Measurement of quantities. Physical quantities and their measurements Physical quantities and their units of measurement physics

1. The concept of magnitude. Basic properties of homogeneous quantities.

2. Measurement of quantity. Numerical value of the quantity.

3. Length, area, mass, time.

4. Dependencies between quantities.

4.1. Concept of magnitude

Quantity is one of the basic mathematical concepts that arose in ancient times and, in the process of long development, underwent a number of generalizations. Length, area, volume, mass, speed and many others are all quantities.

Value - this is a special property of real objects or phenomena. For example, the property of objects “to have extension” is called “length”. A quantity is considered as a generalization of the properties of some objects and as an individual characteristic of the property of a particular object. Values ​​can be quantified based on comparison.

For example, the concept length occurs:

when denoting the properties of a class of objects (“many objects around us have length”);

when denoting a property of a specific object from this class (“this table has a length”);

when comparing objects by this property (“the length of the table is greater than the length of the desk”).

Homogeneous quantities – quantities that express the same property of objects of a certain class.

Heterogeneous quantities express various properties of objects (one object can have mass, volume, etc.).

Properties of homogeneous quantities:

1. Homogeneous quantities can be compare.

For any values ​​a and b, only one of the relations is valid: A < b, A > b, A = b.

For example, the mass of a book is greater than the mass of a pencil, and the length of the pencil is less than the length of the room.

2. Homogeneous quantities can be add and subtract. As a result of addition and subtraction, a quantity of the same kind is obtained.

Quantities that can be added are called additivenym. For example, you can add the lengths of objects. The result is length. There are quantities that are not additive, for example temperature. When water of different temperatures from two vessels is combined, a mixture is obtained, the temperature of which cannot be determined by adding the values.

We will consider only additive quantities.

Let be: A– fabric length, b– the length of the piece that was cut off, then: ( A - b) is the length of the remaining piece.

3. The size can be multiply by a real number. The result is a quantity of the same kind.

Example: “Pour 6 glasses of water into a jar.”

If the volume of water in a glass is V, then the volume of water in the jar is 6V .

4. Homogeneous quantities share. The result is a non-negative real number, it is called attitudequantities

Example: “How many ribbons of length b can be obtained from a ribbon of length a?” ( X = A : b)

5. The size can be measure.

4.2. Quantity measurement

By comparing quantities directly, we can establish their equality or inequality. For example, by comparing strips by length by overlay or application, you can determine whether they are equal or not:

If the ends coincide, then the strips are of equal length;

If the left ends match, and the right end of the bottom strip protrudes, then its length is greater.

To obtain a more accurate comparison result, the values ​​are measured.

The measurement consists of comparing a given value with a certainsecond quantity taken as unity.

When measuring the mass of a watermelon on a scale, compare it with the mass of a weight.

When measuring the length of the room in steps, compare it with the length of the step.

The comparison process depends on the type of quantity: length is measured using a ruler, mass - using scales. Whatever this process, the result of the measurement is a certain number, depending on the chosen unit of value.

Purpose of measurement – obtain a numerical characteristic of a given quantity with the selected unit.

If the quantity a is given and the unit of the quantity e is chosen, then in reAs a result of measuring the quantity a, they find such a realnumber x such that a = x e. This number x is called the numerical valuedecrease in the value of a for a unit of value e.

1) The mass of the melon is 3 kg.

3kg = 3∙1 kg, where 3 is the numerical value of the mass of a melon with a mass unit of 1 kg.

2) The length of the segment is 10cm.

10 cm = 10 1 cm, where 10 is the numerical value of the length of the segment with a length unit of 1 cm.

Quantities determined by one numerical value are called scalar(length, volume, mass, etc.). There are also vector quantities, which are determined by numerical value and direction (speed, force, etc.).

Measurement allows us to reduce the comparison of quantities to the comparison of numbers, and actions with quantities to actions on numbers.

1. If the values A And b measured using a unit of magnitude e, then the relationships between quantities A And b will be the same as the relationships between their numerical values ​​(and vice versa):

Let A= that is,b= n e, Then a=b<= > m = n,

a >b < = > t > p,

A< b < = > T< п.

Example: “The mass of a watermelon is 5 kg. Melon weight 3 kg. The mass of a watermelon is greater than the mass of a melon, because... 5 > 3".

2. If the values A And b measured using a unit of magnitude e, then to find the numerical value of the sum (A+ b), it is enough to add the numerical values ​​of the quantities A And b.

Let a=t e,b=p e, s=ke, Then a +b=c< = > t+p= k.

For example, to determine the mass of purchased potatoes poured into two bags, it is not necessary to pour them together and weigh them; it is enough to add up the numerical values ​​of the mass of each bag.

3. If the values A And b are such that b = x a, Where X - positive real number, and the quantity A measured using a unit of magnitude e, then to find the numerical value of the quantity b for unit e, a sufficient number X multiply by the numerical value of the quantity A.

Let A= that is,b= x a, Then b=(x t) e.

Example: “The length of the blue stripe is 2 inches. The yellow one is 3 times longer. How long is the yellow stripe?

2dm 3 = (2 1dm) 3 = (2 3) 1dm = 6 1dm = 6dm.

Preschoolers first become familiar with measuring quantities using conventional measures. In the process of practical activity, they realize the relationship between a quantity and its numerical value, as well as the numerical value of a quantity from the chosen unit of measurement.

“Measure in steps the length of the path from the house to the tree, and now from the tree to the fence. What is the length of the entire path?

(Children add quantities using their numerical values.)

What is the length of the path measured by Masha's steps? (5 steps of Masha.)

What is the length of the same path, measured by Kolya’s steps? (4 steps Kolya.)

Why did we measure the length of the same track and get different results?

(The length of the path is measured in different steps. Kolya’s steps are longer, so there are fewer of them).

Numerical values ​​for road lengths differ due to the use of different units of measurement.

The need to measure quantities arose in the practical activities of man in the process of his development. The result of the measurement is expressed as a number and makes it possible to better understand the essence of the concept of number. The measurement process itself teaches children to think logically, develops practical skills, and enriches cognitive activity. In the process of measuring, children can get not only natural numbers, but also fractions.

This lesson will not be new for beginners. We have all heard from school such things as centimeter, meter, kilometer. And when it came to mass, they usually said gram, kilogram, ton.

Centimeters, meters and kilometers; grams, kilograms and tons have one common name - units of measurement of physical quantities.

In this lesson we will look at the most popular units of measurement, but we will not delve too deeply into this topic, since units of measurement go into the field of physics. Today we are forced to study part of physics because we need it for further study of mathematics.

Units of length

The following units of measurement are used to measure length:

• millimeters;
• centimeters;
• decimeters;
• meters;
• kilometers.

millimeter(mm). Millimeters can even be seen with your own eyes if you take the ruler that we used at school every day

Small lines running one after another are millimeters. More precisely, the distance between these lines is one millimeter (1 mm):

centimeter(cm). On the ruler, each centimeter is marked with a number. For example, our ruler, which was in the first picture, had a length of 15 centimeters. The last centimeter on this ruler is marked with the number 15.

There are 10 millimeters in one centimeter. You can put an equal sign between one centimeter and ten millimeters, since they indicate the same length:

1 cm = 10 mm

You can see this for yourself if you count the number of millimeters in the previous figure. You will find that the number of millimeters (distances between lines) is 10.

The next unit of length is decimeter(dm). There are ten centimeters in one decimeter. An equal sign can be placed between one decimeter and ten centimeters, since they indicate the same length:

1 dm = 10 cm

You can verify this if you count the number of centimeters in the following figure:

You will find that the number of centimeters is 10.

The next unit of measurement is meter(m). There are ten decimeters in one meter. One can put an equal sign between one meter and ten decimeters, since they indicate the same length:

1 m = 10 dm

Unfortunately, the meter cannot be illustrated in the figure because it is quite large. If you want to see the meter live, take a tape measure. Everyone has it in their home. On a tape measure, one meter will be designated as 100 cm. This is because there are ten decimeters in one meter, and one hundred centimeters in ten decimeters:

1 m = 10 dm = 100 cm

100 is obtained by converting one meter to centimeters. This is a separate topic that we will look at a little later. For now, let's move on to the next unit of length, which is called the kilometer.

The kilometer is considered the largest unit of length. There are, of course, other higher units, such as megameter, gigameter, terameter, but we will not consider them, since a kilometer is enough for us to further study mathematics.

There are a thousand meters in one kilometer. You can put an equal sign between one kilometer and a thousand meters, since they indicate the same length:

1 km = 1000 m

Distances between cities and countries are measured in kilometers. For example, the distance from Moscow to St. Petersburg is about 714 kilometers.

International System of Units SI

The International System of Units SI is a certain set of generally accepted physical quantities.

The main purpose of the international system of SI units is to achieve agreements between countries.

We know that the languages ​​and traditions of the countries of the world are different. There's nothing to be done about it. But the laws of mathematics and physics work the same everywhere. If in one country “twice two is four,” then in another country “twice two is four.”

The main problem was that for each physical quantity there are several units of measurement. For example, we have now learned that to measure length there are millimeters, centimeters, decimeters, meters and kilometers. If several scientists speaking different languages ​​gather in one place to solve some problem, then such a large variety of units of length measurement can give rise to contradictions between these scientists.

One scientist will state that in their country length is measured in meters. The second may say that in their country the length is measured in kilometers. The third may offer his own unit of measurement.

Therefore, the international system of SI units was created. SI is an abbreviation for the French phrase Le Système International d’Unités, SI (which translated into Russian means the international system of units SI).

The SI lists the most popular physical quantities and each of them has its own generally accepted unit of measurement. For example, in all countries, when solving problems, it was agreed that length would be measured in meters. Therefore, when solving problems, if the length is given in another unit of measurement (for example, in kilometers), then it must be converted into meters. We'll talk about how to convert one unit of measurement to another a little later. For now, let's draw our international system of SI units.

Our drawing will be a table of physical quantities. We will include each studied physical quantity in our table and indicate the unit of measurement that is accepted in all countries. Now we have studied the units of length and learned that the SI system defines meters to measure length. So our table will look like this:

Mass units

Mass is a quantity indicating the amount of matter in a body. People call body weight weight. Usually when something is weighed they say “It weighs so many kilograms” , although we are not talking about weight, but about the mass of this body.

However, mass and weight are different concepts. Weight is the force with which the body acts on a horizontal support. Weight is measured in newtons. And mass is a quantity that shows the amount of matter in this body.

But there is nothing wrong with calling body weight weight. Even in medicine they say "person's weight" , although we are talking about the mass of a person. The main thing is to be aware that these are different concepts.

The following units of measurement are used to measure mass:

• milligrams;
• grams;
• kilograms;
• centners;
• tons.

The smallest unit of measurement is milligram(mg). You will most likely never use a milligram in practice. They are used by chemists and other scientists who work with small substances. It is enough for you to know that such a unit of measurement of mass exists.

The next unit of measurement is gram(G). It is customary to measure the amount of a particular product in grams when preparing a recipe.

There are a thousand milligrams in one gram. You can put an equal sign between one gram and a thousand milligrams, since they mean the same mass:

1 g = 1000 mg

The next unit of measurement is kilogram(kg). The kilogram is a generally accepted unit of measurement. It measures everything. The kilogram is included in the SI system. Let us also include one more physical quantity in our SI table. We will call it “mass”:

There are a thousand grams in one kilogram. You can put an equal sign between one kilogram and a thousand grams, since they denote the same mass:

1 kg = 1000 g

The next unit of measurement is hundredweight(ts). In centners it is convenient to measure the mass of a crop collected from a small area or the mass of some cargo.

There are one hundred kilograms in one centner. You can put an equal sign between one centner and one hundred kilograms, since they denote the same mass:

1 c = 100 kg

The next unit of measurement is ton(T). Large loads and masses of large bodies are usually measured in tons. For example, the mass of a spaceship or car.

There are one thousand kilograms in one ton. One can put an equal sign between one ton and a thousand kilograms, since they denote the same mass:

1 t = 1000 kg

Time units

There is no need to explain what time we think is. Everyone knows what time is and why it is needed. If we open the discussion to what time is and try to define it, we will begin to delve into philosophy, and we do not need this now. Let's start with the units of time.

The following units of measurement are used to measure time:

• seconds;
• minutes;
• watch;
• day.

The smallest unit of measurement is second(With). There are, of course, smaller units such as milliseconds, microseconds, nanoseconds, but we will not consider them, since at the moment this makes no sense.

Various parameters are measured in seconds. For example, how many seconds does it take for an athlete to run 100 meters? The second is included in the SI international system of units for measuring time and is designated as "s". Let us also include one more physical quantity in our SI table. We will call it “time”:

minute(m). There are 60 seconds in one minute. One minute and sixty seconds can be equated because they represent the same time:

1 m = 60 s

The next unit of measurement is hour(h). There are 60 minutes in one hour. An equal sign can be placed between one hour and sixty minutes, since they represent the same time:

1 hour = 60 m

For example, if we studied this lesson for one hour and we are asked how much time we spent studying it, we can answer in two ways: “we studied the lesson for one hour” or so “we studied the lesson for sixty minutes” . In both cases, we will answer correctly.

The next unit of time is day. There are 24 hours in a day. You can put an equal sign between one day and twenty-four hours, since they mean the same time:

1 day = 24 hours

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Magnitude is something that can be measured. Concepts such as length, area, volume, mass, time, speed, etc. are called quantities. The value is measurement result, it is determined by a number expressed in certain units. The units in which a quantity is measured are called units of measurement.

To indicate a quantity, a number is written, and next to it is the name of the unit in which it was measured. For example, 5 cm, 10 kg, 12 km, 5 min. Each quantity has countless values, for example the length can be equal to: 1 cm, 2 cm, 3 cm, etc.

The same quantity can be expressed in different units, for example kilogram, gram and ton are units of weight. The same quantity in different units is expressed by different numbers. For example, 5 cm = 50 mm (length), 1 hour = 60 minutes (time), 2 kg = 2000 g (weight).

To measure a quantity means to find out how many times it contains another quantity of the same kind, taken as a unit of measurement.

For example, we want to find out the exact length of a room. This means we need to measure this length using another length that is well known to us, for example using a meter. To do this, set aside a meter along the length of the room as many times as possible. If it fits exactly 7 times along the length of the room, then its length is 7 meters.

As a result of measuring the quantity, we obtain or named number, for example 12 meters, or several named numbers, for example 5 meters 7 centimeters, the totality of which is called compound named number.

Measures

In each state, the government has established certain units of measurement for various quantities. An accurately calculated unit of measurement, adopted as a standard, is called standard or exemplary unit. Model units of the meter, kilogram, centimeter, etc. were made, according to which units for everyday use were made. Units that have come into use and are approved by the state are called measures.

The measures are called homogeneous, if they serve to measure quantities of the same kind. So, gram and kilogram are homogeneous measures, since they are used to measure weight.

Units

Below are units of measurement of various quantities that are often found in mathematics problems:

Weight/mass measures

• 1 ton = 10 quintals
• 1 quintal = 100 kilograms
• 1 kilogram = 1000 grams
• 1 gram = 1000 milligrams

• 1 kilometer = 1000 meters
• 1 meter = 10 decimeters
• 1 decimeter = 10 centimeters
• 1 centimeter = 10 millimeters

• 1 sq. kilometer = 100 hectares
• 1 hectare = 10,000 sq. meters
• 1 sq. meter = 10000 sq. centimeters
• 1 sq. centimeter = 100 square meters millimeters

• 1 cu. meter = 1000 cubic meters decimeters
• 1 cu. decimeter = 1000 cubic meters centimeters
• 1 cu. centimeter = 1000 cubic meters millimeters

Let's consider another quantity like liter. A liter is used to measure the capacity of vessels. A liter is a volume that is equal to one cubic decimeter (1 liter = 1 cubic decimeter).

Measures of time

• 1 century (century) = 100 years
• 1 year = 12 months
• 1 month = 30 days
• 1 week = 7 days
• 1 day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds
• 1 second = 1000 milliseconds

In addition, time units such as quarter and decade are used.

• quarter - 3 months
• decade - 10 days

A month is taken to be 30 days unless it is necessary to specify the date and name of the month. January, March, May, July, August, October and December - 31 days. February in a simple year is 28 days, February in a leap year is 29 days. April, June, September, November - 30 days.

A year is (approximately) the time it takes for the Earth to complete one revolution around the Sun. It is customary to count every three consecutive years as 365 days, and the fourth year following them as 366 days. A year containing 366 days is called leap year, and years containing 365 days - simple. One extra day is added to the fourth year for the following reason. The Earth's revolution around the Sun does not contain exactly 365 days, but 365 days and 6 hours (approximately). Thus, a simple year is shorter than a true year by 6 hours, and 4 simple years are shorter than 4 true years by 24 hours, i.e., by one day. Therefore, one day is added to every fourth year (February 29).

You will learn about other types of quantities as you further study various sciences.

Abbreviated names of measures

Abbreviated names of measures are usually written without a dot:

Measuring instruments

Special measuring instruments are used to measure various quantities. Some of them are very simple and designed for simple measurements. Such instruments include a measuring ruler, tape measure, measuring cylinder, etc. Other measuring instruments are more complex. Such devices include stopwatches, thermometers, electronic scales, etc.

Measuring instruments usually have a measuring scale (or scale for short). This means that there are line divisions on the device, and next to each line division the corresponding value of the quantity is written. The distance between the two strokes, next to which the value of the value is written, can be additionally divided into several smaller divisions; these divisions are most often not indicated by numbers.

It is not difficult to determine what value each smallest division corresponds to. So, for example, the figure below shows a measuring ruler:

The numbers 1, 2, 3, 4, etc. indicate the distances between the strokes, which are divided into 10 identical divisions. Therefore, each division (the distance between the nearest strokes) corresponds to 1 mm. This quantity is called at the cost of a scale division measuring device.

Before you begin measuring a value, you should determine the scale division value of the instrument you are using.

In order to determine the division price, you must:

1. Find the two closest lines on the scale, next to which the values ​​of the quantity are written.
2. Subtract the smaller number from the larger value and divide the resulting number by the number of divisions between them.

As an example, let’s determine the price of the scale division of the thermometer shown in the figure on the left.

Let's take two lines, near which the numerical values ​​of the measured value (temperature) are plotted.

For example, bars indicating 20 °C and 30 °C. The distance between these strokes is divided into 10 divisions. Thus, the price of each division will be equal to:

(30 °C - 20 °C) : 10 = 1 °C

Therefore, the thermometer shows 47 °C.

Each of us constantly has to measure various quantities in everyday life. For example, in order to arrive at school or work on time, you have to measure the time that will be spent on the road. Meteorologists measure temperature, barometric pressure, wind speed, etc. to predict the weather.

Measurements are based on comparison of identical properties of material objects. For properties for which physical methods are used for quantitative comparison, metrology has established a single generalized concept - a physical quantity. Physical quantity- a property that is qualitatively common to many physical objects, but quantitatively individual for each object, for example, length, mass, electrical conductivity and heat capacity of bodies, gas pressure in a vessel, etc. But smell is not a physical quantity, since it is established using subjective sensations.

A measure for quantitative comparison of identical properties of objects is unit of physical quantity - a physical quantity that, by agreement, is assigned a numerical value equal to 1. Units of physical quantities are assigned a full and abbreviated symbolic designation - dimension. For example, mass - kilogram (kg), time - second (s), length - meter (m), force - Newton (N).

The value of a physical quantity is assessment of a physical quantity in the form of a certain number of units accepted for it characterizes the quantitative individuality of objects. For example, the diameter of the hole is 0.5 mm, the radius of the globe is 6378 km, the speed of the runner is 8 m/s, the speed of light is 3 10 5 m/s.

By measuring is called finding the value of a physical quantity using special technical means. For example, measuring the shaft diameter with a caliper or micrometer, liquid temperature with a thermometer, gas pressure with a pressure gauge or vacuum gauge. Physical quantity value x^, obtained during measurement is determined by the formula x^ = ai, Where A- numerical value (size) of a physical quantity; and is a unit of physical quantity.

Since the values ​​of physical quantities are found experimentally, they contain measurement error. In this regard, a distinction is made between true and actual values ​​of physical quantities. True meaning - the value of a physical quantity that ideally reflects the corresponding property of an object in qualitative and quantitative terms. It is the limit to which the value of a physical quantity approaches with increasing measurement accuracy.

Real value - a value of a physical quantity found experimentally that is so close to the true value that it can be used instead for a certain purpose. This value varies depending on the required measurement accuracy. In technical measurements, the value of a physical quantity found with an acceptable error is accepted as the actual value.

Measurement error is the deviation of the measurement result from the true value of the measured value. Absolute error called the measurement error expressed in units of the measured value: Oh = x^- x, Where X- the true value of the measured quantity. Relative error - the ratio of the absolute measurement error to the true value of a physical quantity: 6=Ax/x. The relative error can also be expressed as a percentage.

Since the true value of the measurement remains unknown, in practice only an approximate estimate of the measurement error can be found. In this case, instead of the true value, the actual value of a physical quantity is taken, obtained by measuring the same quantity with a higher accuracy. For example, the error in measuring linear dimensions with a caliper is ±0.1 mm, and with a micrometer - ± 0.004 mm.

The measurement accuracy can be expressed quantitatively as the reciprocal of the relative error modulus. For example, if the measurement error is ±0.01, then the measurement accuracy is 100.

Physical size is a physical property of a material object, process, physical phenomenon, characterized quantitatively.

Physical quantity value expressed by one or more numbers characterizing this physical quantity, indicating the unit of measurement.

The size of a physical quantity are the values ​​of numbers appearing in the value of a physical quantity.

Units of measurement of physical quantities.

Unit of measurement of physical quantity is a quantity of fixed size that is assigned a numerical value equal to one. It is used for the quantitative expression of physical quantities homogeneous with it. A system of units of physical quantities is a set of basic and derived units based on a certain system of quantities.

Only a few systems of units have become widespread. In most cases, many countries use the metric system.

Basic units.

Measure a physical quantity - means to compare it with another similar physical quantity taken as a unit.

The length of an object is compared with a unit of length, the mass of a body with a unit of weight, etc. But if one researcher measures the length in fathoms and another in feet, it will be difficult for them to compare the two values. Therefore, all physical quantities throughout the world are usually measured in the same units. In 1963, the International System of Units SI (System international - SI) was adopted.

For each physical quantity in the system of units there must be a corresponding unit of measurement. Standard units is its physical implementation.

The length standard is meter- the distance between two strokes applied on a specially shaped rod made of an alloy of platinum and iridium.

Standard time serves as the duration of any regularly repeating process, for which the movement of the Earth around the Sun is chosen: the Earth makes one revolution per year. But the unit of time is taken not to be a year, but give me a sec.

For a unit speed take the speed of such uniform rectilinear motion at which the body moves 1 m in 1 s.

A separate unit of measurement is used for area, volume, length, etc. Each unit is determined when choosing a particular standard. But the system of units is much more convenient if only a few units are selected as the main ones, and the rest are determined through the main ones. For example, if the unit of length is a meter, then the unit of area will be a square meter, volume will be a cubic meter, speed will be a meter per second, etc.

Basic units The physical quantities in the International System of Units (SI) are: meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), candela (cd) and mole (mol).

There are also derived SI units that have their own names:

ANDmeasurements. To obtain an accurate, objective and easily reproducible description of a physical quantity, measurements are used. Without measurements, a physical quantity cannot be characterized quantitatively. Definitions such as “low” or “high” pressure, “low” or “high” temperature reflect only subjective opinions and do not contain comparisons with reference values. When measuring a physical quantity, a certain numerical value is assigned to it.

Measurements are carried out using measuring instruments. There are quite a large number of measuring instruments and devices, from the simplest to the most complex. For example, length is measured with a ruler or tape measure, temperature with a thermometer, width with calipers.

Measuring instruments are classified: by the method of presenting information (displaying or recording), by the method of measurement (direct action and comparison), by the form of presentation of readings (analog and digital), etc.

The following parameters are typical for measuring instruments:

Measuring range- the range of values ​​of the measured quantity for which the device is designed during its normal operation (with a given measurement accuracy).

Sensitivity threshold- the minimum (threshold) value of the measured value, distinguished by the device.

Sensitivity- connects the value of the measured parameter and the corresponding change in the instrument readings.

Accuracy- the ability of the device to indicate the true value of the measured indicator.

Stability- the ability of the device to maintain a given measurement accuracy for a certain time after calibration.