Plotting graphs in a physics course based on functional dependence. Constructing graphs in a physics course based on functional dependence. What is the general principle for constructing graphs of physical quantities

Rules for constructing graphs

It is possible to construct two types of graphs: general view without numerical data and with digital data.

Drawing graphs in a “general form” without numerical data helps the student to correctly comprehend the problem and convey the general trend of changes in a particular function based on mathematical analysis of the dependence.

A graph with digital data is constructed in the following sequence:

1. Graphs should only be drawn on suitable specialty paper (eg graph paper).

2. For a given range of argument changes, determine the maximum and minimum values ​​of the function at the boundaries of the required range of argument changes.

So, to plot a graph of X = 4t 2 - 6t + 2 in the range of t from 0 to 2 s, we have:

When determining intervals of function and argument values, you should round their last significant digits downwards from the smallest possible values ​​and upward from the largest possible values. In our example, t changes from 0 to 3 s and X changes from -1 m to +7 m.

3. Select the sheet size for the graph so that there are free margins of 1.5-2 cm wide around the coordinate angle field and scale inscriptions.

4. Select a linear scale of the coordinate axes along the rounded boundaries of the intervals so that the lengths of the axes segments for functions and arguments are approximately the same, but so that the division of the intervals into countable parts forms scales convenient for counting any values ​​of quantities. Determine the scale for plotting such that the sheet area is used to the maximum. To do this, select the sheet size for the graph in such a way that around the field of the coordinate sheet and the scale inscriptions there are free margins 1.5 - 2 cm wide. Next, determine the scale for constructing the graph. For example, for the above example, the field for constructing a graph turned out to be equal to the field of a school notebook, then for constructing a graph you can use 10-12 cm horizontally (x-axis) and 8-10 cm vertically (ordinate axis). Thus, we obtain the scale x and y for the x and y axes respectively:

5. Combine the smallest rounded values ​​of the argument (along the abscissa axis) and the function (along the ordinate axis) with the origin of coordinates.

6. Construct the axes of the graph by plotting a series of numbers on them with a constant step in the form arithmetic progression and are designated by numbers at regular intervals, convenient for reading the values. These symbols should not be spaced too frequently or sparingly. The numbers on the graph axes should be simple; they should not be associated with calculated values. If the numbers are very large or very small, then they are multiplied by a constant factor of type 10 n (n is an integer), moving this factor to the end of the axis. Instead of digital designations, argument and function symbols are placed at the ends of the axes with the names of their units of measurement, separated by a comma. For example, when constructing a pressure axis P in the range from 0 to 0.003 N/m2, it is advisable to multiply P by 10 3 and depict the axis as follows (Fig. 7):

Rice. 7.

The calculated or experimentally obtained values ​​of quantities are plotted on the graph, guided by the table of values ​​of quantities. To construct a smooth curve, it is enough to calculate 5-6 points. In theoretical calculations, the points on the graph are not highlighted (Fig. 8a).

The experimental graph is constructed as an approximated curve point by point (Fig. 8b).

7. When constructing graphs based on experimental data, it is necessary to indicate experimental points on the graph. In this case, each value of the quantity must be shown taking into account the confidence interval. Confidence intervals are plotted from each point in the form of straight line segments (horizontal for arguments and vertical for functions). The total length of these segments on the graph scale should be equal to twice the absolute measurement error. Experimental points can be depicted in the form of crosses, rectangles or ellipses with horizontal dimensions of 2x and vertical dimensions of 2y. When depicting confidence intervals of functions and arguments on graphs, the ends of the vertical and horizontal lines with a dot in the middle represent the axes of the scattering area of ​​the values ​​(Fig. 9).

If the lines of confidence intervals beyond smallness cannot be depicted on the scale of the graph, the value point is surrounded by a small circle, triangle or diamond. Note that the experimental curves should be drawn smooth, with maximum approximation to the confidence intervals of the experimental values. The considered example in Fig. 9 illustrates the most common form of graphs that a student will have to construct when processing experimental data.

Graphic image quantities is a unique language that is visual and highly informative, provided that it is used correctly and undistorted. Therefore, it is useful to familiarize yourself with examples of errors in the design of graphs presented in Fig. 10.

Graphs of two functions of the same argument, for example F() and K(), can be combined on a common x-axis. In this case, the scales of the ordinate axes are plotted on the left for one and on the right for another function. The graph's belonging to one or another function is indicated by arrows (Fig. 11a).

Graphs of one function for different values ​​of the constant are always combined on the same plane of the coordinate angle, the curves are numbered and the values ​​of the constants are written under the graph (Fig. 11b).

Prefixes for forming names of multiples and submultiples

Listed in table. 6 factors and prefixes are used to form multiples and submultiples from units of the International System of Units (SI), the GHS system, as well as from non-systemic units allowed by state standards. It is recommended to select consoles in such a way that numeric values values ​​ranged from 0.1 to 1. 10 3. For example, to express the number 3. 10 8 m/s it is better to choose the prefix mega, not kilo or giga. With the prefix kilo we get: 3. 10 8 m/s = 3. 10 5 km/s, i.e. a number greater than 10 3. With the prefix giga we get: 3. 10 8 m/s = 0.3. Hm/s, a number, although greater than 0.1, is not an integer. With the prefix mega we get: 3. 10 8 m/s = 3. 10 2 mm/s.

Table 6

The names and designations of decimal multiples and submultiples are formed by adding prefixes to the names of the original units. Connecting two or more consoles in a row is not allowed. For example, instead of the micromicroFarad unit, the picoFarad unit should be used.

The designation of the prefix is ​​written together with the designation of the unit to which it is attached. When naming a derived unit with a complex name, the SI prefix is ​​attached to the name of the first unit included in the product or numerator of the fraction. For example: kOhm. m, but not Om. km.

As an exception to this rule, it is allowed to attach a prefix to the name of the second unit included in the product or in the denominator of the fraction, if they are units of length, area or volume. For example: W/cm 3, V/cm, A/mm 2, etc.

In table 6 indicates prefixes for the formation of only decimal multiples and submultiples. In addition to these units, the state standard “Units physical quantities» multiples and submultiples time, plane angle and relative units that are not decimal. For example, time units: minute, hour, day; Angle units: degree, minute, second.

Expressing physical quantities in one system of units

To successfully solve a physical problem, you must be able to express all available numerical data in one system of units of measurement (SI or CGS). Such a translation is most conveniently carried out by replacing each factor in the dimension of a given value with an equivalent factor of the required system of units (SI or CGS), taking into account the conversion factor. If the latter is unknown, then conversion to any other intermediate system of units for which the conversion factor is known is possible.

Example 1. Write a = 0.7 km/min 2 in the SI system.

In this example, the conversion factors are known in advance (1 km = 10 3 m, 1 min = 60 s), therefore,

Example 2. Write P = 10 hp. (horsepower) in the SI system.

It is known that 1 hp. = 75 kgm/s. Conversion factor from hp. into watts is unknown to the student, so they use conversion through intermediate systems of units:

Example 3: Convert specific gravity d = 600 lb/gal (written in English system measures) in the GHS systems.

From the reference literature we find:

1 pound (English) = 0.454 kg (kilogram force).

1 gallon (English) = 4.546 liters (liter).

Hence,

An expression was obtained using non-systemic units, the translation of which into the GHS system, however, may be unknown to the student. Therefore, we use intermediate systems of units:

1 l = 10 -3 m 3 (SI) = 10 -3 (10 2 cm) 3 = 10 3 cm 3, and

1 kg = 9.8 N (SI) = 9.8(10 5 dynes) = 9.8. 10 5 din.

Graphs provide a visual representation of the relationship between quantities, which is extremely important when interpreting the data obtained, since graphic information is easily perceived, inspires more confidence, and has significant capacity. Based on the graph, it is easier to draw a conclusion about the correspondence of theoretical concepts to experimental data.

Graphs are drawn on graph paper. It is allowed to draw graphs on a notebook sheet in a box. The size of the graph is no less than 1012 cm. Graphs are constructed in a rectangular coordinate system, where the argument, an independent physical quantity, is plotted along the horizontal axis (abscissa axis), and the function, the dependent physical quantity, is plotted along the vertical axis (ordinate axis).

Typically, a graph is constructed based on a table of experimental data, from where it is easy to establish the intervals in which the argument and function change. Their smallest and largest values ​​​​specify the values ​​of the scales plotted along the axes. You should not try to place the point (0,0) on the axes, which is used as the origin on mathematical graphs. For experimental graphs, the scales on both axes are chosen independently of each other and, as a rule, are correlated with the error in measuring the argument and function: it is desirable that the value of the smallest division of each scale is approximately equal to the corresponding error.

The scale scale should be easy to read, and for this it is necessary to choose a scale division price that is convenient for perception: one cell should correspond to a multiple of 10 number of units of the physical quantity being set aside: 10 n, 210 n or 510 n, where n is any integer, positive or negative. So, the numbers are 2; 0.5; 100; 0.02 – suitable, and the numbers are 3; 7; 0.15 – not suitable for this purpose.

If necessary, the scale along the same axis for positive and negative values ​​of the plotted quantity can be chosen differently, but only if these values ​​differ by at least an order of magnitude, i.e. 10 times or more. An example is the current-voltage characteristic of a diode, when the forward and reverse currents differ by at least a thousand times: the forward current is milliamps, the reverse is microamps.

Arrows that specify a positive direction are usually not indicated on the coordinate axes if the accepted positive direction of the axes is selected: bottom - up and left - right. The axes are labeled: the abscissa axis is at the bottom right, the ordinate axis is at the top left. Against each axis indicate the name or symbol of the quantity plotted along the axis, and separated by a comma - the units of its measurement, and all units of measurement are given in Russian writing in the SI system. The numerical scale is chosen in the form of “round numbers” equally spaced in value, for example: 2; 4; 6; 8 ... or 1.82; 1.84; 1.86…. Scale risks are placed along the axes at equal distances from each other so that they appear on the graph field. On the abscissa axis, numbers of the numerical scale are written under the marks, on the ordinate axis - to the left of the marks. It is not customary to indicate the coordinates of experimental points near the axes.

Experimental points are carefully plotted on the graph field pencil. They are always marked so that they are clearly visible. If different dependencies are constructed in the same axes, obtained, for example, under changed experimental conditions or at different stages of work, then the points of such dependencies should differ from each other. They should be marked with different icons (squares, circles, crosses, etc.) or applied with pencils of different colors.

The calculated points obtained by calculations are placed evenly on the graph field. Unlike experimental points, they must merge with the theoretical curve after it is plotted. Calculated points, like experimental ones, are applied with a pencil - in case of an error, an incorrectly placed point is easier to erase.

Figure 1.5 shows the experimental dependence obtained point by point, which is plotted on paper with a coordinate grid.

Using a pencil, draw a smooth curve through the experimental points so that the points, on average, are equally located on both sides of the drawn curve. If the mathematical description of the observed dependence is known, then the theoretical curve is drawn in exactly the same way. There is no point in trying to draw a curve through each experimental point - after all, the curve is only an interpretation of the measurement results known from the experiment with an error. In essence, there are only experimental points, and the curve is an arbitrary, not necessarily correct, conjecture of the experiment. Let's imagine that all experimental points are connected and a broken line appears on the graph. It has nothing to do with true physical addiction! This follows from the fact that the shape of the resulting line will not be reproduced in repeated series of measurements.

Figure 1.5 – Dependence of the dynamic coefficient

water viscosity depending on temperature

On the contrary, the theoretical dependence is plotted on a graph in such a way that it passes smoothly through all calculated points. This requirement is obvious, since the theoretical values ​​of the coordinates of points can be calculated as accurately as desired.

A correctly constructed curve should fill the entire field of the graph, which will indicate the correct choice of scales along each of the axes. If a significant part of the field turns out to be unfilled, then it is necessary to re-select the scales and rebuild the dependence.

The measurement results on the basis of which experimental dependencies are constructed contain errors. To indicate their values ​​on a graph, two main methods are used.

The first was mentioned when discussing the issue of choosing scales. It consists in choosing the scale division value of the graph, which should be equal to the error of the value plotted along this axis. In this case, the accuracy of the measurements does not require additional explanation.

If it is not possible to achieve correspondence between the error and the division price, use the second method, which consists in directly displaying the errors on the graph field. Namely, two segments are constructed around the indicated experimental point, parallel to the abscissa and ordinate axes. On the selected scale, the length of each segment should be equal to twice the error of the value plotted along the parallel axis. The center of the segment should be at the experimental point. A sort of “whisker” is formed around the point, defining the range of possible values ​​of the measured value. Errors become visible, although “whiskers” may unwittingly litter the graph field. Note that this method is most often used when errors vary from measurement to measurement. The method is illustrated in Figure 1.6.

Figure 1.6 – Dependence of body acceleration on force,

attached to it

Using the principle of constructing a graph to find the critical sales volume, you can find - using a similar method, or with complications by entering relative indicators - both the critical price level and the critical

At first, conducting technical analysis of the market, especially using such a specific method, seems difficult. But if you thoroughly understand this, at first glance, not very presentable and dynamic method graphic construction, then it will turn out to be the most practical and effective. One of the reasons is that when using “tic-tac-toe” there is no particular need to use various technical market indicators, without which many simply cannot imagine the possibility of conducting analysis. You say this is contradictory common sense, asking the question “Where is technical analysis then?” - “It is in the very principle of constructing a tic-tac-toe chart,” I will answer. After reading the book, you will understand that the method really deserves to write a whole book about it.

Principles of charting

Principles of constructing statistical graphs

Graphic image. Many of the models or principles presented in this book will be expressed graphically. The most important of these patterns are designated as key charts. You should read the appendix to this chapter on graphing and analyzing quantitative relative relationships.

Sections A through C describe the use of corrections as trading tools. Corrections will first be linked to the Fibonacci PHI ratio in principle and then applied as charting tools on daily and weekly data sets for various products.

For these cases effective ways planning are based on the use of methods related to the construction of network diagrams (networks). The simplest and most common principle for constructing a network is the critical path method. In this case, the network is used to identify the impact of one job on another and on the program as a whole. The execution time of each job can be specified for each element of the network schedule.

Activities of subcontractors. Whenever possible, the project manager uses software and work breakdown structure (WBS) principles to schedule the activities of major subcontractors. Data from subcontractors should be capable of Level 1 or 2 scheduling, depending on the level of detail required by the contract.

Analysis is related to statistics and accounting. For a comprehensive study of all aspects of production and financial activity, data from both statistical and accounting data, as well as sample observations, are used. In addition, it is necessary to have a basic knowledge of the theory of groupings, methods for calculating average and relative indicators, indices, principles of constructing tables and graphs.

Of course, here is a graphical representation of one of the possible options for the team’s work. In practice, you will encounter a variety of options. In principle, there are a great many of them. And plotting a graph makes it possible to clearly illustrate each of these options.

Let us consider the principles of constructing universal “verification graphs” that allow graphically interpreting the verification results with a certain (specified) reliability.

On electrified lines, when constructing graphs, it is necessary to take into account the conditions for the most complete and rational use of power supply devices. To obtain the highest speeds for trains on these lines, it is especially important to place trains on the schedule evenly, according to the principle of a paired schedule, occupying the stages by alternately passing even and odd trains, while avoiding condensation of trains on the schedule at certain hours of the day.

Example 4. Graphs on coordinates with a logarithmic scale. The logarithmic scale on the coordinate axes is constructed according to the principle of constructing a slide rule.

The method of representation is material (physical, i.e. coinciding subject-mathematical) and symbolic (linguistic). Material physical models correspond to the original, but may differ from it in size, range of parameter changes, etc. Symbolic models are abstract and are based on their description by various symbols, including in the form of fixing an object in drawings, drawings, graphs, diagrams, texts, mathematical formulas etc. At the same time, they can be based on the principle of construction - probabilistic (stochastic) and deterministic in adaptability - adaptive and non-adaptive in terms of changes in output variables over time - static and dynamic in terms of the dependence of the model parameters on the variables - dependent and independent.

The construction of any model is based on certain theoretical principles and certain means of its implementation. A model built on the principles of mathematical theory and implemented using mathematical means is called a mathematical model. Modeling in the field of planning and management is based on mathematical models. The field of application of these models - economics - determined their commonly used name - economic-mathematical models. In economics, a model is understood as an analogue of any economic process, phenomenon or material object. A model of certain processes, phenomena or objects can be presented in the form of equations, inequalities, graphs, symbolic images, etc.

The principle of periodicity, reflecting the production and commercial cycles of an enterprise, is also important for building a management accounting system. Information for managers is required when it is appropriate, neither sooner nor later. Reducing the time plan can significantly reduce the accuracy of information produced by management accounting. As a rule, the management apparatus sets a schedule for collecting primary data, processing it and grouping it into final information.

Graph in Fig. 11 corresponds to the level of coverage amount of 200 DM per day. It was built as a result of an analysis carried out by an economics specialist, who reasoned as follows: how many cups of coffee at a price of 0.60 DM is enough to sell to obtain a coverage amount of 200 DM? What additional quantity will need to be sold if at a price of 0.45 DM they want to keep the same coverage amount 200 DM To calculate the target number of sales, you need to divide the target coverage amount for the day in the amount of 200 DM by the corresponding coverage amount per unit of product. The if principle applies. .., That... .

The stated principles for constructing scale-free network graphs were presented mainly in relation to site structures. The construction of network models for organizing the construction of the linear part of pipelines has a number of features.

The principles of constructing scale-free soybean graphs and graphs constructed on a time scale are outlined in Section 2, mainly in relation to on-site structures. The variegated network models for organizing the construction of the front part of pipelines have a number of features.

Another fundamental advantage of an intraday point-to-digit chart with single-cell reversal is the ability to identify price targets using a horizontal reference. If you mentally return to the basic principles of constructing a bar chart and price models discussed above, then remember that we have already touched on the topic of price benchmarks. However, almost every method of establishing price targets using a bar chart is based, as we said, on the so-called vertical measurement. It consists of measuring the height of some graphical model (swing range) and projecting the resulting distance up or down. For example, in the “head and shoulders” model, the distance from the “head” to the “neck” line is measured and the reference point is laid off from the breakout point, that is, the intersection of the “neck” line.

Must know the structure of the equipment being serviced, the recipe, types, purpose and features of the materials, raw materials, semi-finished products and finished products to be tested, the rules for conducting physical and mechanical tests of varying complexity with the performance of work on their processing and generalization, the principle of operation of ballistic installations for determining magnetic permeability, the main components of vacuum systems fore-vacuum and diffusion pumps, thermocouple vacuum gauge, basic methods for determining physical properties samples basic properties of magnetic bodies thermal expansion of alloys methods for determining linear expansion coefficients and critical points on dilatometers methods for determining temperature using high- and low-temperature thermometers elastic properties of metals and alloys rules for making corrections for the geometric dimensions of a sample methods for constructing graphs system for recording tests performed and generalization methods test results.

The same principle of constructing a calendar plan underlies schedules for planning production processes that have a complex structure. An example of the most typical schedule of this type is the cyclic schedule for the production of machines, used in single and small-scale mechanical engineering (Fig. 2). It shows in what sequence and with what calendar advance in relation to the planned release date of finished machines, parts and assemblies of this machine must be manufactured and submitted for subsequent processing and assembly, so that the scheduled final date for the series release is met. This schedule is based on technological diagram of the manufacture of parts and the sequence of their assembly during the assembly process, as well as on standard calculations of the duration of the production cycle for the manufacture of parts for the main stages - production of blanks, mechanical. processing, heat treatment, etc. and the assembly cycle of units and machines in general. Hence the graph is called cyclic. The calculation unit of time when constructing it is usually a working day, and the days are counted on the graph from right to left from the final date of the planned release in the reverse order of the machine manufacturing process. In practice, cycle schedules are compiled for a large range of components and parts, dividing the production time of large parts by stages of the production process (blanking, mechanical processing, heat treatment), sometimes highlighting the main mechanical operations. processing. Such graphs are much more cumbersome and complex than the diagram in Fig. 2. But they are indispensable when planning and controlling the production of products in serial production, especially in small-scale production.

The second example of a calendar optimization problem involves constructing a schedule that best matches the timing of product release at several successive stages of production (processing stages) with different processing times for the product at each of them. For example, in a printing house it is necessary to coordinate the work of the typesetting, printing and binding shops, subject to different labor and machine intensity for individual shops different types products (letterhead products, book products of simple or complex type, with or without binding, etc.). The problem can be solved under various optimization criteria and various restrictions. Thus, it is possible to solve the problem of the minimum duration of production, cycle and, therefore, the minimum value of the average balance of products in the work in progress (backlog); in this case, the restrictions should be determined by the available throughput of various workshops (processing areas). Another formulation of the same problem is possible, in which the optimization criterion is the greatest use of available production capacity under restrictions imposed on the production time of certain types of products. An algorithm for an exact solution of this problem (the so-called Johnson problem a) is developed for cases when the product undergoes only 2 operations, and for an approximate solution for three operations. At more operations, these algorithms are unsuitable, which practically depreciates them, since the need to solve the optimization problem calendar schedule Ch. arises. arr. in planning multi-operational processes (for example, in mechanical engineering). E. Bowman (USA) in 1959 and A. Lurie (USSR) in 1960 proposed mathematically rigorous algorithms based on the general ideas of linear programming and allowing, in principle, to solve the problem with any number of operations. However, at the present time (1965) these algorithms cannot be practically applied; they are too computationally cumbersome even for the most powerful existing electronic computers. Therefore, these algorithms have only promising significance; either they can be simplified, or the progress of computer technology will make it possible to implement them on new machines.

For example, if you are going to a car dealership to check out new cars, appearance, interior decoration, etc., then you are unlikely to be interested in graphs explaining the order of fuel injection into the engine cylinders, or discussions on the principles of constructing an engine control system. Most likely you will be interested in engine power, acceleration time to 100 km/h, fuel consumption per 100 km, comfort and equipment of the car. In other words, you will want to imagine what the car will be like to drive, how good you would look in it, when going on a trip with your girlfriend or boyfriend. As you imagine this trip, you will begin to think about all the features and benefits of the car that would be useful to you on your trip. This is a simple example of a use case.

For decades, the principle of flow in construction production has been proclaimed in building codes and regulations, in technological instructions and in textbooks. However, the theory of threading has not yet received a unified basis. Some employees of VNIIST and MINKh and GP express the idea that theoretical constructions and models created by flow are not always adequate to construction processes, and therefore schedules and calculations performed when designing a construction organization, as a rule, cannot be implemented.

Robert Rea studied Dow's writings and spent a lot of time compiling market statistics and adding to Dow's observations. He noticed that indexes were more prone than individual stocks to form horizontal lines or continuation chart formations. He was also one of the first

1. Design of axes, scale, dimension. The results of measurements and calculations are conveniently presented in graphical form. Graphs are drawn on graph paper; The dimensions of the graph should not be less than 150*150 mm (half a page of the laboratory journal). First of all, they are applied to the sheet coordinate axes. For the results of direct measurements, as a rule, they are plotted on the abscissa axis. At the ends of the axes, designations of physical quantities and their units of measurement are applied. Then scale divisions are applied to the axes so that the distance between divisions is 1, 2, 5 units or 1;2;5*10 ± n, where n is an integer. The intersection point of the axes does not have to correspond to zero along one or more axes. The origin of the axes and the scale should be chosen so that: 1) the curve (straight line) occupies the entire field of the graph; 2) the angles between the tangents to the curve and the axes should be close to 45º (or 135º) if possible in most of the graph.

2. Graphic representation of physical quantities. After selecting and applying scales to the axes, the values ​​of physical quantities are plotted on the sheet. They are designated by small circles, triangles, squares, and numerical values ​​corresponding to plotted points are not plotted on the axis. Then, from each point up and down, to the right and to the left, the corresponding errors on the scale of the graph are plotted in the form of segments.

After plotting the points, a graph is built, i.e. a smooth curve or straight line predicted by the theory is drawn so that it intersects all error areas or, if this is not possible, the sums of deviations of the experimental points below and above the curve should be close. In the upper right or left corner (sometimes in the middle) the name of the relationship that is depicted by the graph is written.

The exception is calibration graphs, on which points plotted without errors are connected by successive straight segments, and the calibration accuracy is indicated in the upper right corner, under the name of the graph. However, if during the calibration of the device the absolute measurement error changed, then the errors of each measured point are plotted on the calibration graph. (This situation is realized when calibrating the “amplitude” and “frequency” scales of the GSK generator using an oscilloscope). Calibration graphs are used to find intermediate values linear interpolations.

The graphs are drawn in pencil and pasted into the laboratory notebook.

3. Linear approximations. In experiments, it is often necessary to construct a graph of the dependence of the physical quantity obtained in the work Y from the obtained physical quantity X, approximating Y(x) linear function, where k, b– permanent. The graph of such a dependence is a straight line, and the slope k, is often itself the main goal of the experiment. It is natural that k in this case, it is also a physical parameter that must be determined with the accuracy inherent in a given experiment. One of the methods for solving this problem is the paired point method, described in detail in. However, it should be borne in mind that the paired point method is applicable if there is large number points n ~ 10, in addition, it is quite labor-intensive. The following graphical method for determining:

1) Based on experimental points plotted with errors, it is carried out

straight line using the least squares method (LSM).

The fundamental idea of ​​least squares approximation is minimization

total standard deviation of experimental points from

the desired straight line

In this case, the coefficients are determined from the minimization conditions:

Here are experimentally measured values, n is the number

experimental points.

As a result of solving this system, we have expressions for calculating

coefficients based on experimentally measured values:

2) After calculating the coefficients, the desired straight line is drawn. Then the experimental point is selected that has the largest, taking into account its error, deviation from the graph in the vertical direction DY max as indicated in Fig. 2. Then the relative error Dk/k, due to the inaccuracy of the Y values, , where the measuring interval of Y values ​​is from max to min. Moreover, both sides of the equality contain dimensionless quantities, so DY max can be simultaneously calculated in mm according to the graph or taken simultaneously taking into account the Y dimension.

3) Similarly, the relative error is calculated due to the error in determining X.

.

4) If one of the errors, for example, , or the value X has very small errors D X, invisible on the graph, then we can consider d k=d k y.

5) Absolute error D k=d k*k. As a result.

Literature:

1. Svetozarov V.V. Elementary processing of measurement results, M., MEPhI, 1983.

2. Svetozarov V.V. Statistical processing of measurement results. M.: MEPhI.1983.

3. Hudson. Statistics for physicists. M.: Mir, 1967.

4. Taylor J.Z. Introduction to error theory. M.: Mir.1985.

5. Burdun G.D., Markov B.N. Fundamentals of metrology. M.: Publishing house of standards, 1967.

6. Laboratory workshop « Measuring instruments"/ ed. Nersesova E.A., M., MEPhI, 1998.

7. Laboratory workshop “Electrical measuring instruments. Electromagnetic vibrations and alternating current” / Ed. Aksenova E.N. and Fedorova V.F., M., MEPhI, 1999.

Annex 1

Student's Coefficient Table

Sections: Physics

The graphical method, which is based on mathematics, is used in physics courses at various stages of its study. This is natural, since the graph allows you to show the specifics of what is happening, predict the expected result, and clearly explain the answer.

It is used in physics to form and analyze the physical concepts being studied by revealing their connections with other concepts, to solve problems of generalization and systematization of knowledge.

Graphics tasks are divided into two large groups:

• Graphing problems
• Tasks to obtain information from graphs

In turn, tasks for constructing graphs are divided (according to the method of assignment) into two types:

• Tabular method of specifying dependencies
• Functional way to specify a dependency
• Tasks for obtaining information from a graph are divided (according to the nature of the information) into three types:
• Verbal description of processes
• Analytical expression of a functional relationship represented by a graph
• Determination of unknown quantities from a graph

Most often, when constructing graphs of the dependence of some quantities on others, students remember the appearance of the graph, without going into detail, why it goes this way and not otherwise. When enough dependencies accumulate, errors begin in plotting. In my work, when constructing graphs for various dependencies of physical quantities, I use a functional approach. In a school physics course, only seven functions are used to construct graphs. Almost all physical quantities are positive, so we will consider function graphs only in the first quarter.

№	Function name	Schedule
	Direct proportionality y = k x
	Linear y = k x + b
	Inverse proportionality y = k\x
	Exponential y = k a x
	Function y =
	Quadratic function y = ax 2 + b x + c, y = ax 2
	Trigonometric function y = k sin x

Students study graphs of these functions in a mathematics course. They know these graphs or know how to plot them point by point. My task comes down to teaching students to see the dependence in a physical formula, determine its type, and then establish the corresponding graph.

I'll show this with an example:

Example No. 1. It is necessary to plot the dependence of current on voltage, which is expressed by the dependence I =. Students must understand that if it is necessary to plot the dependence of current on voltage, then only the voltage and, depending on it, the current will change, and the remaining quantities will be constant, in particular resistance. Then our function (formula) can be represented as . If R is the resistance is a constant value, then the unit divided by the resistance is a constant value. Replace this value with k, we get I = k U. We determine the type of function, this is direct proportionality. The graph will be a straight line passing through the origin.

Example No. 2. It is necessary to construct a graph of the dependence of current on resistance, which is expressed by the dependence I =. In the bottom example, the resistance will change and, depending on it, the current strength, and the voltage will be a constant value. Let's make the following replacements: I = y; U = k; R = x; We obtain the function y = k\ x, the graph of which is the branch of the hyperbola