t gives v = 1 m/s. Therefore, the unit of speed is coherent. 12
Principles for constructing a system of units of quantities n n n n When constructing a system of physical quantities, a sequence of defining equations is selected in which each subsequent equation contains only one new derived quantity, which allows this quantity to be expressed through a set of previously determined quantities, and, ultimately, through the basic quantities of the system quantities To find the dimension of the derivative of a physical quantity in a certain system of quantities, it is necessary to substitute their dimensions into the right side of the defining equation of this quantity instead of the designations of quantities. So, for example, putting in the defining equation of the speed of uniform motion v = ds/dt instead of ds the dimension of length L and instead of dt the dimension of time T, we obtain dim v = L / T =LT-1 Substituting in the defining equation of acceleration a = dv/dt instead dt dimension of time T and instead of dv the dimension of speed LT-1 found above, we obtain dim a = LT-2 Knowing the dimension of acceleration from the defining equation of force F = ma, we obtain: dim F = M∙LT-2 = LMT-2 Knowing the dimension of force , you can find the dimension of work, then the 13th dimension of power, etc.
Principles for constructing a system of units of quantities Note: If the communication equation contains a numerical coefficient different from one, then to form a coherent SI unit, quantities with values in SI units are substituted into the right side of the equation, giving, after multiplication by the coefficient, a total numerical value equal to one. 14
Principles for constructing a system of units of quantities For example, if the equation is used to form a coherent unit of energy where m is the mass of the body; v is its speed, then the coherent unit of energy can be formed in two ways: Therefore, the coherent SI unit is the joule, equal to newton times the meter. In the cases considered, it is equal to the kinetic energy of a body weighing 2 kg moving at a speed of 1 m/s, or a body weighing 1 kg moving at a speed of m/s. 15
International system of units (SI) On the territory of the Russian Federation, the system of units (SI) has been in effect since 01. 1982. In accordance with GOST 8. 417 -81 (Now GOST 8. 417 -2002) Currently includes 7 main units 16
Definition and content of basic SI units n n n Definition and content of basic SI units. In accordance with the decisions of the General Conference on Weights and Measures (GCPM), adopted in different years, the following definitions of the basic SI units are currently in force. The unit of length is the meter - the length of the path traveled by light in a vacuum in 1/299792458 of a second (decision of the XVII CGPM in 1983). The unit of mass is the kilogram - a mass equal to the mass of the international prototype of the kilogram (decision of the 1st CGPM in 1889). The unit of time is a second - the duration of 9192631770 periods of radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom, not perturbed by external fields (decision of the XIII CGPM in 1967). The unit of electric current is the ampere, the force of unchanging current which, when passing through two parallel conductors of infinite length and negligible circular cross-section, located at a distance of 1 m from each other in a vacuum, would create a force between these conductors equal to 2 10 -7 N for every meter of length (approved by the IX GCPM in 1948). 17
Definition and content of basic SI units n n n The unit of thermodynamic temperature is the kelvin (until 1967 it was called the degree Kelvin) - 1/273.16 part of the thermodynamic temperature of the triple point of water. It is permissible to express thermodynamic temperature in degrees Celsius (resolution XIII of the CGPM in 1967). The unit of luminous intensity is the candela - the luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540∙1012 Hz, the energetic luminous intensity of which in this direction is 1/683 W/sr (resolution XVI GCPM in 1979). Unit of quantity of a substance - mole - amount of substance of a system containing the same number of structural elements as there are atoms contained in a carbon-12 nuclide weighing 0.012 kg (resolution XIV GCPM in 1971) 18
Definition and content of SI base units n n The mole is not a pure base unit, since it is related to another base unit - the kilogram. Generally speaking, the unit of quantity of substance has not received widespread use, like other basic SI units. Mole standards have not yet been created. One of the reasons here is that the mass of one mole is different for different substances (structural elements). In recent years, metrologists at scientific conferences have proposed excluding the mole from the number of basic SI units, transferring it to the category of a special unit of mass or a derivative quantity. However, in recent years there has been a “turn” in the activity of estimating the quantity of a substance associated with the use of metrology in medicine, chemistry, pharmaceuticals, the food industry, and environmental protection: the International Committee of Weights and Measures has created a new Advisory Committee on the quantity of a substance, an international “project” is underway Avogadro" in order to create a new mass standard based on the pure isotope of silicon, since 1999. A new derivative of the SI unit - catal (mol per second) was officially introduced to measure the catalytic activity of enzymes. The unit was adopted at the request of the Advisory Committee on Units (UCU), the International Federation of Clinical Chemistry and Laboratory Medicine, and the International Union of Biochemists. 19
GOST 8.417 -2002 GSI. Units of quantities Formation of derived units of quantities: 1. from the names and designations of basic units: Designation Unit of measurement international Russian Dimension Expression through basic units Area square meter m 2 m 2 L 2 m 2 Volume cubic meter m 3 m 3 L 3 m 3 Speed meter per second m/s m/s LT-1 m-1∙kg∙s-2 Density cubic meter per kilogram m 3/kg m 3/kg L 3 M-1 m 3∙kg-1 Name of value 20
GOST 8.417 -2002 GSI. Units of quantities Formation of derived units of quantities: 2. using special names and symbols: Designation Name of quantity Unit of measurement international Russian Dimension Expression through basic units Frequency hertz Hz Hz T-1 s-1 Force newton N N LMT-2 m∙ kg∙s-2 Pressure pascal Pa Pa L-1 MT-2 m-1∙kg∙s-2 Energy, work, joule amount of heat J J L 2 MT-2 m 2∙kg∙s-2 Power watt W W L 2 MT-3 m 2∙kg∙s-3 Electric charge coulomb C Cl TI s∙A 21
GOST 8.417 -2002 GSI. Units of quantities Formation of derived units of quantities: 3. from the names and designations of basic and special names and designations of derived units: Designation Unit of measurement international Russian Dimension Expression through basic units Moment of force newton meter N∙m N∙m L 2 M -2 T m 2∙kg-2∙s Heat capacity joule per kelvin J/ K J/K L 2 MT-2 -1 m∙kg∙s-2 Electric field strength volts per meter V/m V/m LMT-3 I-1 m∙kg∙s-3∙A-1 Brightness candelas per square meter kd/m 2 cd/m 2 L-2 J m-2∙kd Name of value 22
GOST 8.417 -2002 GSI. Units of quantities Formation of derived units of quantities: 4. using multiple and submultiple prefixes and factors: Decimal factor Prefix Designation Decimal Russian decimal plural Prefix Designation international Russian 1015 peta R P 10 -1 deci d d 1012 tera T T 10 - 2 centi c s 109 giga G G 10 -3 milli m m 106 mega M M 10 -6 micro μ mk 103 kilo k k 10 -9 nano n n 102 hecto h g 10 -12 pico p p 101 deca da yes 10 - 15 femto f f 23
GOST 8.417 -2002 GSI. Units of quantities From the rules for writing units of quantities: Rule Correct Incorrect 100 k. W 20 ° C 80% 100 k. W 20 ° C 80% 30 ° 30 ° If there is a decimal fraction in a numerical value, the designation is placed after all numbers 423, 06 m 423 meters a character raised above a line without leaving a space before it 24
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 French National Assembly in 1791. It included units of length, area, volume, capacity and mass, the main of which were two units: the 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 based on the units of length (millimeter), mass (milligram) and time (second), 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 were calibrated in appropriate 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 proportionality coefficients. The properties of materials and processes were expressed in various numerical values. The International Committee on Weights and Measures selected from its members a commission to develop a unified International System of Units. The Commission developed a draft International System of Units, which was approved by the XI General Conference on Weights and Measures in I960. The adopted system was called the International System of Units, abbreviated SI (SI - the initial letters of the name System International).
Taking into account the need for the International System of Units to cover all fields of science and technology, seven units have been selected as the main ones. In mechanics these are the units of length, mass and time, in electricity a unit of electric current 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 amount of matter. These seven units are respectively: meter, kilogram, second, ampere, Kelvin, candela and mole - and are chosen as the basic SI units (Table 2.1).
Unit of length (meter) - the length of the path traveled by light in a vacuum in 1/299,792,458 of a second.
Unit of mass (kilogram) - mass equal to the mass of the international prototype of the kilogram.
Unit of time (second) - 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 (ampere) is the strength of a constant current, which, passing through two normal straight conductors of infinite length and a negligibly small circular cross-sectional area, located at a distance of I m from each other in a vacuum, causes an interaction force between the conductors equal to 2- Yu~7N for each meter of length.
The unit of thermodynamic temperature (Kelvin) is 1/273.16 of the thermodynamic temperature of the triple point of water. It is also possible to use the Celsius scale.
Luminous intensity unit (candela) is the luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540-1012 Hz, the luminous energy intensity of which in this direction is 1/683 W/sr.
Unit of quantity of a substance (mol) - the amount of substances in a system containing the same number of structural elements as there are atoms contained in carbon-12 weighing 0.012 kg.
The basic units of the International System have dimensions convenient for practical purposes and are widely used in their respective fields of measurement.
The International System of Units also contains two additional units: for a plane angle - radians and for a solid angle - steradians (Table 2.1).
Radian (rad) - a unit of 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 = 57° 1744.8".
Steradian (sr) is a unit equal to the solid angle with the vertex at 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 plane angle a at the vertex of the cone, followed by calculation using the formula
The solid angle in I sr corresponds to a flat angle equal to 65°32", the angle l-sr corresponds to a flat angle of 120°, the angle 2 sr corresponds to a flat angle of 180°. Additional units are used only for theoretical calculations and the formation of derivative units, for example, angular velocity, angular acceleration. Angular degrees, minutes and seconds are used to measure angles. There are no devices 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 (circular arc, area 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 units of length, the dimensions of the radian and steradian remain unchanged.
From seven basic units and two additional ones, units for measuring physical quantities in all fields of science and technology are derived as derivatives.
In the decisions of the XI and XII General Conferences on weights and measures, 33 derived SI units are given. Examples of derived units that have their own names are given in Table. 2.2.
An important principle that is observed in the International System of Units is its coherence (consistency). Thus, the choice of the main units of the system ensured complete consistency between the mechanical and electrical units. For example, a watt, a unit of mechanical power (equal to a joule per second), is equal to the power generated by an electric current of I ampere at a voltage of I volt.
In SI, like other coherent systems of units, the coefficients of proportionality in the physical equations that define the 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 quantities are assumed to be 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; / is the length of the path traveled; / is time. Substituting for /, / and K their SI units gives [ V = [/]/M = I m/s.
Therefore, the SI unit of speed is meter per second. It is equal to the speed of a rectilinearly and uniformly moving point, at which this point moves to a distance of 1 m in time / I s.
For example, to form a unit of energy, the equation T = mU is used where T is kinetic energy; t - body weight; V is the speed of motion of a point, then the coherent SI unit of energy is formed as follows:
That is, the SI unit of energy is the joule (equal to the newton meter). It is equal to the kinetic energy of a body weighing 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 (letter) designation of the dependence of derived quantities (or units) on basic ones.
For example, if any physical quantity is expressed through the length L, mass M and time Г (which are the main quantities in the LMT type system of units) by the formula X = f(L, M, 7), then it can be shown that the measurement results will be independent of choice of units if the function / is a homogeneous function of length, mass and time. Let X = LpM"Tr. The dimension of quantity A is expressed by the formula 6mX = 11MYT where dim is an abbreviation for the word dimension - dimension.
This formula shows how a derived quantity is related to the basic quantities and is called the dimensional formula.
Since any quantity can be represented as the product of its numerical value (A) by a unit X X = (ШХ, it can be represented in the form (ХХ = ШП(М)Я(Т)Г1ЛРМЯТГ.
The equality of quantities in this formula breaks down into two equalities: equality of numerical values
Dimension serves as a qualitative characteristic of a quantity and is expressed by the product of powers of basic quantities through which it can be determined.
Dimension does not fully reflect all the qualitative features of quantities. There are various quantities that have the same dimension. For example, work and torque, current strength and magnetomotive force, etc.
Dimension plays an important role in checking the correctness of complex calculation formulas in similarity theory and dimensional theory.
Advantages 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 and a system for the formation of multiples and submultiples of it using multipliers are established (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 convenient for practical use. The system distinguishes between units of mass (kilogram) and force (newton);
- - it simplifies the writing of equations and formulas in various fields of science and technology. The SI has one common unit for all types of energy (mechanical, thermal, electrical, etc.) - the joule.
Physical quantities are introduced to quantify and describe various properties and processes.
Physical quantity(PV) is a property that is general in qualitative terms for many physical objects, but individual in quantitative terms for each of them.
Unit of physical quantity- a physical quantity of a fixed size, which is conventionally assigned a numerical value equal to 1, and is used for the quantitative expression of physical quantities similar to it.
In order to organize the entire set of used units of physical quantities, it is necessary to systematize the used physical quantities, i.e. create a system of physical quantities. Then, on the basis of the system of physical quantities, a system of units of physical quantities is built.
System of physical quantities- a set of physical quantities formed in accordance with accepted principles, when some quantities are taken as independent, while others are determined as functions of independent quantities.
The main PV is PV. included in a system of quantities and conventionally accepted as independent of other quantities of this system.
The derivative of PV is PV, which is part of a system of quantities and is determined through the basic quantities of this system.
The relationship between various physical quantities is expressed by coupling equations. There are two types of such equations: equations of connection between quantities and equations of connection between numerical values. Equation, connections between quantities - an equation reflecting the relationship between quantities determined by the laws of nature, in which the corresponding physical quantities are understood by letter symbols. Such equations represent relationships in a general form that does not depend on units of measurement. An equation for the connection between numerical values is an equation that reflects the relationship between quantities determined by the laws of nature, in which the alphabetic symbols are understood as the values of the corresponding physical quantities.
System of units of physical quantities- a set of basic and derived units of physical quantities, formed in accordance with the principles for a given system of physical quantities.
The basic unit of a system of units of physical quantities is the unit of the main physical function in a given system of units.
A derived unit of a system of units of physical quantities is a unit of a derivative of a PV system of units, formed in accordance with an equation connecting it with the basic units or with the basic and already defined derivatives.
Construction of systems of units of physical quantities
To build a system of units, we must select several basic units and establish, using defining equations (equations of relationships between numerical values), the derived units of all other quantities of interest to us.
The main requirement for a system of units is that the system should be as convenient as possible for practical purposes. In this regard, the number of basic units cannot be arbitrary. Here you need to keep the following in mind. It is advisable to build systems of units suitable for various fields of science and technology, in which the number of basic units is five to seven. Such systems of units include the International System of Units SI and with some additions GHS.
In a universal system of units, which is suitable for various measurements in science and technology, the quantities whose units are taken as basic must reflect the most general properties of matter. The number of basic units of such a system has now reached seven - these are units of length, mass, time, temperature, current strength, light intensity and amount of matter.
Having chosen the basic units, you need to decide on their sizes.
FV unit size- quantitative determination of the PV unit reproduced or stored by the measuring instrument.
Basic units are established in two ways: by prototypes and by measuring natural quantities. The first method is based on establishing a unit using some body (weight, ruler). The second method involves carrying out some measurement procedure.
Throughout the development of humanity, different systems have been used.
1. 1791 National Assembly of France adopted a matrix system of measures. It included units of length, area, volume, weight, which were based on two units: m, kg.
2. The concept of a system of units as a set of basic and derived quantities was first proposed by a German scientist Gauss in 1832. The basic ones in this system were: the unit of length is mm, the unit of mass is mg, the unit of time is s. This system of units was called absolute.
3. In 1881 was accepted GHS system of units of physical quantities. It is a system of mechanical units. It was constructed using a system of equations of classical mechanics. The basic units of which were: cm - unit of length, g - unit of mass, s - unit of time. The derived units were the unit of force - dyne and the unit of work - erg. The GHS also included 2 additional units: for a flat angle - radians; for solid angle - steradian. There is one constant in the system - the speed of light.
The Italian scientist Giorgi proposed another system of units, called MCSA and quite widespread in the world. This is a system of units of electrical and magnetic quantities. Basic units: m, kg, s, A, and derivatives: unit of force - N, unit of energy - J, unit of power - W. ICSA is included as an integral part of the international system of SI units.
4. MKGSS– a system of measurement units in which the main units are m, kgf and s. The MKGSS system is still partially used, at least in determining engine power in horsepower (Horsepower - power = 75 kgf m/s)
5. MTS system– a system of mechanical units. The following basic units were chosen: meter (unit of length), ton (unit of mass), second (unit of time). The unit of mass, the ton, has proven to be the most convenient in a number of industries dealing with large masses. However, the size of the derived units of most PVs turned out to be inconvenient for practice and the system was canceled.
The most widely used system of units throughout the world is the SI. The International System of Units SI was adopted by the General Conference on Weights and Measures in 1960. Its advantages are:
Versatility
Possibility of reproducing units with high accuracy of compliance with their definition and the smallest error
Unification of all areas and types of measurements
Simplifying the writing of formulas and reducing the number of allowed units
A unified system for the formation of multiples and submultiples that have their own name. The SI system is based on 7 basic and 2 additional physical units (meter, kg, second, ampere, kelvin, mole, candela) additional (radian, steradian)