Physical bodies use quantities that characterize space, time and the body in question: length l, time t and mass m. Length l is defined as the geometric distance between two points in space.

IN International system units (SI) The unit of length is the meter (m).

\[\left=m\]

The meter was originally defined as ten millionths of a quarter of the earth's meridian. By this, the creators of the metric system sought to achieve invariance and accurate reproducibility of the system. The meter standard was a ruler made of a platinum alloy with 10% iridium, the cross-section of which was given a special X-shape to increase bending rigidity with a minimum volume of metal. In the groove of such a ruler there was a longitudinal flat surface, and the meter was defined as the distance between the centers of two strokes applied across the ruler at its ends, at a standard temperature of 0$()^\circ$ C. Currently, due to increased requirements for accuracy measurements, the meter is defined as the length of the path traveled in a vacuum by light in 1/299,792,458 of a second. This definition was adopted in October 1983.

The time t between two events at a given point in space is defined as the difference in the readings of a clock (a device whose operation is based on a strictly periodic and uniform physical process).

The International System of Units (SI) uses the second (s) as the unit of time.

\[\left=c\]

According to modern ideas, 1 second is a time interval equal to 9,192,631,770 periods of radiation corresponding to the transition between two hyperfine levels of the ground (quantum) state of the cesium-133 atom at rest at 0° K in the absence of disturbance by external fields. This definition was adopted in 1967 (clarification regarding temperature and resting state appeared in 1997).

The mass m of a body characterizes the force that must be applied to bring it out of the equilibrium position, as well as the force with which it is capable of attracting other bodies. This indicates the dualism of the concept of mass - as a measure of the inertia of a body and a measure of its gravitational properties. As experiments show, gravitational and inert mass bodies are equal at least, within the limits of measurement accuracy. Therefore, except for special cases, they simply talk about mass - without specifying whether it is inertial or gravitational.

The International System of Units (SI) uses the kilogram as a unit of mass.

$\left=kg\ $

The international prototype of the kilogram is taken to be the mass of a cylinder made of a platinum-iridium alloy, with a height and diameter of about 3.9 cm, stored in the Breteuil Palace near Paris. The weight of this reference mass, equal to 1 kg at sea level at latitude 45$()^\circ$, is sometimes called kilogram-force. Thus, it can be used either as a standard of mass for an absolute system of units, or as a standard of force for a technical system of units in which one of the basic units is the unit of force. In practical measurements, 1 kg can be considered equal to the weight of 1 liter clean water at a temperature of +4°C.

In continuum mechanics, the main units of measurement are thermodynamic temperature and amount of matter.

The SI unit of temperature is Kelvin:

$\left[T\right]=K$.

1 Kelvin is equal to 1/273.16 of the thermodynamic temperature of the triple point of water. Temperature is a characteristic of the energy that molecules possess.

The amount of substance is measured in moles: $\left=Mole$

1 mole is equal to the amount of substance in a system containing the same number of structural elements as there are atoms in carbon-12 weighing 0.012 kg. When using mole structural elements must be specified and may be atoms, molecules, ions, electrons and other particles or specified groups of particles.

Other units of measurement of mechanical quantities are derived from the basic ones, representing their linear combination.

Derivatives of length are area S and volume V. They characterize areas of space, respectively, of two and three dimensions, occupied by extended bodies.

Units of measurement: area - square meter, volume - cubic meter:

\[\left=m^2 \left=m^3\]

The SI unit of speed is meters per second: $\left=m/s$

The SI unit of force is newton: $\left=Н$ $1Н=1\frac(kg\cdot m)(s^2)$

The same derived units of measurement exist for all other mechanical quantities: density, pressure, momentum, energy, work, etc.

Derived units are obtained from basic units using algebraic operations such as multiplication and division. Some of the derived units in the SI are given their own names, for example, the unit radian.

Prefixes can be used before unit names. They mean that a unit must be multiplied or divided by a certain integer, a power of 10. For example, the prefix “kilo” means multiplied by 1000 (kilometer = 1000 meters). SI prefixes are also called decimal prefixes.

IN technical systems of measurements, instead of the unit of mass, the unit of force is considered the main one. There are a number of other systems that are close to the SI, but use different base units. For example, in the GHS system, generally accepted before the advent of the SI system, the basic unit of measurement is the gram, and the basic unit of length is the centimeter.

Topic: QUANTITIES AND THEIR MEASUREMENTS

Target: Give the concept of quantity and its measurement. Introduce the history of the development of the system of units of quantities. Summarize knowledge about quantities that preschoolers become familiar with.

Plan:

The concept of quantities, their properties. The concept of measuring a quantity. From the history of the development of the system of units of quantities. International system of units. Quantities that preschoolers become familiar with and their characteristics.

1. The concept of quantities, their properties

Quantity is one of the basic mathematical concepts that arose in ancient times and was subjected to a number of generalizations in the process of long-term development.

The initial idea of ​​size is associated with the creation of a sensory basis, the formation of ideas about the size of objects: show and name length, width, height.

By magnitude we mean special properties real objects or phenomena of the surrounding world. The size of an object is its relative characteristic, emphasizing the extent individual parts and determining its place among homogeneous ones.

Quantities characterized only by numerical value are called scalar(length, mass, time, volume, area, etc.). In addition to scalar quantities, mathematics also considers vector quantities, which are characterized not only by number, but also by direction (force, acceleration, electric field strength, etc.).

Scalar quantities can be homogeneous or heterogeneous. Homogeneous quantities express the same property of objects of a certain set. Heterogeneous quantities express various properties objects (length and area)

Properties of scalar quantities:

§ any two quantities of the same kind are comparable, either they are equal, or one of them is less (greater) than the other: 4t5ts…4t 50kgÞ 4t5ts=4t500kg Þ 4t500kg>4t50kg, because 500kg>50kg, which means

4t5ts >4t 50kg;

§ quantities of the same kind can be added, the result is a quantity of the same kind:

2km921m+17km387mÞ 2km921m=2921m, 17km387m=17387m Þ 17387m+2921m=20308m; Means

2km921m+17km387m=20km308m

§ the value can be multiplied by real number, the result will be a quantity of the same kind:

12m24cm× 9 Þ 12m24m=1224cm, 1224cm×9=110m16cm, that means

12m24cm× 9=110m16cm;

4kg283g-2kg605gÞ 4kg283g=4283g, 2kg605g=2605g Þ 4283g-2605g=1678g, which means

4kg283g-2kg605g=1kg678g;

§ quantities of the same kind can be divided, resulting in a real number:

8h25min: 5 Þ 8h25min=8×60min+25min=480min+25min=505min, 505min : 5=101min, 101min=1h41min, that means 8h25min: 5=1h41min.

Magnitude is a property of an object, perceived by different analyzers: visual, tactile and motor. In this case, most often the value is perceived simultaneously by several analyzers: visual-motor, tactile-motor, etc.

The perception of magnitude depends on:

§ the distance from which the object is perceived;

§ the size of the object with which it is compared;

§ its location in space.

Basic properties of the quantity:

§ Comparability– determination of a value is possible only on the basis of comparison (directly or by comparison with a certain image).

§ Relativity– the characteristic of size is relative and depends on the objects chosen for comparison; one and the same object can be defined by us as larger or smaller depending on the size of the object with which it is compared. For example, a bunny is smaller than a bear, but larger than a mouse.

§ Variability– the variability of quantities is characterized by the fact that they can be added, subtracted, multiplied by a number.

§ Measurability– measurement makes it possible to characterize a quantity by comparing numbers.

2. Concept of quantity measurement

The need to measure all kinds of quantities, as well as the need to count objects, arose in the practical activities of man at the dawn of human civilization. Just as to determine the number of sets, people compared different sets, different homogeneous quantities, determining first of all which of the compared quantities was larger or smaller. These comparisons were not yet measurements. Subsequently, the procedure for comparing values ​​was improved. One value was taken as a standard, and other values ​​of the same kind were compared with the standard. When people acquired knowledge about numbers and their properties, magnitude, the number 1 was assigned to the standard and this standard began to be called a unit of measurement. The purpose of measurement has become more specific – to evaluate. How many units are contained in the measured quantity. the measurement result began to be expressed as a number.

The essence of measurement is the quantitative division of measured objects and establishing the value of a given object in relation to to the extent taken. Through the measurement operation, the numerical relationship of the object is established between the measured quantity and a pre-selected unit of measurement, scale or standard.

The measurement includes two logical operations:

the first is the process of separation, which allows the child to understand that the whole can be split into parts;

the second is a substitution operation consisting of connecting individual parts (represented by the number of measures).

The measurement activity is quite complex. It requires certain knowledge, specific skills, knowledge of the generally accepted system of measures, and the use of measuring instruments.

In the process of developing measurement activities in preschoolers using conventional measures, children must understand that:

§ measurement gives an accurate quantitative description of a quantity;

§ for measurement it is necessary to choose an adequate standard;

§ the number of measurements depends on the quantity being measured (the larger the quantity, the greater its numerical value and vice versa);

§ the measurement result depends on the selected measure (the larger the measure, the smaller the numerical value and vice versa);

§ to compare quantities, they must be measured with the same standards.

3. From the history of the development of the system of units of quantities

Man has long realized the need to measure different quantities, and to measure as accurately as possible. The basis for accurate measurements are convenient, clearly defined units of quantities and accurately reproducible standards (samples) of these units. In turn, the accuracy of the standards reflects the level of development of science, technology and industry of the country and speaks of its scientific and technical potential.

In the history of the development of units of quantities, several periods can be distinguished.

The most ancient period is when units of length were identified with the names of parts of the human body. Thus, the units of length used were the palm (the width of four fingers without the thumb), the cubit (the length of the elbow), the foot (the length of the foot), the inch (the length of the joint thumb) etc. The units of area during this period were: a well (an area that can be watered from one well), a plow or a plow (the average area processed per day by a plow or a plow), etc.

In the XIV-XVI centuries. In connection with the development of trade, so-called objective units of measurement of quantities appear. In England, for example, an inch (the length of three barley grains placed side by side), a foot (the width of 64 barley grains placed side by side).

Gran (weight of grain) and carat (weight of seed of one type of bean) were introduced as units of mass.

The next period in the development of units of quantities is the introduction of units interconnected with each other. In Russia, for example, these were the units of length: mile, verst, fathom and arshin; 3 arshins was a fathom, 500 fathoms was a verst, 7 versts was a mile.

However, the connections between units of quantities were arbitrary; not only individual states, but also individual regions within the same state used their own measures of length, area, and mass. Particular disparity was observed in France, where each feudal lord had the right to establish his own measures within the boundaries of his possessions. Such a variety of units of quantities hampered the development of production, interfered with scientific progress and the development of trade relations.

The new system of units, which later became the basis for the international system, was created in France at the end of the 18th century, during the era of the Great french revolution. The basic unit of length in this system was meter- one forty millionth of the length of the earth's meridian passing through Paris.

In addition to the meter, the following units were installed:

§ ar- the area of ​​a square whose side length is 10 m;

§ liter- volume and capacity of liquids and bulk solids, equal to the volume of a cube with an edge length of 0.1 m;

§ gram- the mass of pure water occupying the volume of a cube with an edge length of 0.01 m.

Decimal multiples and submultiples were also introduced, formed using prefixes: miria (104), kilo (103), hecto (102), deca (101), deci, centi, milli

The unit of mass, kilogram, was defined as the mass of 1 dm3 of water at a temperature of 4 °C.

Since all units of quantities turned out to be closely related to the unit of length meter, the new system of quantities was called metric system.

In accordance with accepted definitions, platinum standards of the meter and kilogram were made:

§ the meter was represented by a ruler with strokes applied to its ends;

§ kilogram - a cylindrical weight.

These standards were transferred to the National Archives of France for storage, and therefore they received the names “archival meter” and “archival kilogram”.

The creation of the metric system of measures was a great scientific achievement - for the first time in history, measures appeared that formed a coherent system, based on a model taken from nature, and closely related to decimal system Reckoning.

But soon changes had to be made to this system.

It turned out that the length of the meridian was not determined accurately enough. Moreover, it became clear that as science and technology develop, the value of this quantity will become more precise. Therefore, the unit of length taken from nature had to be abandoned. The meter began to be considered the distance between the strokes marked on the ends of the archival meter, and the kilogram the mass of the standard archival kilogram.

In Russia, the metric system of measures began to be used on a par with Russian national measures since 1899, when a special law was adopted, the draft of which was developed by an outstanding Russian scientist. Special resolutions Soviet state The transition to the metric system of measures was legalized first in the RSFSR (1918), and then in the entire USSR (1925).

4. International system of units

International System of Units (SI) is a single universal practical system of units for all branches of science, technology, national economy and teaching. Since the need for such a system of units, which is uniform throughout the world, was great, in a short time it received wide international recognition and distribution throughout the world.

This system has seven basic units (meter, kilogram, second, ampere, kelvin, mole and candela) and two additional units (radian and steradian).

As is known, the unit of length meter and unit of mass kilogram were also included in the metric system of measures. What changes did they undergo when they entered the new system? A new definition of the meter has been introduced - it is considered as the distance that a plane electromagnetic wave travels in a vacuum in a fraction of a second. The transition to this definition of the meter is caused by increasing requirements for measurement accuracy, as well as the desire to have a unit of magnitude that exists in nature and remains unchanged under any conditions.

The definition of the kilogram unit of mass has not changed; the kilogram is still the mass of a platinum-iridium alloy cylinder manufactured in 1889. This standard is stored at the International Bureau of Weights and Measures in Sevres (France).

The third basic unit of the International System is the time unit, the second. She is much older than a meter.

Before 1960, the second was defined as 0 " style="border-collapse:collapse;border:none">

Prefix names

Prefix designation

Factor

Prefix names

Prefix designation

Factor

For example, a kilometer is a multiple of a unit, 1 km = 103×1 m = 1000 m;

A millimeter is a submultiple unit, 1 mm = 10-3 × 1 m = 0.001 m.

In general, for length, the multiple units are kilometer (km), and the subunit are centimeter (cm), millimeter (mm), micrometer (µm), nanometer (nm). For mass, the multiple unit is megagram (Mg), and the subunit is gram (g), milligram (mg), microgram (mcg). For time, the multiple unit is the kilosecond (ks), and the subunit is the millisecond (ms), microsecond (µs), nanosecond (not).

5. Quantities that preschoolers become familiar with and their characteristics

The goal of preschool education is to introduce children to the properties of objects, teach them to differentiate them, highlighting those properties that are commonly called quantities, and introduce them to the very idea of ​​measurement through intermediate measures and the principle of measuring quantities.

Length- this is a characteristic of the linear dimensions of an object. In preschool methods of forming elementary mathematical concepts, it is customary to consider “length” and “width” as two different qualities of an object. However, in school, both linear dimensions of a flat figure are more often called “side length”; the same name is used when working with a volumetric body that has three dimensions.

The lengths of any objects can be compared:

§ by eye;

§ application or overlay (combination).

In this case, it is always possible to either approximately or accurately determine “how much one length is greater (smaller) than another.”

Weight- This physical property an object measured by weighing. It is necessary to distinguish between the mass and weight of an object. With the concept item weight children get acquainted in the 7th grade in a physics course, since weight is the product of mass and acceleration free fall. The terminological incorrectness that adults allow themselves in everyday life often confuses a child, since we sometimes, without thinking, say: “The weight of an object is 4 kg.” The very word “weighing” encourages the use of the word “weight” in speech. However, in physics, these quantities differ: the mass of an object is always constant - this is a property of the object itself, and its weight changes if the force of attraction (acceleration of free fall) changes.

To prevent the child from learning incorrect terminology, which will confuse him in the future, elementary school, you should always say: object mass.

In addition to weighing, the mass can be approximately determined by an estimate on the hand (“baric feeling”). Mass is a difficult category from a methodological point of view for organizing classes with preschoolers: it cannot be compared by eye, by application, or measured by an intermediate measure. However, any person has a “baric feeling”, and using it you can build a number of tasks that are useful for a child, leading him to understand the meaning of the concept of mass.

Basic unit of mass – kilogram. From this basic unit other units of mass are formed: gram, ton, etc.

Square- this is a quantitative characteristic of a figure, indicating its dimensions on a plane. The area is usually determined for flat closed figures. To measure the area, you can use any flat shape that fits tightly into the given figure (without gaps) as an intermediate measure. In elementary school, children are introduced to palette - piece transparent plastic with a grid of squares of equal size applied to it (usually 1 cm2 in size). Placing the palette on a flat figure makes it possible to count the approximate number of squares that fit in it to determine its area.

IN preschool age children compare the areas of objects without naming this term, by superimposing objects or visually, by comparing the space they occupy on the table or ground. Area is a convenient quantity from a methodological point of view, since it allows the organization of various productive exercises in comparing and equalizing areas, determining the area by laying down intermediate measures and through a system of tasks for equal composition. For example:

1) comparison of the areas of figures by the superposition method:

Area of ​​a triangle less area circle, and the area of ​​the circle more area triangle;

2) comparison of the areas of figures by the number of equal squares (or any other measurements);

The areas of all figures are equal, since the figures consist of 4 equal squares.

When performing such tasks, children indirectly become acquainted with some area properties:

§ The area of ​​a figure does not change when its position on the plane changes.

§ Part of an object is always smaller than the whole.

§ The area of ​​the whole is equal to the sum of the areas of its constituent parts.

These tasks also form in children the concept of area as number of measures contained in a geometric figure.

Capacity- this is a characteristic of liquid measures. At school, capacity is examined sporadically during one lesson in 1st grade. Children are introduced to the measure of capacity - the liter, in order to later use the name of this measure when solving problems. The tradition is that capacity is not associated with the concept of volume in elementary school.

Time- this is the duration of the processes. The concept of time is more complex than the concept of length and mass. In everyday life, time is what separates one event from another. In mathematics and physics, time is considered as a scalar quantity, because time intervals have properties similar to the properties of length, area, mass:

§ Time periods can be compared. For example, a pedestrian will spend more time on the same path than a cyclist.

§ Time periods can be added together. Thus, a lecture in college lasts the same amount of time as two lessons in school.

§ Time intervals are measured. But the process of measuring time is different from measuring length. To measure length, you can use a ruler repeatedly, moving it from point to point. A period of time taken as a unit can be used only once. Therefore, the unit of time must be a regularly repeating process. Such a unit in the International System of Units is called second. Along with the second, others are also used. units of time: minute, hour, day, year, week, month, century.. Units such as year and day were taken from nature, and hour, minute, second were invented by man.

A year is the time it takes for the Earth to revolve around the Sun. A day is the time the Earth rotates around its axis. A year consists of approximately 365 days. But a year in a person’s life is made up of a whole number of days. Therefore, instead of adding 6 hours to each year, they add a whole day to each fourth year. This year consists of 366 days and is called a leap year.

A calendar with such an alternation of years was introduced in 46 BC. e. Roman Emperor Julius Caesar in order to streamline the very confusing calendar existing at that time. That's why the new calendar is called Julian. According to it, the new year begins on January 1 and consists of 12 months. It also preserved such a measure of time as a week, invented by Babylonian astronomers.

Time dares both physical and philosophical meaning. Since the sense of time is subjective, it is difficult to rely on the senses in assessing and comparing it, as can be done to some extent with other quantities. In this regard, at school, almost immediately, children begin to become familiar with instruments that measure time objectively, that is, regardless of human sensations.

When introducing the concept of “time” at first, it is much more useful to use an hourglass than a watch with arrows or an electronic one, since the child sees the sand pouring in and can observe the “passage of time.” Hourglasses are also convenient to use as an intermediate measure when measuring time (in fact, this is exactly what they were invented for).

Working with the quantity “time” is complicated by the fact that time is a process that is not directly perceived by the child’s sensory system: unlike mass or length, it cannot be touched or seen. This process is perceived by a person indirectly, compared to the duration of other processes. At the same time, the usual stereotypes of comparisons: the course of the sun across the sky, the movement of hands on a clock, etc. - as a rule, are too long for a child of this age to really follow them.

In this regard, “Time” is one of the most difficult topics both in preschool mathematics teaching and in primary school.

The first ideas about time are formed in preschool age: the change of seasons, the change of day and night, children become familiar with the sequence of concepts: yesterday, today, tomorrow, the day after tomorrow.

By the beginning of school, children develop ideas about time as a result of practical activities related to taking into account the duration of processes: performing routine moments of the day, maintaining a weather calendar, becoming familiar with the days of the week, their sequence, children become familiar with the clock and orienting themselves by it in connection with a visit kindergarten. It is quite possible to introduce children to such units of time as year, month, week, day, to clarify the idea of ​​the hour and minute and their duration in comparison with other processes. The tools for measuring time are the calendar and the clock.

Speed- this is the path traveled by the body per unit of time.

Speed ​​is a physical quantity, its names contain two quantities - units of length and units of time: 3 km/h, 45 m/min, 20 cm/s, 8 m/s, etc.

It is very difficult to give a child a visual idea of ​​speed, since it is the ratio of path to time, and it is impossible to depict it or see it. Therefore, when getting acquainted with speed, we usually turn to comparing the time of movement of objects over an equal distance or the distances covered by them in the same time.

Named numbers are numbers with names of units of measurement of quantities. When solving problems at school, you have to perform arithmetic operations with them. Preschoolers are introduced to named numbers in the School 2000 programs (“One is a step, two is a step...”) and “Rainbow.” In the School 2000 program, these are tasks of the form: “Find and correct errors: 5 cm + 2 cm - 4 cm = 1 cm, 7 kg + 1 kg - 5 kg = 4 kg.” In the Rainbow program these are tasks of the same type, but by “naming” there we mean any name when numerical values, and not just the names of measures of quantities, for example: 2 cows + 3 dogs + + 4 horses = 9 animals.

You can mathematically perform an operation with named numbers in the following way: perform actions with the numerical components of named numbers, and add a name when writing the answer. This method requires compliance with the rule of a single name in action components. This method is universal. In elementary school, this method is also used when performing actions with compound named numbers. For example, to add 2 m 30 cm + 4 m 5 cm, children replace the composite named numbers with numbers of the same name and perform the action: 230 cm + 405 cm = 635 cm = 6 m 35 cm or add the numerical components of the same names: 2 m + 4 m = 6 m, 30 cm + 5 cm = 35 cm, 6 m + 35 cm = 6 m 35 cm.

These methods are used when performing arithmetic operations with numbers of any kind.

Units of some quantities

Units of length 1 km = 1,000 m 1 m = 10 dm = 100 m 1 dm = 10 cm 1 cm = 10 mm	Units of mass 1 t = 1,000 kg 1 kg = 1,000 g 1 g = 1,000 mg	Ancient measures length 1 verst = 500 fathoms = 1,500 arshins = = 3,500 feet = 1,066.8 m 1 fathom = 3 arshins = 48 vershoks = 84 inches = 2.1336 m 1 yard = 91.44cm 1 arshin = 16 vershka = 71.12 cm 1 vershok = 4.450 cm 1 inch = 2.540 cm 1 weave = 2.13 cm
Area units 1 m2 = 100 dm2 = cm2 1 ha = 100 a = m2 1 a (ar) = 100m2	Volume units 1 m3 = 1,000 dm3 = 1,000,000 cm3 1 dm3 = 1,000 cm3 1 bbl (barrel) = 158.987 dm3 (l)	Measures of mass 1 pood = 40 pounds = 16.38 kg 1 lb = 0.40951 kg 1 carat = 2×10-4 kg

According to their purpose and requirements, the following types of standards are distinguished.

Primary standard – ensures the reproduction and storage of a unit of physical quantity with the highest accuracy in the country (compared to other standards of the same quantity). Primary standards are unique measuring systems created taking into account the latest achievements science and technology and ensuring the uniformity of measurements in the country.

Special standard - ensures the reproduction of a unit of physical quantity in special conditions, in which direct transfer of the unit size from the primary standard with the required accuracy is not feasible, and serves as the primary standard for these conditions.

The primary or special standard, officially approved as the source for the country, is called the state standard. State standards are approved by Gosstandart, and for each of them a state standard. State standards are created, stored and applied by the country's central scientific metrological institutes.

Secondary standard – stores the dimensions of a unit of a physical quantity obtained by comparison with the primary standard of the corresponding physical quantity. Secondary standards refer to the subordinate means of storing units and transferring their sizes during verification work and ensure the safety and least wear of state primary standards.

According to their metrological purpose, secondary standards are divided into copy standards, comparison standards, witness standards and working standards.

Reference copy – designed to convey the size of a unit of physical quantity as a working standard for a large volume of verification work. It is a copy of the state primary standard for metrological purposes only, but is not always a physical copy.

Standard of comparison – used for comparing standards that, for one reason or another, cannot be directly compared with each other.

Standard witness – designed to check the safety and immutability of the state standard and replace it in case of damage or loss. Since most government standards are created based on the use of the most stable physical phenomena and are therefore indestructible; currently only the kilogram standard has a witness standard.

Working standard – used to convey the size of a unit of physical quantity using a working measuring instrument. This is the most common type of standards that are used for verification work by territorial and departmental metrological services. Working standards are divided into categories that determine the order of their subordination in accordance with the verification scheme.

Standards of basic SI units.

Standard unit of time. The unit of time - the second - has long been defined as 1/86400 of the average solar day. Later it was discovered that the rotation of the Earth around its axis occurs unevenly. Then the definition of the unit of time was based on the period of rotation of the Earth around the Sun - the tropical year, i.e. the time interval between two spring equinoxes, following one after the other. The size of a second was defined as 1/31556925.9747 of a tropical year. This made it possible to increase the accuracy of determining the unit of time by almost 1000 times. However, in 1967, the 13th General Conference on Weights and Measures adopted a new definition of the second as the time interval during which 9192631770 oscillations occur, corresponding to the resonant frequency of the energy transition between the levels of the hyperfine structure of the ground state of the cesium-133 atom in the absence of disturbance by external fields. This definition implemented using cesium frequency references.

In 1972, the transition to the Universal Coordinated Time system was made. Since 1997, state primary control and the state verification scheme for time and frequency measuring instruments are determined by the rules of interstate standardization PMG18-96 “Interstate verification scheme for time and frequency measuring instruments.”

The state primary standard of a time unit, consisting of a set of measuring instruments, ensures the reproduction of time units with a standard deviation of the measurement result not exceeding 1 * 10 -14 for three months.

Standard unit of length. In 1889, the meter was adopted as equal to the distance between two lines marked on a metal rod of an X-shaped cross-section. Although the international and national meter standards were made of an alloy of platinum and iridium, which is distinguished by significant hardness and great resistance to oxidation, it was not completely certain that the length of the standard would not change over time. In addition, the error in comparing platinum-iridium line meters with each other is + 1.1 * 10 -7 m (+0.11 microns), and since the lines have a significant width, the accuracy of this comparison cannot be significantly increased.

After studying the spectral lines of a number of elements, it was found that the orange line of the krypton-86 isotope provides the greatest accuracy in reproducing a unit of length. In 1960, the 11th General Conference on Weights and Measures adopted the expression of the size of the meter in these wavelengths as its most accurate value.

The krypton meter made it possible to increase the accuracy of reproducing a unit of length by an order of magnitude. However, further research made it possible to obtain a more accurate meter standard based on the wavelength in vacuum of monochromatic radiation generated by a stabilized laser. The development of new reference complexes for reproducing the meter led to the definition of the meter as the distance that light travels in a vacuum in 1/299792458 of a second. This definition of the meter was enshrined in law in 1985.

The new standard complex for reproducing the meter, in addition to increasing the accuracy of measurement in necessary cases, also makes it possible to monitor the constancy of the platinum-iridium standard, which has now become a secondary standard used to convey the size of the unit as a working standard.

Standard unit of mass. When establishing the metric system of measures, the mass of one cubic decimeter of pure water at the temperature of its highest density (4 0 C) was taken as a unit of time.

During this period there were carried out precise definitions mass of a known volume of water by successively weighing in air and water an empty bronze cylinder, the dimensions of which have been carefully determined.

Based on these weighings, the first prototype of the kilogram was a platinum cylindrical weight with a height of 39 mm equal to its diameter. Like the prototype of the meter, it was transferred to the National Archives of France for storage. In the 19th century, several careful measurements of the mass of one cubic decimeter of pure water at a temperature of 4 0 C were repeated. It was found that this mass was slightly (approximately 0.028 g) less than the prototype kilogram of the Archive. In order not to change the value of the original unit of mass during further, more accurate weighings, the International Commission on the Prototypes of the Metric System in 1872. it was decided to take the mass of the prototype kilogram of the Archive as a unit of mass.

In the production of platinum-iridium kilogram standards, the international prototype was taken to be the one whose mass differed least from the mass of the Archive kilogram prototype.

Due to the adoption of the conventional prototype of the unit of mass, the liter turned out to be not equal to the cubic decimeter. The value of this deviation (1l=1.000028 dm3) corresponds to the difference between the mass of the international prototype of a kilogram and the mass of a cubic decimeter of water. In 1964, the 12th General Conference on Weights and Measures decided to equate the volume of 1 liter to 1 dm 3.

It should be noted that at the time the metric system of measures was established there was no clear distinction between the concepts of mass and weight, therefore the international prototype of the kilogram was considered the standard of the unit of weight. However, already with the approval of the international prototype of the kilogram at the 1st General Conference on Weights and Measures in 1889, the kilogram was approved as the prototype of mass.

A clear distinction between the kilogram as a unit of mass and the kilogram as a unit of force was given in the decisions of the 3rd General Conference on Weights and Measures (1901).

The state primary standard and verification scheme for means of changing mass are determined by GOST 8.021 - 84. The state standard consists of a set of measures and measuring instruments:

· national prototype of the kilogram - copy No. 12 of the international prototype of the kilogram, which is a weight made of a platinum-iridium alloy and is intended to transfer the size of a unit of mass to the weight R1;

· national prototype of the kilogram - copy No. 26 of the international prototype of the kilogram, which is a weight made of a platinum-iridium alloy and is intended to verify the invariance of the size of a unit of mass, reproduced by the national prototype of the kilogram - copy No. 12, and replacing the latter during its comparisons at the International Bureau of Measures and scales;

· weights R1 and a set of weights made of platinum-iridium alloy and designed to transfer the size of a unit of mass to standards - copies;

· standard scales.

The nominal mass value reproduced by the standard is 1 kg. The state primary standard ensures the reproduction of a unit of mass with a standard deviation of the measurement result when compared with the international prototype of the kilogram, not exceeding 2 * 10 -3 mg.

Standard scales, with the help of which the mass standard is compared, with a weighing range of 2 * 10 -3 ... 1 kg have an average standard deviation observation result on scales 5*10 -4 ... 3*10 -2 mg.

The study of physical phenomena and their patterns, as well as the use of these patterns in human practice is associated with measurement physical quantities.

A physical quantity is a property that is qualitatively common to many physical objects (physical systems, their states and processes occurring in them), but quantitatively individual for each object.

A physical quantity is, for example, mass. Different physical objects have mass: all bodies, all particles of matter, particles of the electromagnetic field, etc. Qualitatively, all specific realizations of mass, i.e., the masses of all physical objects, are the same. But the mass of one object can be a certain number of times greater or less than the mass of another. And in this quantitative sense, mass is a property that is individual for each object. Physical quantities are also length, temperature, electric field strength, oscillation period, etc.

Specific implementations of the same physical quantity are called homogeneous quantities. For example, the distance between the pupils of your eyes and the height Eiffel Tower there are specific realizations of the same physical quantity - length and therefore are homogeneous quantities. The mass of this book and the mass of the Earth satellite “Cosmos-897” are also homogeneous physical quantities.

Homogeneous physical quantities differ from each other in size. The size of a physical quantity is

the quantitative content in a given object of a property corresponding to the concept of “physical quantity”.

The sizes of homogeneous physical quantities of different objects can be compared with each other if the values ​​of these quantities are determined.

The value of a physical quantity is an assessment of a physical quantity in the form of a certain number of units accepted for it (see p. 14). For example, the value of the length of a certain body, 5 kg is the value of the mass of a certain body, etc. An abstract number included in the value of a physical quantity (in our examples 10 and 5) is called a numerical value. In general, the value X of a certain quantity can be expressed as the formula

where is the numerical value of the quantity, its unit.

It is necessary to distinguish between the true and actual values ​​of a physical quantity.

The true value of a physical quantity is the value of the quantity that in an ideal way would reflect in qualitative and quantitative terms the corresponding property of the object.

The actual value of a physical quantity is the value of a quantity found experimentally and so close to the true value that it can be used instead for a given purpose.

Finding the value of a physical quantity experimentally using special technical means called measurement.

The true values ​​of physical quantities are usually unknown. For example, no one knows the true values ​​of the speed of light, the distance from the Earth to the Moon, the mass of an electron, a proton, and others elementary particles. We do not know the true value of our height and body weight, we do not know and cannot find out the true value of the air temperature in our room, the length of the table at which we work, etc.

However, using special technical means, it is possible to determine the actual

the values ​​of all these and many other quantities. Moreover, the degree of approximation of these actual values ​​to the true values ​​of physical quantities depends on the perfection of the technical measuring instruments used.

Measuring instruments include measures, measuring instruments, etc. A measure is understood as a measuring instrument designed to reproduce a physical quantity of a given size. For example, a weight is a measure of mass, a ruler with millimeter divisions is a measure of length, a measuring flask is a measure of volume (capacity), a normal element is a measure of electromotive force, a quartz oscillator is a measure of the frequency of electrical oscillations, etc.

A measuring device is a measuring instrument designed to generate a signal of measuring information in a form accessible to direct perception by observation. TO measuring instruments include dynamometer, ammeter, pressure gauge, etc.

There are direct and indirect measurements.

Direct measurement is a measurement in which the desired value of a quantity is found directly from experimental data. Direct measurements include, for example, measuring mass on an equal-arm scale, temperature - with a thermometer, length - with a scale ruler.

Indirect measurement is a measurement in which the desired value of a quantity is found on the basis of a known relationship between it and quantities subjected to direct measurements. Indirect measurements are, for example, finding the density of a body by its mass and geometric dimensions, finding the electrical resistivity of a conductor by its resistance, length and cross-sectional area.

Measurements of physical quantities are based on various physical phenomena. For example, to measure temperature, the thermal expansion of bodies or the thermoelectric effect is used, to measure the mass of bodies by weighing, the phenomenon of gravity, etc. The set of physical phenomena on which measurements are based is called the measurement principle. Measurement principles are not covered in this manual. Metrology studies the principles and methods of measurements, types of measuring instruments, measurement errors and other issues related to measurements.

Physical quantities and their dimensions

FORMATION OF STUDENTS' CONCEPTS ABOUT PHYSICAL QUANTITIES AND LAWS

Classification of physical quantities

Units of measurement of physical quantities. Systems of units.

Problems of formation among students physical concepts

Formation of students' concepts of physical quantities using the method of frame supports

Formation of students' concepts of physical laws using the method of frame supports

Physical quantities and their dimensions

Physical size name a property that is qualitatively common to many physical objects, but quantitatively individual for each object (Bolsun, 1983)/

The set of physical functions interconnected by dependencies is called a system of physical quantities. The PV system consists of basic quantities, which are conditionally accepted as independent, and from derived quantities, which are expressed through the basic quantities of the system.

Derived physical quantities- these are physical quantities included in the system and determined through the basic quantities of this system. The mathematical relationship (formula) through which the derivative of the PV of interest to us is expressed explicitly through other quantities of the system and in which the direct connection between them is manifested is called defining equation. For example, the defining equation for speed is the relation

V = (1)

Experience shows that the PV system, covering all branches of physics, can be built on seven basic quantities: mass, time, length, temperature, luminous intensity, amount of matter, force electric current.

Scientists have agreed to denote the main PVs with symbols: length (distance) in any equations and any systems with the symbol L (it starts with this letter in English and German languages the word length), and time - the symbol T (this letter begins with English word time). The same applies to the dimensions of mass (symbol M), electric current (symbol I), thermodynamic temperature (symbol Θ), amount of matter (symbol

N), luminous intensity (symbol J). These symbols are called dimensions length and time, mass, etc., regardless of the size of length or time. (Sometimes these symbols are called logical operators, sometimes radicals, but most often dimensions.) Thus, Dimension of the main PV -This just FV symbol in the form of a capital letter of the Latin or Greek alphabet.
So, for example, the dimension of speed is a symbol of speed in the form of two letters LT −1 (according to formula (1)), where T represents the dimension of time, and L - length. These symbols denote the PV of time and length, regardless of their specific size (second , minute, hour, meter, centimeter, etc.). The dimension of force is MLT −2 (according to the equation of Newton’s second law F = ma). Any derivative of the PV has a dimension, since there is an equation that determines this quantity. There is an extremely useful mathematical procedure in physics called dimensional analysis or checking a formula by dimension.

There are still two opposing opinions regarding the concept of “dimension”. Prof. Kogan I. Sh., in the article Dimension of a physical quantity(Kogan,) gives the following arguments regarding this dispute.. For more than a hundred years, disputes have continued about physical sense dimensions. Two opinions - dimension refers to a physical quantity, and dimension refers to a unit of measurement - have been dividing scientists into two camps for a century. The first point of view was defended by the famous physicist of the early twentieth century A. Sommerfeld. The second point of view was defended outstanding physicist M. Planck, who considered the dimension of a physical quantity to be some kind of convention. The famous metrologist L. Sena (1988) adhered to the point of view according to which the concept of dimension does not refer to a physical quantity at all, but to its unit of measurement. The same point of view is presented in the popular textbook on physics by I. Savelyev (2005).

However, this confrontation is artificial. The dimension of a physical quantity and its unit of measurement are different physical categories and should not be compared. This is the essence of the answer that solves this problem.

We can say that a physical quantity has dimension insofar as there is an equation that determines this quantity. As long as there is no equation, there is no dimension, although this does not make the physical quantity cease to exist objectively. There is no objective need for the existence of dimension in a unit of measurement of a physical quantity.

Again, dimensions physical quantities for the same physical quantities must be the same on any planet in any star system. At the same time, the units of measurement of the same quantities may turn out to be anything and, of course, not similar to our earthly ones.

This view of the problem suggests that Both A. Sommerfeld and M. Planck are right. Each of them just meant something different. A. Sommerfeld meant the dimensions of physical quantities, and M. Planck meant units of measurement. Contrasting their views to each other, metrologists groundlessly equate the dimensions of physical quantities with their units of measurement, thereby artificially contrasting the points of view of A. Sommerfeld and M. Planck.

In this manual, the concept of “dimension,” as expected, refers to PV and is not identified with PV units.