What is the general principle for constructing graphs of physical quantities. Rules for constructing graphs

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 of graphic construction, you will find that it is 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 will say that this is contrary to 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 him.

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 planning methods are based on the use of methods associated with 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. Moreover, according to the principle of construction, they can be probabilistic (stochastic) and deterministic according to adaptability - adaptive and non-adaptive in terms of changes in output variables over time - static and dynamic in terms of the dependence of model parameters on 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 forevacuum and diffusion pumps, thermocouple vacuum gauges basic methods for determining the physical properties of 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 introducing geometric corrections sample dimensions, methods for constructing graphs, a system of recording tests performed and a methodology for summarizing 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 of different types of products (form 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. For a larger number of operations, these algorithms are unsuitable, which practically depreciates them, since the need to solve the problem of optimizing the calendar schedule 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 visit a car showroom in order to get acquainted with new cars, their 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 construction engine control systems. 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

2. Ott V.D., Fesenko M.E. and others. Diagnosis and treatment of obstructive bronchitis in young children. Kyiv-1991.

3. Rachinsky S.V., Tatochenko V.K. Respiratory diseases in children. M.: Medicine, 1987.

4. Rachinsky S.V., Tatochenko V.K. Bronchitis in children. Leningrad: Medicine, 1978.

5. Smiyan I.S. Pediatrics (course of lectures). Ternopil: Ukrmedkniga, 1999.

What is the general principle of constructing a system of units of physical quantities?

A physical quantity is a property that is qualitatively common to many physical objects, but quantitatively individual for each object. Physical quantities are objectively interrelated. Using equations of physical quantities, you can express the relationships between physical quantities. A group of basic quantities is distinguished (the units corresponding to these quantities are called basic units) (their number in each field of science is determined as the difference between the number of independent equations and the number of physical quantities included in them) and derived quantities (the units corresponding to these quantities are called derivative units), which are formed using basic quantities and units using equations of physical quantities. The values ​​and units that can be reproduced with the greatest accuracy are chosen as the main ones. The set of selected basic physical quantities is called a system of quantities, and the set of units of basic quantities is called a system of units of physical quantities. This principle for constructing systems of physical quantities and their units was proposed by Gauss in 1832.

Mechanical movement is represented graphically. The dependence of physical quantities is expressed using functions. Designate

Uniform motion graphs

Dependence of acceleration on time. Since during uniform motion the acceleration is zero, the dependence a(t) is a straight line that lies on the time axis.

Dependence of speed on time. The speed does not change over time, the graph v(t) is a straight line parallel to the time axis.

The numerical value of the displacement (path) is the area of ​​the rectangle under the speed graph.

Dependence of the path on time. Graph s(t) - sloping line.

The rule for determining speed from the graph s(t): The tangent of the angle of inclination of the graph to the time axis is equal to the speed of movement.

Graphs of uniformly accelerated motion

Dependence of acceleration on time. Acceleration does not change with time, has a constant value, the graph a(t) is a straight line parallel to the time axis.

Dependence of speed on time. With uniform motion, the path changes according to a linear relationship. In coordinates. The graph is a sloping line.

The rule for determining the path using the graph v(t): The path of a body is the area of ​​the triangle (or trapezoid) under the velocity graph.

The rule for determining acceleration using the graph v(t): The acceleration of a body is the tangent of the angle of inclination of the graph to the time axis. If the body slows down, the acceleration is negative, the angle of the graph is obtuse, so we find the tangent of the adjacent angle.

Dependence of the path on time. During uniformly accelerated motion, the path changes according to

Graphic representation of information can be very useful precisely because of its clarity. Using the graphs, you can determine the nature of the functional dependence and determine the values ​​of quantities. Graphs allow you to compare experimental results with theory. It's easy to find highs and lows on charts, easy to spot misses, etc.

1. The graph is drawn on paper marked with a grid. For student practical work, it is best to take graph paper.

2. Special mention should be made about the size of the graph: it is determined not by the size of the piece of graph paper you have, but by the scale. The scale is chosen primarily taking into account the measurement intervals (it is selected separately for each axis).