Physical quantities. Units of quantities
Physical quantity- this is a property that is qualitatively common to many physical objects, but quantitatively individual for each of them.
Meaning physical quantity - This quantification the size of a physical quantity, presented in the form of a certain number of units accepted for it (for example, the value of a conductor resistance is 5 Ohms).
Distinguish true the value of a physical quantity that ideally reflects the property of an object, and real, found experimentally to be close enough to the true value that it can be used instead, and measured value measured by the reading device of the measuring instrument.
A set of quantities interconnected by dependencies form a system of physical quantities, in which there are basic and derived quantities.
Main a physical quantity is a quantity included in a system and conventionally accepted as independent of other quantities of this system.
Derivative a physical quantity is a quantity included in a system and determined through the basic quantities of this system.
An important characteristic of a physical quantity is its dimension (dim). Dimension- this is an expression in the form of a power monomial, composed of products of symbols of basic physical quantities and reflecting the relationship of a given physical quantity with physical quantities accepted in a given system of quantities as basic ones with a proportionality coefficient equal to one.
Unit of physical quantity - it is a specific physical quantity, defined and agreed upon, to which other quantities of the same kind are compared.
In accordance with the established procedure, units of quantities of the International System of Units (SI), adopted by the General Conference on Weights and Measures, recommended by the International Organization of Legal Metrology are allowed to be used.
There are basic, derivative, multiple, submultiple, coherent, systemic and non-systemic units.
Basic unit of the system of units- a unit of a basic physical quantity chosen when constructing a system of units.
Meter- the path length traveled by light in a vacuum in a time interval of 1/299792458 of a second.
Kilogram- unit of mass, equal to mass international prototype of the kilogram.
Second- time equal to 9192631770 periods of radiation corresponding to the transition between two hyperfine levels of the ground state of the Cesium-133 atom.
Ampere- the strength of a constant current, which, when passing through two parallel straight conductors infinite length and insignificant small area circular cross section, located in a vacuum at a distance of 1 m from one another, would cause an interaction force equal to 2 ∙ 10 -7 N on each section of a conductor 1 m long.
Kelvin- a unit of thermodynamic temperature equal to 1/273.16 of the thermodynamic temperature of the triple point of water.
Mole- the amount of substance in a system containing the same amount structural elements, how many atoms are there in carbon-12 weighing 0.012 kg.
Candela- luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540 ∙ 10 12 Hz, the energetic luminous intensity of which in this direction is 1/683 W/sr.
Two additional units are also provided.
Radian- the angle between two radii of a circle, the length of the arc between which is equal to the radius.
Steradian- a solid angle with a vertex at the center of the sphere, cutting out an area on the surface of the sphere, equal to the area square with side equal to the radius spheres.
Derived unit of system of units- a unit of a derivative of a physical quantity of a system of units, formed in accordance with an equation connecting it with the basic units or with the basic and already defined derivatives. For example, the unit of power expressed in SI units is 1W = m 2 ∙ kg ∙ s -3.
Along with SI units, the Law “On Ensuring the Uniformity of Measurements” allows the use of non-system units, i.e. units not included in any of the existing systems. It is customary to distinguish several types non-systemic units:
Units accepted on a par with SI units (minute, hour, day, liter, etc.);
Units used in special fields of science and technology
(light year, parsec, diopter, electron volt, etc.);
Units retired (millimeter of mercury,
horsepower, etc.)
Non-systemic units also include multiple and submultiple units of measurement, which sometimes have their own names, for example, the unit of mass - ton (t). IN general case decimals, multiples and submultiples are formed using multipliers and prefixes.
Measuring instruments
Under measuring instrument(SI) is understood as a device intended for measurements and having standardized metrological characteristics.
According to their functional purpose, measuring instruments are divided into: measures, measuring instruments, measuring transducers, measuring installations, measuring systems.
Measure- a measuring instrument designed to reproduce and store a physical quantity of one or more sizes with the required accuracy. A measure can be represented as a body or a device.
Measuring device(IP) - a measuring instrument designed to extract measurement information and convert
it into a form that can be directly perceived by the operator. Measuring instruments, as a rule, include
measure. Based on the principle of operation, power supplies are distinguished between analog and digital. According to the method of presenting measurement information, measuring instruments are either indicating or recording.
Depending on the method of converting the measurement information signal, a distinction is made between direct conversion devices (direct action) and balancing conversion devices (comparison). In direct conversion devices, the measurement information signal is converted required amount times in one direction without application feedback. In balancing conversion devices, along with a direct conversion circuit, there is a circuit inverse conversion and the measured quantity is compared with a known quantity that is homogeneous with the measured one.
Depending on the degree of averaging of the measured value, instruments are distinguished that give readings instantaneous values measured quantity, and integrating devices, the readings of which are determined by the time integral of the measured quantity.
Transducer- a measuring instrument designed to convert a measured value into another value or measuring signal, convenient for processing, storage, further transformations, indication or transmission.
Depending on their location in the measuring circuit, primary and intermediate converters are distinguished. Primary transducers are those to which the measured value is supplied. If primary converters are placed directly on the research object, remote from the processing site, then they are sometimes called sensors.
Depending on the type of input signal, converters are divided into analog, analog-to-digital and digital-to-analog. Widely used are large-scale measuring transducers designed to change the size of a quantity in given number once.
Measuring setup is a set of functionally combined measuring instruments (measures, measuring instruments, measuring transducers) and auxiliary devices (interface, power supply, etc.), designed for one or more physical quantities and located in one place.
Measuring system - a set of functionally combined measures, measuring transducers, computers and other technical means located in different points controlled object for the purpose of measuring one or more physical quantities.
Types and methods of measurements
In metrology, measurement is defined as a set of operations performed using a technical+ means that stores a unit of physical quantity, allowing one to compare the measured quantity with its unit and obtain the value of this quantity.
Classification of types of measurements according to the main ones classification criteria presented in table 2.1.
Table 2.1 – Types of measurements
Direct measurement - measurement in which the initial value of a quantity is found directly from experimental data as a result of performing a measurement. For example, measuring current with an ammeter.
Indirect measurement - a measurement in which the desired value of a quantity is found based on known dependence between this quantity and the quantities that are subject to direct measurements. For example, measuring the resistance of a resistor using an ammeter and a voltmeter using a relationship that relates resistance to voltage and current.
Joint measurements are measurements of two or more quantities of different names to find the relationship between them. Classic example joint measurements is to find the dependence of the resistor resistance on temperature;
Aggregate measurements are measurements of several quantities of the same name, in which the desired values of quantities are found by solving a system of equations obtained from direct measurements and various combinations these quantities.
For example, finding the resistances of two resistors based on the results of measuring the resistances of series and parallel connections these resistors.
Absolute measurements - measurements based on direct measurements of one or more quantities and the use of values of physical constants, for example, measurements of current in amperes.
Relative measurements - measuring the ratio of the value of a physical quantity to a quantity of the same name or a change in the value of a quantity in relation to a quantity of the same name, taken as the initial one.
TO static measurements include measurements in which the SI operates in static mode, i.e. when its output signal (eg pointer deflection) remains unchanged during the measurement time.
TO dynamic measurements include measurements performed by SI in dynamic mode, i.e. when its readings depend on dynamic properties. The dynamic properties of SI are manifested in the fact that the level of variable influence on it at any point in time determines the output signal of SI at a subsequent point in time.
Measurements with the highest possible accuracy achieved at the current level of development of science and technology. Such measurements are carried out when creating standards and measuring physical constants. Characteristic of such measurements are the assessment of errors and analysis of the sources of their occurrence.
Technical measurements are measurements taken in given conditions By a certain technique and carried out in all industries National economy, with the exception of scientific research.
The set of techniques for using the principle and measuring instruments is called measurement method(Fig. 2.1).
Without exception, all measurement methods are based on comparison of the measured value with the value reproduced by the measure (single-valued or multi-valued).
The direct assessment method is characterized by the fact that the values of the measured quantity are read directly from the reading device measuring instrument direct action. The scale of the device is calibrated in advance using a multi-valued measure in units of the measured value.
Methods of comparison with a measure involve comparison of the measured value and the value reproduced by the measure. The most common comparison methods are: differential, zero, substitution, coincidence.
Figure 2.1 – Classification of measurement methods
With the zero measurement method, the difference between the measured value and the known value is reduced to zero during the measurement process, which is recorded by a highly sensitive zero indicator.
At differential method The difference between the measured value and the value reproduced by the measure is counted on the scale of the measuring device. The unknown quantity is determined from the known quantity and the measured difference.
The substitution method involves alternately connecting the measured and known quantities to the input of the indicator, i.e. measurements are carried out in two steps. The smallest measurement error is obtained when, as a result of selection known quantity the indicator gives the same reading as for an unknown value.
The coincidence method is based on measuring the difference between the measured value and the value reproduced by the measure. When measuring, coincidences of scale marks or periodic signals are used. The method is used, for example, when measuring frequency and time using reference signals.
Measurements are performed with single or multiple observations. Observation here refers to an experimental operation performed during the measurement process, as a result of which one value of a quantity is obtained, which is always random in nature. When making measurements with multiple observations, statistical processing of the observation results is required to obtain the measurement result.
Objects and phenomena of the world around us are characterized by various properties that can manifest themselves to a greater or lesser extent and, therefore, can be quantitatively assessed. For quantitative description various properties of processes and physical bodies the concept of a physical quantity is introduced.
Under physical quantity understand one of the properties physical object (physical system, phenomenon or process), common in qualitative terms for many physical objects, but quantitatively individual for each of them. So, all bodies have mass and temperature, but for each of them these properties are different. The same can be said about other quantities - electrical conductivity, strength, radiation flux, etc.
Usually, when talking about measurement, they mean the measurement of physical quantities, i.e. quantities characteristic material world. These quantities are studied in natural and technical sciences(physics, chemistry, biology, electrical engineering, thermal engineering, etc.), they are the object of control and management in production (in metallurgy, mechanical engineering, instrument making, etc.). For example, the object of measurement may be the diameter of the shaft being turned, the amount of product released, the speed of liquid flow through the pipeline, the content of alloying components in the alloy, the temperature of the melt, etc.
For a more detailed study of physical quantities, they are classified into groups (Fig. 1.1). According to affiliation various groups physical phenomena physical quantities are divided into spatiotemporal, mechanical, thermal, electrical and magnetic, acoustic, light, physicochemical, etc.
Rice. 1.1. Classification of physical quantities
According to the degree of conditional independence from other quantities, physical quantities are divided into basic and derivative. Currently in International system units use seven quantities chosen as basic (independent of one another): length, time, mass, temperature, force electric current, the amount of matter and the intensity of light. Other quantities, such as density, force, energy, power, etc., are derivative (i.e., dependent on other quantities).
Based on the presence of dimension, physical quantities are divided into dimensional ones, i.e. having dimension and dimensionless.
Size a physical quantity characterizes the quantitative content of a property in each object. Meaning a physical quantity is an expression of its size in the form of a certain number of units of measurement accepted for it. For example, 0.001km; 1m; 100 cm; 1000mm – four options for representing the same size value, in in this case length.
Numeric value a physical quantity is a number expressing the ratio of the value of a quantity to the corresponding unit of measurement.
Unit is a fixed-size quantity that is conventionally assigned a numerical value of 1 and is used for quantitative expression physical quantities homogeneous with it. A unit of measurement may belong to any system of units or be non-systemic or conventional.
Obviously, the numerical value of a quantity directly depends on the chosen unit of measurement.
Units of the same quantity can differ in size, for example, meter, foot and inch, being units of length, have different sizes: 1 foot = 0.3048 m, 1 inch = 0.0254 m.
Thus, in order to measure any physical quantity, i.e. to determine its value, it is necessary to compare (compare) it with the unit of measurement of this value, and determine how many times it is greater or less than the unit of measurement.
Currently installed following definition measurements:
measurement is a set of operations to apply technical means, which stores a unit of a physical quantity, providing for finding the relationship (in explicit or implicit form) of the measured quantity with its unit and obtaining the value of this quantity.
In other words, a measurement is a physical experiment carried out using measuring instruments. Without physical experience there is no measurement. The founder of Russian metrology D.I. Mendeleev wrote: “Science begins as soon as they begin to measure; exact science is inconceivable without measure.”
It is appropriate to cite the definition of the concept of “measurement” given by the outstanding philosopher P.A. Florensky (“Technical Encyclopedia” 1931): “Measurement is the main cognitive process science and technology, through which an unknown quantity is quantitatively compared with another, homogeneous with it and considered known.”
So, if there is a certain quantity Q, the unit of measurement accepted for it is equal to [Q], then the size of the physical quantity
Q = q×[Q], (1.1)
where q is the numerical value of Q.
The expression q×[Q] is measurement result, it is composed of two parts: the numerical value q, which is the ratio of the measured quantity to the unit of measurement (it can be integer or fractional), and the unit of measurement [Q]. Typically, a unit of physical quantity is stored by the technical device used for measurement - the measuring instrument.
Let’s say that when measuring the length of a part, the measurement result is 101.6 mm. In this case, the unit of length is taken to be q = 101.6. If we take q as a unit, then q = 10.16, if we use q as a unit, then q = 40.
Equation (1.1) is called basic measurement equation, because it describes measurement as the process of comparing a physical quantity with its unit of measurement.
Different units can be chosen to measure a quantity, i.e.
Q = q 1 ×[Q] 1 = q 2 ×[Q] 2 (1.2)
From this expression it follows that the numerical value of a quantity is inversely proportional to the size of the unit: than larger size units, the smaller the numerical value of the quantity, and vice versa:
In addition, equation (1.3) shows that the size of the physical quantity Q does not depend on the choice of unit of measurement.
Thus, the numerical values of the measured quantities depend on which units of measurement are used. The choice of units has great importance to ensure comparability of measurement results; to allow arbitrariness in the choice of units means to violate the unity of measurements. That is why in most countries of the world the sizes of units of measurement are fixed by law (i.e. legalized). In Russia, in accordance with the Law “On Ensuring the Uniformity of Measurements,” units of the International System of Units are allowed to be used.
IN real world There are no units of measurement; they are the result of human activity. A unit of measurement is a certain model, according to which a certain size of a physical quantity is accepted as a unit by agreement and established by law. In addition, this model is implemented in a measuring instrument, which stores it and transmits it to all other measuring instruments that use this unit. This process of formation, storage and use of units of physical quantities has developed over the last two centuries.
A measurement is significant only when its result can be used to estimate the true value of the quantity. When analyzing measurements, one should clearly distinguish between these two concepts: the true value of a physical quantity and its empirical manifestation - the result of the measurement.
Any measurement result contains an error due to imperfection of measurement tools and methods, the influence of external conditions and other reasons. The true value of the measured quantity remains unknown. It can only be imagined theoretically. The result of measuring a quantity only approaches its true value to a greater or lesser extent, i.e. represents his assessment. For more information about measurement error, see Chapter. 2 “Measurement errors.”
Measurement scales
Measurement scale serves as the initial basis for measuring this quantity. It represents an ordered collection of quantity values.
Practical activities led to the formation various types measurement scales of physical quantities, the main ones of which are four, discussed below.
1. Scale of order (ranks) represents a ranked series – a sequence of quantities, ordered in ascending or descending order, characterizing the property being studied. It allows you to establish an order relationship based on increasing or decreasing quantities, but there is no way to judge how many times (or how much) one quantity is larger or smaller compared to another. In order scales, in some cases there may be a zero (zero mark); the fundamental thing for them is the absence of a unit of measurement, because its size cannot be determined; in these scales it is impossible to carry out mathematical operations(multiplication, summation).
An example of an order scale is the Mohs scale for determining the hardness of bodies. This is a scale with reference points, which contains 10 reference (reference) minerals with different hardness numbers. Examples of such scales are also the Beaufort scale for measuring wind force (speed) and the Richter earthquake scale (seismic scale).
2. Interval (difference) scale differs from the order scale in that for measured quantities, not only order relations are introduced, but also summations of intervals (differences) between various quantitative manifestations of properties. Difference scales may have conventional reference zeros and measurement units established by agreement. Using an interval scale, you can determine how much one value is greater or less than another, but you cannot tell by how many times. Interval scales measure time, distance (if the beginning of the journey is not known), temperature in Celsius, etc.
Interval scales are more advanced than order scales. In these scales, additive mathematical operations (addition and subtraction) can be performed on quantities, but multiplicative ones (multiplication and division) cannot be performed.
3.Relationship scale describes the properties of quantities for which the relations of order, summation of intervals and proportionality are applicable. In these scales there is a natural zero and the unit of measurement is established by agreement. The ratio scale serves to present measurement results obtained in accordance with the basic measurement equation (1.1) by experimentally comparing the unknown quantity Q with its unit [Q]. Examples of ratio scales are scales of mass, length, speed, and thermodynamic temperature.
The relationship scale is the most advanced and most widespread of all measuring scales. This is the only scale on which you can set the value of the measured size. Any mathematical operations are defined on the ratio scale, which allows you to make multiplicative and additive corrections to the readings on the scale.
4. Absolute scale has all the features of a ratio scale, but in addition there is a natural, unambiguous definition of the unit of measurement. Such scales are used for measurements relative values(gain, attenuation, useful action, reflection, absorption, amplitude modulation, etc.). A number of such scales have boundaries between zero and one.
Interval and ratio scales are combined under the term “metric scales.” The order scale is classified as a conditional scale, i.e. to scales in which the unit of measurement is not defined and is sometimes called non-metric. Absolute and metric scales are classified as linear. Practical implementation measurement scales is carried out by standardizing both the scales and units of measurement themselves, and, if necessary, the methods and conditions for their unambiguous reproduction.
2.1 Physical quantity, its qualitative and quantitative characteristics. Unit of physical quantity
In the broad sense of the word “magnitude” is a multi-species concept. For example, quantities such as price and cost of goods are expressed in monetary units. Another example is the value of the biological activity of medicinal substances, which is expressed in the corresponding units, denoted by the letters I.E. For example, recipes indicate the amount of many antibiotics and vitamins in these units.
Modern metrology is interested in physical quantities. Physical magnitude - this is a property that is qualitatively common for many objects (systems, their states and processes occurring in them), but quantitatively individual for each object. Individuality in quantitative terms should be understood in the sense that a property can be for one object a certain number of times greater or less than for another. All electrical and radio engineering quantities are typical examples of physical quantities.
A formalized reflection of the qualitative difference between measured quantities is their dimension. Dimension is denoted by the symbol dim, which comes from the word dimension, which, depending on the context, can be translated as both size and dimension. The dimensions of basic physical quantities are indicated by the corresponding capital letters. For example, for length, mass and time
dim l = L; dim m = M; dim t = T. (2.1)
The dimensions of derived physical quantities can be expressed through the dimensions of basic physical quantities using a power monomial:
where dim z is the dimension of the derivative of the physical quantity z;
L, M, T, … - dimensions of the corresponding basic physical quantities;
α, β, γ, … - indicators of dimension.
Each dimension indicator can be positive or negative, integer or fractional number, zero. If all dimension indicators are equal to zero, then such a quantity is called dimensionless. It can be relative, if defined as the ratio of quantities of the same name (for example, relative dielectric constant), and logarithmic, if defined as the logarithm of a relative quantity (for example, the logarithm of the voltage ratio).
So, dimension is a qualitative characteristic of a physical quantity.
Dimensional theory is widely used to quickly check the correctness of complex formulas. If the dimensions of the left and right sides of the equation do not coincide, then an error should be looked for in the derivation of the formula, no matter what field of knowledge it belongs to.
A quantitative characteristic of a physical quantity is its size . Obtaining information about the size of a physical or non-physical quantity
is the content of any dimension. The simplest way to obtain such information, which allows one to get some idea of the size of the measured value, is to compare it with another according to the principle “which is larger (smaller)?” or “which is better (worse)?” More detailed information how much more (less) or how many times better (worse) is sometimes not even required. In this case, the number of sizes compared with each other can be quite large. Arranged in ascending or descending order, the sizes of the measured quantities form order scale . For example, at many competitions and competitions, the skill of performers and athletes is determined by their place in the final table. The latter, therefore, is a scale of order - a form of representation of measurement information that reflects the fact that the skill of some is higher than the skill of others, although it is not known to what extent (how much or how many times). Having arranged people by height, it is possible, using a scale of order, to draw a conclusion about who is taller than whom, but it is impossible to say how much taller. The arrangement of sizes in ascending or descending order in order to obtain measurement information on an order scale is called ranking .
To facilitate measurements on the order scale, some points on it can be fixed as reference points (reference) . Knowledge, for example, is measured on a reference scale of order, which has the following form: unsatisfactory, satisfactory, good, excellent. Reference scale points can be assigned numbers called points . For example, the intensity of earthquakes is measured on the twelve-point international seismic scale MSK-64, and the strength of wind is measured on the Beaufort scale. The strength of sea waves, the hardness of minerals, the sensitivity of photographic films and many other quantities are also measured using reference scales. Reference scales are especially widespread in the humanities, sports, and art.
The disadvantage of reference scales is the uncertainty of the intervals between reference points. Therefore, points cannot be added, subtracted, multiplied, divided, etc. More advanced in this regard are scales composed of strictly defined intervals. It is generally accepted, for example, to measure time on a scale divided into intervals equal to the period of revolution of the Earth around the Sun. These intervals (years) are in turn divided into smaller ones (days), equal to the period of rotation of the Earth around its axis. The day is in turn divided into hours, hours into minutes, minutes into seconds. This scale is called interval scale . Using the interval scale, one can already judge not only that one size is larger than another, but also how much larger, i.e. on the interval scale the following are defined mathematical operations like addition and subtraction. Regardless of the chronology, the fundamental turning point in the course of the Second World War occurred at Stalingrad 700 years after Alexander Nevsky defeated the German knights of the Livonian Order on the ice of Lake Peipsi. But if we ask the question of “how many times” later this event occurred, it turns out that according to our Gregorian style - in 1942/1242 = 1.56 times, according to the Julian calendar, counting time from the “creation of the world” - in 7448/6748 = 1.10 times, according to the Jewish calendar, where time is counted “from the creation of Adam,” - 5638/4938 = 1.14 times, and according to the Mohammedan chronology, which began from the date of Mohammed’s flight from Mecca to the holy city of Medina , - in 1320/620 = 2.13 times. Therefore, it is impossible to say on an interval scale how many times one size is larger or smaller than another. This is explained by the fact that the scale is known from the interval scale, and the origin can be chosen arbitrarily.
Interval scales are sometimes obtained by proportionally dividing the interval between two reference points. Thus, on the Celsius temperature scale, one degree is a hundredth part of the interval between the melting temperature of ice, taken as the starting point, and the boiling point of water. On the Reaumur temperature scale, the same interval is divided into 80 degrees, and on the Fahrenheit temperature scale - into 180 degrees, with the starting point shifted by 32 degrees Fahrenheit towards low temperatures.
If one of the two reference points is chosen as one in which the size is not accepted equal to zero(which leads to negative values), and equal to zero in fact, using such a scale it is already possible to count the absolute value of the size and determine not only how much one size is larger or smaller than another, but also how many times it is larger or smaller. This scale is called relationship scale. An example of this is the Kelvin temperature scale. In it, absolute zero temperature is taken as the starting point, at which the thermal motion of molecules stops. There cannot be a lower temperature. The second reference point is the melting temperature of ice. On the Celsius scale, the interval between these reference points is approximately 273 degrees Celsius. Therefore, on the Kelvin scale it is divided by 273 equal parts, each of which is called Kelvin and is equal to a degree Celsius, which greatly facilitates the transition from one scale to another.
The relationship scale is the most advanced of all the scales considered. It defines the largest number of mathematical operations: addition, subtraction, multiplication, division. But, unfortunately, constructing a relationship scale is not always possible. Time, for example, can only be measured on an interval scale.
Depending on what intervals the scale is divided into, the same size is presented differently. For example, 0.001 km; 1m; 10 dm; 100 cm; 1000 mm - five versions of the same size. They are called values physical quantity. Thus, the value of a physical quantity is an expression of its size in certain units of physical quantity. The abstract number included in the expression is called numerical values eat. It shows how many units the measured size is greater than zero or how many times it is more than one measurements. Thus, the value of a physical quantity z is determined by its numerical value(z) and some size [z], taken as unit of physical quantity
z=(z)·[z]. (2.3)
Equation (2.3) is called the basic measurement equation. From this equation it follows that the value of (z) depends on the size of the selected unit [z]. The smaller the selected unit, the larger the numerical value for a given measured quantity. If, when measuring the value z, instead of the unit [z] we take another unit, then expression (2.3) will take the form
z=(z 1 )·.
Taking into account equation (2.3), we obtain
(z)·[z]=(z 1 )·,
(z 1 )=(z)·[z]/.
From this formula it follows that to move from the value (z) expressed in one unit [z] to the value (z 1) expressed in another unit, it is necessary to multiply (z) by the ratio of the accepted units.
2.2 Emergence, development and unification of units
physical quantities. Creating Metric Measures
Units of physical quantities began to appear from the moment when a person had a need to express something quantitatively. This "something" could be a number of objects. In this case, the measurement was extremely simple, since it consisted of counting the number of objects, and the unit was one object. But then the task became more complicated, since it became necessary to determine the number of objects (liquids, granular bodies, etc.) that could not be counted by piece. Volume measures have appeared. The need to measure lengths and weights gave rise to the appearance of length and weight measures. For example, the first measures of length were parts of the human body: span, foot, elbow, as well as step, etc. In addition to the quantitative determination of the properties of the body and substances, a non-
the need to quantitatively characterize processes. This is how the need to measure time arose. The first unit of time was the day - the change of day and night.
The second stage in the development of units was associated with the development of science and the progress of scientific experiment technology. It was discovered that the properties of physical objects, which were the basis for the creation of measures that reproduce units of value, do not have the degree of constancy and reproducibility that are required in science, technology and other branches of human activity. The second stage is characterized by the rejection of units of quantities reproduced by nature and their consolidation in “material” samples. The most characteristic of the transition from the first stage to the second is the history of the creation of metric measures. Beginning with precise measurements of the “natural” unit - the length of the Earth's meridian - it ended with the creation of a material standard for the unit of length - the meter.
The third stage in the development of units of physical quantities was a consequence of the rapid development of science and increased requirements for measurement accuracy. It turned out that man-made material (object) standards of units of physical quantities cannot ensure the storage and transmission of these units with the accuracy that has become necessary. The discovery of new physical phenomena, the emergence and development of atomic and nuclear physics made it possible to find ways to more accurately reproduce units of physical quantities. However, the third stage is not a return to the principles of the first stage. The difference between the third stage and the first is the separation of units of physical quantities from the measure, from the quantitative characteristics of the properties of physical objects that serve to reproduce them. The units of measurement remained overwhelmingly the same as they were established in the second stage. A typical example is the unit of length. The discovery of the ability to reproduce length using the wavelength of monochromatic light did not change the unit of length, the meter. The meter remained a meter, but the use of light wavelengths made it possible to increase the accuracy of its reproduction by one decimal place.
However, now even this definition of the meter does not allow reproducing the meter with sufficient accuracy to solve some problems. Therefore, at the XVII General Conference of Weights and Measures (1983), a new definition of the meter was adopted, allowing the latter to be reproduced with greater accuracy.
The prospect for the development of metrology in terms of units of physical quantities is to further increase the accuracy of reproduction of existing ones. The need to establish new units may arise when fundamentally new physical objects are discovered.
Initially, units of physical quantities were chosen arbitrarily, without any connection with each other, which created great difficulties. A significant number of arbitrary units of the same quantity made it difficult to compare the results of measurements made by different observers. Each country, and sometimes each city, created its own units. Converting one unit to another was very difficult and led to a significant decrease in accuracy.
In addition to the indicated variety of units, which can be called “territorial,” there was a variety of units used in various areas of human activity. Within the same industry, different units of the same size were also used.
With the development of technology, as well as international relations, the difficulties of using and comparing measurement results due to differences in units increased and hampered further scientific and technological progress. For example, in the second half of the 18th century. in Europe there were up to a hundred feet of different lengths, about fifty different miles, over 120 different pounds. In addition, the situation was further complicated by the fact that the relationships between submultiples and multiples were unusually diverse. For example, 1 foot = 12 inches = 304.8 mm.
In 1790, France decided to create a system of new measures “based on an unchangeable prototype taken from nature, so that all nations could adopt it.” It was proposed to consider the length of the ten-millionth part of the quarter of the Earth's meridian passing through Paris as a unit of length. This unit was called the meter. To determine the size of the meter, measurements were taken from 1792 to 1799 of the arc of the Parisian meridian. The mass of 0.001 m3 of pure water at the temperature of the highest density (+4 °C) was taken as a unit of mass; this unit was called the kilogram. With the introduction of the metric system, not only was the basic unit of length taken from nature established, but also a decimal system for the formation of multiples and submultiples was adopted, corresponding to the decimal system of numerical counting. The decimalization of the metric system is one of its most important advantages.
However, as subsequent measurements showed, a quarter of the Parisian meridian contains not 10,000,000, but 10,000,856 originally determined meters. But this number cannot be considered final, since even more precise measurements give a different meaning. In 1872, the International Prototype Commission decided to move from units of length and mass based on natural standards to units based on conventional material standards (prototypes).
In 1875, a diplomatic conference was convened at which 17 states signed the Meter Convention. According to this convention:
International prototypes of the meter and kilogram were installed;
the International Bureau of Weights and Measures was created - a scientific institution, the funds for the maintenance of which were pledged to be allocated by the states that signed the convention;
the International Committee of Weights and Measures was established, consisting of scientists from different countries, one of whose functions was to manage the activities of the International Bureau of Weights and Measures;
the convening of the General Conference on Weights and Measures was established once every six years.
Samples of the meter and kilogram were made from an alloy of platinum and iridium. The prototype of the meter was a platinum-iridium line measure with a total length of 102 cm, at distances of 1 cm from the ends of which strokes were applied, defining the unit of length - the meter.
In 1889, the First General Conference on Weights and Measures met in Paris, which approved international prototypes from among the newly manufactured samples. The prototypes of the meter and kilogram were deposited with the International Bureau of Weights and Measures. The remaining samples of the meter and kilogram were distributed by the General Conference by lot among the states that signed the Meter Convention. Thus, in 1899 the establishment of metric measures was completed.
2.3 Principles of formation of a system of units of physical quantities
The concept of a system of units of physical quantities was first introduced by the German scientist K. Gauss. According to his method, when forming a system of units, several quantities independent of each other are first established or chosen arbitrarily. The units of these quantities are called main , since they are the basis for building the system. Basic units are established in such a way that, using the mathematical relationship between quantities, it would be possible to form units of other quantities. Units expressed in terms of base units are called derivatives . The complete set of basic and derived units established in this way is a system of units of physical quantities.
You can select following features the described method for constructing a system of units of physical quantities.
First, the method of constructing the system is not related to the specific sizes of the basic units. For example, as one of the basic units we can
choose a unit of length, but which one is indifferent. It could be a meter, or an inch, or a foot. But the derived unit will depend on the choice of the base unit. For example, the derived unit of area would be square meter, or square inch, or square foot.
Secondly, in principle, the construction of a system of units is possible for any quantities between which there is a relationship expressed in mathematical form in the form of an equation.
Thirdly, the choice of quantities whose units should become basic is limited by considerations of rationality, and primarily by the fact that the optimal choice is the minimum number of basic units, which would allow the formation of the maximum number of derivative units.
Fourthly, they strive for the system of units to be coherent. The derived unit [z] can be expressed in terms of the basic [L], [M], [T], ... using the equation
where K is the proportionality coefficient.
Coherence (consistency) of the system of units lies in the fact that in all formulas that determine derived units depending on the basic ones, the proportionality coefficient is equal to one. This provides a number of significant advantages, simplifies the formation of units of various quantities, as well as carrying out calculations with them.
2.4 Systems of units of physical quantities. International System of Units SI
Initially, unit systems were created based on three units. These systems covered a wide range of quantities, conventionally called mechanical. They were built on the basis of those units of physical quantities that were adopted in one country or another. Of all these systems, preference can be given to systems built on units of length - mass - time as the main ones. One of the systems built according to this scheme for metric units is the meter-kilogram-second (MKS) system. In physics, it was convenient to use the centimeter-gram-second (CGS) system. The ISS and SGS systems are coherent in terms of units of mechanical quantities. Serious difficulties were encountered when using these systems to measure electrical and magnetic quantities.
For some time, the so-called technical system of units was used, built according to the length - force - time scheme. When using metric units, the main units of this system were meter - kilogram-force - second (MCGSS). The convenience of this system was that the use of the force unit as one of the main ones simplified the calculations and derivations of dependencies for many quantities used in technology. Its disadvantage was that the unit of mass in it was numerically equal to 9.81 kg, and this violates the metric principle of decimal measures. The second drawback is the similarity of the name of the unit of force - kilogram-force and the metric unit of mass - kilogram, which often leads to confusion. The third disadvantage of the MKGSS system is its inconsistency with practical electrical units.
Since systems of mechanical units did not cover all physical quantities, for individual industries Science and technology unit systems were expanded by adding another basic unit. This is how the system of thermal units meter - kilogram - second - degree temperature scale (MCSG) appeared. The system of units for electrical and magnetic measurements is obtained by adding the unit of current - ampere (MCSA). The system of luminous units contains as the fourth basic unit the unit of luminous intensity - the candela.
The presence of a number of systems of units of measurement of physical quantities and big number non-systemic units, the inconveniences that arise in practice in connection with recalculations when moving from one system to another have caused the need to create a single universal system of units that would cover all branches of science and technology and would be accepted on an international scale.
In 1948, at the IX General Conference on Weights and Measures, proposals were made to adopt a unified practical system of units. The International Committee of Weights and Measures conducted an official survey of the opinions of the scientific, technical and pedagogical communities of all countries and, based on the responses received, made recommendations for establishing a unified practical system of units. The X General Conference (1954) adopted as basic units new system the following: length - meter; mass - kilogram; time - second; current strength - ampere; thermodynamic temperature - kelvin; luminous intensity - candela. Subsequently, the seventh basic unit was adopted - the amount of substance - the mole. After the conference, a list of derived units of the new system was prepared. In 1960, the XI General Conference on Weights and Measures finally adopted the new system, giving it the name International System of Units (System International) with the abbreviation "SI", in Russian transcription "SI".
The adoption of the International System of Units served as an incentive for the transition to metric units in a number of countries that retained national units (England, USA, Canada, etc.). In 1963, GOST 98567-61 “International System of Units” was introduced in the USSR, according to which SI was recognized as preferable. Along with this, the USSR had eight state standards for units. In 1981, GOST 8.417-81 "GSI. Units of physical quantities" was put into effect, covering all branches of science and technology and based on the International System of Units.
SI is the most advanced and universal of all that have existed to date. The need for a single International System of Units is so great, and its advantages so convincing, that this system a short time received wide international recognition and distribution. The International Organization for Standardization (ISO) has adopted the International System of Units in its recommendations for units. The United Nations Educational, Scientific and Cultural Organization (UNESCO) has called on all member countries of the organization to adopt the International System of Units. The International Organization of Legal Metrology (OIML) recommended that member states of the organization introduce the International System of Units by law and calibrate measuring instruments in SI units. SI entered into unit recommendations International Union pure and applied physics, the International Electrotechnical Commission and other international organizations.
2.5 Basic, supplementary and derived units
The basic SI units have the following definitions.
The unit of length is the meter (m) - the length of the path traveled by light in a vacuum in 1/299792458 of a second.
The unit of mass is the kilogram (kg) - a mass equal to the mass of the international prototype of the kilogram.
The unit of time is a second (s) - a time equal to 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 is the ampere (A) - the strength of a constant 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 cause between these conductors a force equal to 2- 10" 7 N per meter of length.
The unit of thermodynamic temperature is kelvin (K) - 1/273.16 part of the thermodynamic temperature of the triple point of water. The International Committee of Weights and Measures has allowed the expression of thermodynamic temperature in degrees Celsius: t = T-273.15 K, where t is the temperature Celsius; T - Kelvin temperature.
The unit of luminous intensity - candela (cd) - is equal to the luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540-10 12 Hz, energetic force of light in this direction is 1/683 W/sr.
The unit of quantity of a substance - mole - is the amount of substance of a system containing the same number of structural elements as there are atoms in a 12C nuclide weighing 0.012 kg.
The SI includes two additional units for plane and solid angles, which are necessary to form the derived units associated with angular quantities. Angular units cannot be included among the basic ones; at the same time, they cannot be considered derivatives, since they do not depend on the size of the basic units.
The unit of plane angle is radian (rad) - the angle between two radii of a circle, the length of the arc between which is equal to the radius. In degrees, a radian is equal to 57° 17" 44.8".
The unit of solid angle - steradian (sr) - is equal to the solid angle with its 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.
Derived SI units are formed on the basis of laws establishing relationships between physical quantities or on the basis of definitions of physical quantities. The corresponding derived SI units are derived from the relationship equations between quantities (defining equations) expressing a given physical law or definition, if all other quantities are expressed in SI units.
More detailed information about derived SI units is given in the works.
2.6 Dimension of physical quantities
The dimension of the derived SI unit of the physical quantity z in general form is determined from the expression
,
(2.5)
where L, M, T, I, θ, N, J are the dimensions of physical quantities, the units of which are taken as basic;
α, β, γ, ε, η, μ, λ are exponents of the degree to which the corresponding quantity is included in the equation that determines the derived quantity z.
Expression (2.5) determines the dimension of the physical quantity z; it reflects the relationship of the quantity z with the basic quantities of the system, in which the proportionality coefficient is taken equal to 1.
Here are examples of the dimensions of derived units in relation to SI units:
for unit area;
for unit of speed;
for a unit of acceleration;
for unit of power;
for a unit of heat capacity;
for a unit of heat capacity;
for the unit of illumination.
Dimensions determine the connections between physical quantities, but they do not yet determine the nature of the quantities. You can find a number of quantities whose dimensions of derived units coincide, although these quantities are different in nature. For example, the dimensions of work (energy) and moment of force are the same and equal to L 2 M T 2.
2.7 Multiples and submultiples
The sizes of metric units, including SI units, are inconvenient for many practical cases: either too large or very small. Therefore, they use multiple and submultiple units, i.e. units that are an integer number of times larger or smaller than the units of a given system. Decimal multiples and submultiples are widely used, which are obtained by multiplying the original units by the number 10 raised to a power. To form the names of decimal multiples and submultiples, appropriate prefixes are used. In table 2.1 provides a list of currently used decimal factors and their corresponding prefixes. The designation of the prefix is written together with the designation of the unit to which it is attached. Moreover, prefixes can only be attached to simple names of units that do not contain prefixes. Connecting two or more consoles in a row is not allowed. For example, the name “micromicrofarad” cannot be used, but the name “picofarad” must be used.
When forming the name of a decimal multiple or submultiple unit from a unit of mass - a kilogram, a new prefix is added to the name "gram" (megagram 1 Mg = 10 3 kg = 10 6 kg, milligram 1 mg = 
kg== 
G).
In multiple and submultiple units of area and volume, as well as other quantities formed by raising to a power, the exponent refers to the entire unit taken together with the prefix, for example: 1 
=
=
;
=
. It is incorrect to attribute the prefix to the original unit raised to a power.
Decimal multiples and submultiples, the names of which are formed using prefixes, are not included in the coherent system of units. Their application in relation to the system should be considered as a rational way of representing small and large numerical values. When substituting prefixes into a formula, they are replaced by their corresponding factors. For example, the value 1 pF (1 picofarad) when substituted into the formula is written 
F.
Table 2.1
|
Factor |
Console |
||
|
Name |
Designation |
||
|
international |
|||
|
1 000 000 000 000 000 000= 1 000 000 000 000 000= 1 000 000 000 000= 1 000 000 000= 1 000 000= 1 000= 100= 10= 0,1= 0,01= 0,001= 0,000
001= 0,000
000 001= 0,000
000 000 001= 0,000
000 000 000 001= 0,000
000 000 000 000 001= |
exa peta tera giga mega kilo hecto deca deci santi micro nano pico femto atto | ||
The prefixes deca, hecto, deci and santi are used relatively rarely, since in most cases they do not create noticeable advantages. Thus, they abandoned the use of the hectowatt unit when recording the power of electrical devices, since it is more convenient to keep records in kilowatts, but in some cases these prefixes are very firmly rooted, for example, a centimeter, a hectare. The unit are (100 m2) is practically not used, but the hectare has found very wide use everywhere. It successfully replaced the Russian tithe: 1 hectare = 0.9158 tithe.
When choosing prefixes for the name of a particular unit, a certain moderation should be observed. For example, the names decameter and hectometer have not been used, and only kilometer is widely used. But further the use of prefixes to the names of units that are multiples of the meter did not come into practice: neither megameter, nor gigameter, nor terameter are used.
The choice of a decimal multiple or submultiple SI unit is dictated primarily by the convenience of its use. From the variety of multiples and submultiples that can be formed using prefixes, a unit is selected that leads to numerical values of the quantity acceptable in practice. In most cases, multiples and submultiples are chosen so that the numerical values of the quantity are in the range from 0.1 to 1000.
Some submultiples and multiple units received special names at one time, which have been preserved to this day. For example, as units that are multiples of a second, not decimal multiples are used, but historically established units: 1 min = 60 s; 1 hour = 60 min = 3600 s; 1 day = 24 hours = 86400 s; 1 week = 7 days = 604800 s. To form fractional units of a second, decimal coefficients are used with the corresponding prefixes to the name: millisecond (ms), microsecond (μs), nanosecond (not).
2.8 Relative and logarithmic quantities and
Relative and logarithmic quantities and their units are widely used in science and technology, which characterize the composition and properties of materials, the ratio of energy and force quantities, etc. Such characteristics are, for example, relative elongation, relative density, relative dielectric and magnetic permeability, gain and weakening of capacities, etc.
Relative value
represents a dimensionless ratio of a physical quantity to a physical quantity of the same name, taken as the initial one. The number of relative quantities also includes the relative atomic or molecular masses of chemical elements, expressed in relation to one twelfth (1/12) of the mass of carbon - 2. Relative quantities can be expressed either in dimensionless units (when the ratio of two quantities of the same name is equal to 1), or in percent (when the ratio is 
), or in ppm (the ratio is ), or in parts per million (the ratio is ).
Logarithmic value
represents the logarithm (decimal, natural, or base 2) of the dimensionless ratio of two physical quantities of the same name. Sound pressure levels, gain, attenuation, frequency interval, etc. are expressed as logarithmic values. The unit of the logarithmic value is Bel (B), determined by the following relationship: 1 B = log (P2/Pl) with P2 = 10 P1, where PI, P2 are energy quantities of the same name (power, energy, energy density, etc.) . If a logarithmic value is taken for the ratio of two “power” quantities of the same name (voltage, current, pressure, field strength, etc.), Bel is determined by the formula 1 B = 2·lg(F2/Fl) with F2= 
·F1. The subunit of white is the decibel (dB), equal to 0.1 B.
For example, in the case of the characteristics of amplification of electrical powers with a ratio of the received power P2 to the original one equal to 10, the amplification will be equal to 1 B or 10 dB, with a change in power of 1000 - 3 B or 30 dB.
2.9 Units of physical quantities of the GHS system
The GHS system still retains its independent significance in theoretical physics. One basic unit of this system - the second - coincides with the SI basic unit of time, and the other two basic GHS units - the centimeter and the gram - are submultiples of the SI units. However, it is impossible to consider the GHS system as some kind of derivative or part of the International system. Firstly, the ratios of the proportions of the basic units are not the same (0.01; 0.001; 1). Secondly, when forming CGS units for electrical and magnetic quantities, as a rule, the equations of electromagnetism are used in a non-rationalized form. In connection with this, the sizes of the units have changed, and in cases where GHS units had special names, and the names also changed. Thus, the CGS unit of magnetomotive force - hilbert - in SI units is equal to 10/(4 
) ampere, and the CGS unit of magnetic field strength - örstad - in SI units is equal to 10 3 /(4· ) ampere per meter.
Some other GHS units have special names, but they are decimal submultiples of SI units and therefore the transition from units of one system to units of another is not difficult. Such GHS units include the units given in Table 2.2. Many GHS units do not have special names. The most commonly used GHS units are given in the works.
Table 2.2
|
Magnitude |
Name of SI unit |
Unit name |
Value in SI units |
|
Work, energy Dynamic viscosity Kinematic viscosity Magnetic flux Magnetic induction |
Square meter per second |
Maxwell |
|
N
J
Wb2.10 Non-system units
Non-systemic are those units of physical quantities that are not included in the system of units used in each specific case, either as basic or as derivatives. Non-system units, to one degree or another, are always some obstacle to the implementation of a system of units. When carrying out calculations using theoretical formulas, it is necessary to reduce all non-system units to the corresponding units of the system. In some cases this is not difficult, as, for example, with decimal multiples or fractions. In other cases, translating units is complex and laborious and is often a source of error. In addition, individual non-systemic units, due to their size, turn out to be very convenient for some branches of science, technology, or for everyday use, and abandoning them is associated with a number of inconveniences. Examples of such units can be: for length - astronomical unit, light year, parsec; for mass - atomic mass unit; for square - bari; for strength - dyna; for work - erg; for magnetic flux - maxwell; for magnetic induction - gauss.
2.11 Names and designations of units
Several types can be distinguished in the names of units. First of all, these are names that, to one degree or another, succinctly reflect the physical essence of the quantity. These names include: meter (measure), candela (candle), dina (force), calorie (from the word heat), etc. It should be recognized that such names are the most convenient. Next come the names of derived units formed in strict accordance with physical laws. For example, joule per kilogram kelvin [J/(kg K)] - unit
specific heat capacity; kilogram-meter squared per second (kg m 2 /s) - unit of angular momentum, etc.
The cumbersomeness of naming derived units, and in some cases the difficulty of finding a name for a unit that reflects the physical essence of the quantity, led to the assignment of short and easy-to-pronounce names to many units. It was decided to assign names to such units after the names of outstanding scientists. As examples, we can point to such names as kelvin, ampere, volt, watt, hertz, etc.
The names of some units are related to the graduation of the scale. These units include: temperature degree, angular degree (minute, second), millimeter of mercury, millimeter of water.
The names of some units are abbreviations, i.e. abbreviations according to initial letters. For example, the unit of reactive power is called "var" from the first letters of the words "volt-ampere reactive". The unit of equivalent radiation dose is called "rem" from the first letters of the words "biological equivalent of rad".
When designating, writing these designations and reading them, the following rules are used.
In most cases, abbreviated unit notations are used to designate units after a numerical expression. These abbreviations consist of one, two or three first letters of the unit name. The designations of derived units that do not have a special name are compiled from the designations of other units according to the formula for their formation (not necessarily from the designations of the basic units).
The abbreviated designation of units, the name of which is derived from the surname of the scientist, is written with a capital letter. For example: ampere - A; newton -N; pendant - Cl; joule - J, etc. In the designation of units, a dot as an abbreviation sign is not used, except in cases of abbreviation of words that are included in the name of the unit, but are not themselves names of units, for example, mmHg. (millimeters of mercury).
In the presence of decimal in the numerical value of the quantity, the unit designation should be placed after all numbers, for example: 53.24 m; 8.5 s; -17.6 °C.
When indicating the values of quantities with maximum deviations, the numerical value with maximum deviations should be enclosed in brackets and the unit designation should be placed after the brackets or the unit designation should be placed after the numerical value of the quantity and after its maximum deviations, for example: (25±10) °C or 25 °C ± 10 °C; (120±5) s or 120 s ± 5 s.
In calculations, when repeating the equal sign, the unit designation is given only in the final result, for example:
.
When writing notations for derived units, the notations for the units included in the product are separated by dots on the center line as multiplication signs, for example: N m (newton meter); N·s/m2 (newton-second per square meter). To indicate the operation of dividing one unit by another, a slash is usually used, for example: m/s. It is allowed to use a horizontal line (for example, 
) or representing a unit as a product of unit symbols raised to positive or negative powers (for example, 
). When using a slash, the product of units in the denominator should be enclosed in parentheses, for example: W/(m K).
The use of more than one oblique or horizontal line in the designation of a derived unit is not allowed: for example, the unit of heat transfer coefficient - watt per square meter kelvin - should be designated W/( 
·TO), 
or 
.
The designations of units by case and number do not change, with the exception of the designation “light year”, which in the genitive plural takes the form “light years”.
When the name corresponds to a product of units, the prefix is attached to the name of the first unit included in the work.
For example, 
Nm should be called kilonewton meter (kNm) rather than newton kilometer (Nkm).
When the name corresponds to the ratio of units, the prefix is also attached to the name of the first unit included in the numerator. An exception to this rule is the basic SI unit, the kilogram, which can be included in the denominator without limitation.
In the names of units of area and volume, the adjectives “square” and “cubic” are used, for example, square meter, cubic centimeter. If the second or third power of length does not represent area or volume, then in the name of the unit, instead of the words “square” or “cubic”, the expressions “squared”, “to the third power”, etc. should be used, for example, unit of moment momentum - kilogram-meter per
square per second (kg m 2 /s).
To form the name of multiple and submultiple units from a unit that represents a degree of some original unit, a prefix is attached to the name of the original unit. For example, square meter ( ), square kilometer ( ) and so on.
In products of derived units formed as products of units, only the last name and the adjectives related to it “square” and “cubic” are declined. The names of the units in the denominator are written and read with the preposition “on”, for example, meter per second squared. The exception is units of quantities that depend on time to the first power; in this case, the name of the unit in the denominator is written and read with the preposition “in”, for example, meter per second. When declension of the names of units containing a denominator, only the part corresponding to the numerator changes.