Direct measurements refers to measurements that are obtained directly with a measuring device. Direct measurements include measuring length with a ruler, vernier calipers, measuring voltage with a voltmeter, measuring temperature with a thermometer, etc. Various factors can affect the results of direct measurements. Therefore, the measurement error has a different form, i.e. there is a device error, systematic and random errors, rounding errors when reading off the scale of the device, misses. In this regard, it is important to identify in each specific experiment which of the measurement errors is the largest, and if it turns out that one of them exceeds all the others by an order of magnitude, then the latter errors can be neglected.
If all the errors taken into account are of the same order of magnitude, then it is necessary to evaluate the combined effect of several different errors. In general, the total error is calculated using the formula:
where - random error, - device error, - rounding error.
In most experimental studies, a physical quantity is not measured directly, but through other quantities, which in turn are determined by direct measurements. In these cases, the measured physical quantity is determined through directly measured quantities by means of formulas. Such measurements are called indirect. In the language of mathematics, this means that the desired physical quantity f related to other quantities x 1, x 2, x 3, … ,. x n functional dependence, i.e.
F= f(x 1 , x 2 ,…., X n )
An example of such dependencies is the volume of a sphere
.
In this case, the indirectly measured quantity is V- ball, which is determined by direct measurement of the radius of the ball R. This measured value V is a function of one variable.
Another example would be the density of a solid
.
(8)
Here - is an indirectly measured quantity that is determined by direct measurement of body weight m and an indirect value V... This measured value is a function of two variables, i.e.
= (m, V)
The theory of errors shows that the error of a function is estimated by the sum of the errors of all arguments. The error of a function will be the smaller, the smaller the errors of its arguments.
4.Plotting graphs from experimental measurements.
An essential aspect of the experimental study is graphing. When plotting graphs, the first step is to select a coordinate system. The most common is a rectangular coordinate system with a coordinate grid formed by equidistant parallel straight lines (for example, graph paper). On the coordinate axes, at regular intervals, divisions are drawn in a certain scale for the function and argument.
In laboratory work, in the study of physical phenomena, it is necessary to take into account changes in some quantities depending on changes in others. For example: when considering body movement, a functional dependence of the distance traveled on time is established; when studying the electrical resistance of a conductor from temperature. There are many more examples.
Variable Havecalled a function of another variable X(argument) if each value Have will correspond to a well-defined value of the quantity X, then the dependence of the function can be written in the form Y \u003d Y (X).
From the definition of the function it follows that to define it, you must specify two sets of numbers (values \u200b\u200bof the argument X and functions Have), as well as the law of interdependence and correspondence between them ( X and Y). The function can be experimentally specified in four ways:
Table; 2. Analytically, in the form of a formula; 3. Graphically; 4. Verbally.
For example: 1. Tabular way of setting the function - the dependence of the DC current I from voltage value U, i.e. I= f(U) .
table 2
2. The analytical way of defining a function is established by a formula, with the help of which the corresponding values \u200b\u200bof the function can be determined from the given (known) values \u200b\u200bof the argument. For example, the functional dependence shown in Table 2 can be written by the formula:
(9)
3.Graphic way of setting the function.
Function graph I= f(U) in the Cartesian coordinate system is called the locus of points, built from the numerical values \u200b\u200bof the coordinate point of the argument and function.
In fig. 1 plotted dependence I= f(U) given by the table.
The points found experimentally and plotted on the graph are clearly marked in the form of circles, crosses. On the graph, for each plotted point, it is necessary to indicate the errors in the form of "hammers" (see Fig. 1). The sizes of these "hammers" should be equal to twice the value of the absolute errors of the function and argument.
The scale of the graphs should be chosen so that the smallest distance, measured according to the graph, would not be less than the largest absolute measurement error. However, this choice of scale is not always convenient. In some cases, it is more convenient to take a slightly larger or smaller scale along one of the axes.
If the investigated interval of values \u200b\u200bof the argument or function is spaced from the origin of coordinates by an amount comparable to the value of the interval itself, then it is advisable to move the origin to a point close to the beginning of the interval under study, both along the abscissa and along the ordinate.
Drawing a curve (i.e. connecting experimental points) through points is usually done in accordance with the ideas of the method of least squares. In the theory of probability, it is shown that the best approximation to the experimental points will be such a curve (or straight line) for which the sum of the least squares of the vertical deviations from the point to the curve will be minimal.
The points marked on the coordinate paper are connected by a smooth curve, and the curve should pass as close as possible to all experimental points. The curve should be drawn so that it lies as close as possible to the points of the not exceeded errors and that on both sides of the curve there are approximately equal numbers of them (see Fig. 2).
If, when constructing a curve, one or more points go beyond the range of permissible values \u200b\u200b(see Fig. 2, points AND and AT), then the curve is drawn along the remaining points, and the dropped out points AND and AT as misses are not taken into account. Then repeated measurements are carried out in this area (points AND and AT) and the reason for such a deviation is established (either it is a mistake or a legitimate violation of the found dependence).
If the investigated, experimentally constructed function detects "special" points (for example, points of extremum, inflection, break, etc.). Then the number of experiments increases at small values \u200b\u200bof the step (argument) in the region of singular points.
- measuring the resistance of delta-connected resistors. In this case, the value of resistance between the peaks is measured. The results are used to determine the resistance of the resistors.
- determination of masses of weights of a set of weights (1, 2, 2, 5) kg using one standard weight of 1 kg and a mass comparator ("scales" designed to determine the difference in the masses of two weights). Comparison, for example:
Standard with 1 kg weight from the set; - standard + 1 kg weight from a set with a 2 kg weight from a set; - standard + 1 kg weight from the set with another 2 kg weight from the set; - weights 1 + 2 + 2 kg from the set with the remaining weight 5 kg from the set.
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An excerpt describing the types of measurements
“I’ll do it,” said Prince Andrey, moving away from the map.- And what do you care about, gentlemen? - said Bilibin, still listening to their conversation with a cheerful smile and now, apparently, intending to joke. - Whether there will be victory or defeat tomorrow, the glory of Russian weapons is insured. Apart from your Kutuzov, there is not a single Russian leader of the columns. Chiefs: Herr general Wimpfen, le comte de Langeron, le prince de Lichtenstein, le prince de Hohenloe et enfin Prsch ... prsch ... et ainsi de suite, comme tous les noms polonais. [Wimpfen, Count Lanzheron, Prince of Liechtenstein, Hohenloe and also Prishprishiprsh, like all Polish names.]
- Taisez vous, mauvaise langue, [Restrain your malice.] - said Dolgorukov. - It's not true, now there are two Russians: Miloradovich and Dokhturov, and he would have been 3rd, Count Arakcheev, but his nerves are weak.
- However, Mikhail Ilarionovich, I think, came out, - said Prince Andrey. - I wish you happiness and success, gentlemen, - he added and left, shaking hands with Dolgorukov and Bibilin.
Returning home, Prince Andrey could not resist asking Kutuzov, who was silently sitting next to him, what he thinks about tomorrow's battle.
Kutuzov looked sternly at his adjutant and, after a pause, answered:
- I think that the battle will be lost, and I told Count Tolstoy so and asked him to convey this to the Emperor. What do you think he answered me? Eh, mon cher general, je me mele de riz et des et cotelettes, melez vous des affaires de la guerre. [And, dear general! I'm busy with rice and cutlets, and you are doing military affairs.] Yes ... That's what they answered me!
At 10 o'clock in the evening, Weyrother with his plans moved to Kutuzov's apartment, where a military council was appointed. All the leaders of the columns were called upon to the commander-in-chief, and, with the exception of Prince Bagration, who refused to come, all appeared at the appointed hour.
Weyrother, who was the complete controller of the proposed battle, represented with his liveliness and haste a sharp contrast with the disgruntled and sleepy Kutuzov, who reluctantly played the role of chairman and leader of the military council. Weyrother evidently felt himself at the head of a movement that was already unstoppable. He was like a harnessed horse running downhill with a cart. Whether he was driving or was being driven, he did not know; but he rushed as fast as possible, no longer having time to discuss what this movement would lead to. Weyrother that evening was twice for a personal examination in the enemy chain and twice with the sovereigns, Russian and Austrian, for a report and explanations, and in his office, where he dictated the German disposition. He, exhausted, has now come to Kutuzov.
He, apparently, was so busy that he even forgot to be respectful with the commander-in-chief: he interrupted him, spoke quickly, indistinctly, without looking into the face of the interlocutor, without answering the questions put to him, was stained with dirt and looked pathetic, exhausted, confused and at the same time arrogant and proud.
Measurements are distinguished by the method of obtaining information, by the nature of changes in the measured value during the measurement, by the amount of measuring information, in relation to the basic units.
According to the method of obtaining information, measurements are divided into direct, indirect, aggregate and joint.
Direct measurements Is a direct comparison of a physical quantity with its measure. For example, when determining the length of an object with a ruler, the required value (a quantitative expression of the length value) is compared with a measure, i.e. a ruler.
Indirect measurements - differ from direct ones in that the sought-for value of the quantity is established from the results of direct measurements of such quantities that are associated with the sought-for specific dependence. So, if you measure the current with an ammeter, and the voltage with a voltmeter, then using the known functional relationship of all three quantities, you can calculate the power of the electrical circuit.
Aggregate measurements - are associated with the solution of a system of equations compiled from the results of simultaneous measurements of several homogeneous quantities. The solution of the system of equations makes it possible to calculate the desired value.
Joint measurements Are measurements of two or more inhomogeneous physical quantities to determine the relationship between them.
Aggregate and shared measurementsoften used in measurements of various parameters and characteristics in the field of electrical engineering.
By the nature of the change in the measured value during the measurement process, there are statistical, dynamic and static measurements.
Statistical measurementsassociated with the determination of the characteristics of random processes, sound signals, noise levels, etc. Static measurements take place when the measured value is practically constant.
Dynamic measurementsassociated with such quantities that undergo certain changes in the course of measurements. Ideal static and dynamic measurements are rare in practice.
According to the amount of measurement information, single and multiple measurements are distinguished.
Single measurements - this is one measurement of one quantity, i.e. the number of measurements is equal to the number of measured quantities. The practical application of this type of measurement is always associated with large errors, therefore, at least three single measurements should be carried out and the final result should be found as the arithmetic mean.
Multiple measurementscharacterized by an excess of the number of measurements of the number of measured values. The advantage of multiple measurements is in a significant reduction in the influence of random factors on the measurement error.
According to the used measurement method - a set of techniques for using principles and measuring instruments are distinguished:
- direct assessment method;
- method of comparison with a measure;
- the method of opposition;
- differential method;
- method zero;
- substitution method;
- method of coincidences.
According to the conditions that determine the accuracy of the result, measurements are divided into three classes: measurements of the highest possible accuracy achievable with the existing level of technology; control and verification measurements, the error of which should not exceed a certain specified value; technical (working) measurements, in which the error of the measurement result is determined by the characteristics of the measuring instruments.
Definition 1
Measurement is a set of specific actions in order to identify the ratio of one homogeneous quantity that is being measured to another stored in the measuring instrument. The resulting value is the numerical value of the measured physical quantity.
Measurement concept in physics
The process of measuring the indicator of a physical quantity in practice is carried out through the use of a variety of measuring instruments and special devices, installations and systems.
The measurement of a physical quantity includes two basic steps:
- comparison of a quantity that is measured with a unit;
- different display methods for converting to a comfortable form.
The measurement principle is considered to be a physical phenomenon (effect) underlying the measurement. A measurement method is one technique or a set of certain measuring actions carried out in accordance with the implemented measurement principles.
The obtained error characterizes the measurement accuracy. In a more simplified format, by applying a ruler with divisions to a certain part, in essence, its size is compared with a unit on the ruler, and after performing the appropriate calculations, the value of the quantity (thickness, length, height and other parameters of the measured part) is obtained.
Remark 1
In cases where it is impossible to perform measuring actions, in practice, such values \u200b\u200bare assessed based on conventional scales (for example, the Mohs and Richter scales, characterizing the hardness of metals and earthquakes).
The importance of existence and classification of measurements in physics
Definition 2
The science responsible for the study of all aspects of measurement is called metrology.
Measurements in physics occupy an important position, since they allow comparing the results of theoretical and experimental research. All measurements are classified in a specific way:
- according to the types of measurements (indirect, direct, aggregate (when a complex measurement of several quantities of the same name is made, where the desired value is determined by solving a system of corresponding equations for various combinations of quantities), joint (in order to determine the relationship between several non-identical quantities);
- according to measurement methods (direct assessment (the value of a quantity is established by calculations exclusively using the indicating measuring instrument), comparison with a measure, measurement by substitution (where the measured quantity is replaced by a measure with an already known quantity value), zero, differential (the measured quantity is compared with a homogeneous quantity with already known value, insignificantly different from it, and where the difference between these two values \u200b\u200bis established), measurement by complement);
- by appointment (metrological and technical);
- by accuracy (deterministic and random);
- in relation to changes in the measured value (dynamic and static);
- based on the quantitative indicator of measurements (multiple and single);
- by the final indicators of measurements (relative (characterized by measuring the ratio of a physical quantity to the unit of the same name (initial) value), and absolute (based on direct measurements of one or several key quantities and the use of values \u200b\u200bof physical constants (constants).
The concept of direct and indirect measurements in physics
Remark 2
The values \u200b\u200bobtained according to the measurement results of different quantities may in fact be dependent on each other. In physics, a relationship is established between similar quantities and is expressed in the format of certain formulas that demonstrate the process of finding the numerical values \u200b\u200bof some quantities by analogous values \u200b\u200bof others.
According to the classification criterion, measurements can be divided into direct and indirect, which is a direct characteristic of their type.
Direct measurement is a measurement according to which the sought-for values \u200b\u200bof physical quantities are obtained directly. In the case of direct measurements, specialized instruments are used for measuring purposes, which are responsible for changing the investigated quantity itself. So, the mass of bodies, for example, can be found using the indicator on the scales, the length is recognized by measuring with a ruler, and the time is recorded using a stopwatch.
Indirect measurement is considered in physics to establish the desired value of a quantity based on the results of direct measurement of the remaining physical quantities obtained from the measurement, which are functionally interconnected with the original quantity.
The same values \u200b\u200bin other cases can be found exclusively due to indirect measurements - recalculation of other important values, whose values \u200b\u200bwere obtained in the process of direct measurements.
This is how physicists calculate the distance from our planet to the Sun, the mass of the Earth, or, for example, the duration of geological periods. Measurement of the density of bodies, according to the indicators of their volumes and mass, the speed of trains (according to the value of the distance traveled for a known time), should also be attributed to an indirect measurement.
Since physics is not an exact science like mathematics, it is not absolutely precise. So, within the framework of physical experiments, any kind of measurement (both indirect and direct) can give not an exact, but only an approximate value of the measured physical quantity.
Remark 3
When measuring, for example, the length, the result obtained will depend on the accuracy of the selected device (for example, a vernier caliper allows measurements with an accuracy of 0.1 mm, and a ruler only up to 1 mm); on the quality of external conditions, such as temperature, humidity, tendency to deformation states, etc.
Consequently, the results of indirect measurements, calculated from the approximate results obtained from direct measurements, will also be approximate. For this reason, in parallel with the result, an indication of its accuracy is always required, called the absolute error of the results.
Metrology called the science of measurements, methods and means of ensuring their unity and ways to achieve the required accuracy.
Measurement is called finding the value physical quantity empirically with special technical means ... The measurement result is a quantitative characteristic of a physical quantity in the form of the number of units of the measured quantity and the error with which this number is obtained.
Types of measurements. Depending on the method of obtaining the numerical value of the measured value, measurements are divided into direct, indirect and cumulative measurements.
Straight measurements are called in which the desired value of the quantity is obtained from experimental data. In direct measurements, experimental operations are performed on the measured value itself. The numerical value of the measured quantity is obtained in an experimental comparison with a measure or according to instrument readings. For example, measuring current with an ammeter, voltage with a voltmeter, temperature with a thermometer, weight on a scale.
Indirect are called such measurements in which the numerical value of the measured quantity is determined by the known functional dependence through other quantities that can be directly measured. In indirect measurements, the numerical value of the measured quantity is obtained with the participation of an operator on the basis of direct measurements - by solving one equation. Indirect measurements are used when it is inconvenient or impossible to automatically calculate the known relationship between one or more input quantities and the measured quantity. For example, the power in DC circuits is determined by the operator by multiplying the voltage by the current, measured directly with an ammeter and voltmeter.
The deviation of the measurement result from the true value of the measured quantity is called measurement error .
Absolute measurement error is equal to the difference between the measurement result and the true value of the measured quantity: 
.
Relative measurement error is the ratio of the absolute measurement error to the true value of the measured quantity. Usually the relative error is expressed as a percentage 
%.
25. Basic concepts and definitions: information, algorithm, program, command, data, technical devices.
Information - from the Latin word "information", which means information, clarification, presentation.
With regard to computer data processing, information is understood as a certain sequence of symbolic designations (letters, numbers, encoded graphics and sounds, etc.), which carries a semantic load and is presented in a computer-understandable form. Each new character in such a sequence of characters increases the information volume of the message.
Algorithm - a sequence of clearly defined actions, the implementation of which leads to the solution of the problem. An algorithm written in the language of a machine is a program for solving a problem.
Algorithm properties: discreteness, comprehensibility, efficiency, certainty, mass character.
Program - a sequence of actions, instructions, prescriptions for some computing device; the file containing this sequence of actions.
A command is an instruction to a computer program to act as an interpreter for solving a problem. More generally, a command is a hint to some command line interface.
Data is information presented in a formalized form, which makes it possible to store, process and transfer it.
Technical devices (informatization means) are a set of systems, machines, instruments, mechanisms, devices and other types of equipment designed to automate various technological processes of informatics, and those whose output product is precisely information (information, knowledge) or data used for satisfaction of information needs in various areas of the objective activity of society.