Interactive experiment on the physics of the model. Research on interactive computer models. Computer physical experiment
Computer experiment Computer experiment To give life to new design developments, to introduce new technical solutions into production or to test new ideas, an experiment is needed. In the recent past, such an experiment could be carried out either in laboratory conditions on installations specially created for it, or in situ, i.e. on a real sample of the product, subjecting it to all kinds of tests. This requires large material costs and time. Computer studies of models came to the rescue. When conducting a computer experiment, the correctness of the models is checked. The behavior of the model is studied under various object parameters. Each experiment is accompanied by an understanding of the results. If the results of a computer experiment contradict the meaning of the problem being solved, then the error must be looked for in an incorrectly chosen model or in the algorithm and method for solving it. After identifying and eliminating errors, the computer experiment is repeated. To give life to new design developments, introduce new technical solutions into production, or test new ideas, an experiment is needed. In the recent past, such an experiment could be carried out either in laboratory conditions on installations specially created for it, or in situ, i.e. on a real sample of the product, subjecting it to all kinds of tests. This requires large material costs and time. Computer studies of models came to the rescue. When conducting a computer experiment, the correctness of the models is checked. The behavior of the model is studied under various object parameters. Each experiment is accompanied by an understanding of the results. If the results of a computer experiment contradict the meaning of the problem being solved, then the error must be looked for in an incorrectly chosen model or in the algorithm and method for solving it. After identifying and eliminating errors, the computer experiment is repeated.
A mathematical model is understood as a system of mathematical relationships of formulas, inequalities, etc., reflecting the essential properties of an object or process. A mathematical model is understood as a system of mathematical relationships of formulas, inequalities, etc., reflecting the essential properties of an object or process.
Modeling problems from various subject areas Modeling problems from various subject areas Economics Economics Economics Astronomy Astronomy Astronomy Physics Physics Physics Ecology Ecology Ecology Biology Biology Biology Geography Geography Geography
The machine-building plant, selling products at negotiated prices, received a certain revenue, having spent a certain amount of money on production. Determine the ratio of net profit to invested funds. The machine-building plant, selling products at negotiated prices, received a certain revenue, having spent a certain amount of money on production. Determine the ratio of net profit to invested funds. Statement of the problem Statement of the problem The purpose of the simulation is to study the process of production and sales of products in order to obtain the greatest net profit. Using economic formulas, find the ratio of net profit to invested funds. The purpose of the simulation is to explore the process of production and sales of products in order to obtain the greatest net profit. Using economic formulas, find the ratio of net profit to invested funds.
The main parameters of the modeling object are: revenue, cost, profit, profitability, profit tax. The main parameters of the modeling object are: revenue, cost, profit, profitability, profit tax. Input data: Input data: revenue B; revenue B; costs (cost) S. costs (cost) S. We will find other parameters using the basic economic dependencies. The profit value is defined as the difference between revenue and cost P=B-S. We will find other parameters using the basic economic dependencies. The profit value is defined as the difference between revenue and cost P=B-S. Profitability r is calculated using the formula:. Profitability r is calculated using the formula:. The profit corresponding to the marginal level of profitability of 50% is 50% of the cost of production S, i.e. S*50/100=S/2, therefore, the profit tax N is determined as follows: Profit corresponding to the marginal level of profitability of 50% is 50% of the cost of production S, i.e. S*50/100=S/2, so the profit tax N is determined as follows: if r
Analysis of results Analysis of results The resulting model allows, depending on profitability, to determine the profit tax, automatically recalculate the amount of net profit, and find the ratio of net profit to invested funds. The resulting model allows, depending on profitability, to determine the profit tax, automatically recalculate the amount of net profit, and find the ratio of net profit to invested funds. A computer experiment shows that the ratio of net profit to invested funds increases with increasing revenue and decreases with increasing production costs. A computer experiment shows that the ratio of net profit to invested funds increases with increasing revenue and decreases with increasing production costs.
Task. Task. Determine the speed of the planets in orbit. To do this, create a computer model of the solar system. Statement of the problem The purpose of the simulation is to determine the speed of the planets in orbit. Modeling object: Solar system, the elements of which are planets. The internal structure of the planets is not taken into account. We will consider planets as elements with the following characteristics: name; R - distance from the Sun (in astronomical units; astronomical units. average distance from the Earth to the Sun); t is the period of revolution around the Sun (in years); V is the orbital speed (astro units/year), assuming that the planets move around the Sun in circles at a constant speed.
Analysis of results Analysis of results 1. Analyze the calculation results. Is it possible to say that planets located closer to the Sun have a higher orbital speed? 1. Analyze the calculation results. Is it possible to say that planets located closer to the Sun have a higher orbital speed? 2. The presented model of the Solar System is static. When constructing this model, we neglected changes in the distance from the planets to the Sun during their orbital motion. To know which planet is further away and what the approximate relationships between the distances are, this information is quite enough. If we want to determine the distance between Earth and Mars, then we cannot neglect temporary changes, and here we will have to use a dynamic model. 2. The presented model of the Solar system is static. When constructing this model, we neglected changes in the distance from the planets to the Sun during their orbital motion. To know which planet is further away and what the approximate relationships between the distances are, this information is quite enough. If we want to determine the distance between Earth and Mars, then we cannot neglect temporary changes, and here we will have to use a dynamic model.
Computer experiment Enter the initial data into the computer model. (For example: =0.5; =12) Find the friction coefficient at which the car will go down the mountain (at a given angle). Find the angle at which the car will stand on the mountain (for a given friction coefficient). What will be the result if the friction force is neglected? Analysis of the results This computer model allows you to conduct a computational experiment instead of a physical one. By changing the values of the source data, you can see all the changes occurring in the system. It is interesting to note that in the constructed model the result does not depend either on the mass of the car or on the acceleration of gravity.
Task. Task. Imagine that there will be only one source of fresh water left on Earth, Lake Baikal. For how many years will Baikal provide the population of the whole world with water? Imagine that there will be only one source of fresh water left on Earth, Lake Baikal. For how many years will Baikal provide the population of the whole world with water?
Model development Model development To build a mathematical model, we determine the initial data. We denote: To build a mathematical model, we define the initial data. Let us denote: V - volume of Lake Baikal km3; V is the volume of Lake Baikal km3; N - Earth population 6 billion people; N - Earth population 6 billion people; p - water consumption per day per person (on average) 300 l. p - water consumption per day per person (on average) 300 l. Since 1l. = 1 dm3 of water, it is necessary to convert V of the lake water from km3 to dm3. V (km3) = V * 109 (m3) = V * 1012 (dm3) Since 1l. = 1 dm3 of water, it is necessary to convert V of the lake water from km3 to dm3. V (km3) = V * 109 (m3) = V * 1012 (dm3) The result is the number of years during which the population of the Earth uses the waters of Lake Baikal, let us denote it as g. So, g=(V*)/(N*p*365) The result is the number of years during which the Earth's population uses the waters of Lake Baikal, let's denote it as g. So, g=(V*)/(N*p*365) This is what the spreadsheet looks like in formula display mode: This is what the spreadsheet looks like in formula display mode:
Task. Task. To produce the vaccine, it is planned to grow a bacterial culture at the plant. It is known that if the mass of bacteria is x g, then after a day it will increase by (a-bx)x g, where coefficients a and b depend on the type of bacteria. The plant will daily collect m bacteria for vaccine production. To draw up a plan, it is important to know how the mass of bacteria changes after 1, 2, 3,..., 30 days. To produce the vaccine, it is planned to grow a bacterial culture at the plant. It is known that if the mass of bacteria is x g, then after a day it will increase by (a-bx)x g, where coefficients a and b depend on the type of bacteria. The plant will daily collect m bacteria for vaccine production. To draw up a plan, it is important to know how the mass of bacteria changes after 1, 2, 3,..., 30 days..
Statement of the problem Statement of the problem The object of modeling is the process of population change depending on time. This process is influenced by many factors: the environment, the state of medical care, the economic situation in the country, the international situation and much more. Having summarized the demographic data, scientists derived a function expressing the dependence of the population on time: The object of modeling is the process of changing the population depending on time. This process is influenced by many factors: the environment, the state of medical care, the economic situation in the country, the international situation and much more. Having generalized the demographic data, scientists derived a function expressing the dependence of the population on time: f(t)=where the coefficients a and b are different for each state, f(t)=where the coefficients a and b are different for each state, e is the base of the natural logarithm. e is the base of the natural logarithm. This formula only approximately reflects reality. To find the values of coefficients a and b, you can use a statistical reference book. Taking the values f(t) (population size at time t) from the reference book, you can approximately select a and b so that the theoretical values of f(t) calculated using the formula do not differ much from the actual data in the reference book. This formula only approximately reflects reality. To find the values of coefficients a and b, you can use a statistical reference book. Taking the values f(t) (population size at time t) from the reference book, you can approximately select a and b so that the theoretical values of f(t) calculated using the formula do not differ much from the actual data in the reference book.
The use of a computer as a tool for educational activities makes it possible to rethink traditional approaches to the study of many issues in natural sciences, strengthen the experimental activities of students, and bring the learning process closer to the real process of cognition based on modeling technology. The use of a computer as a tool for educational activities makes it possible to rethink traditional approaches to the study of many issues in natural sciences, strengthen the experimental activities of students, and bring the learning process closer to the real process of cognition based on modeling technology. Solving problems from various areas of human activity on a computer is based not only on students’ knowledge of modeling technology, but, naturally, also on knowledge of a given subject area. In this regard, it is more expedient to conduct the proposed lessons on modeling after students have studied the material in a general education subject; a computer science teacher needs to collaborate with teachers from different educational fields. There is known experience in conducting binary lessons, i.e. lessons taught by a computer science teacher together with a subject teacher. Solving problems from various areas of human activity on a computer is based not only on students’ knowledge of modeling technology, but, naturally, also on knowledge of a given subject area. In this regard, it is more expedient to conduct the proposed lessons on modeling after students have studied the material in a general education subject; a computer science teacher needs to collaborate with teachers from different educational fields. There is known experience in conducting binary lessons, i.e. lessons taught by a computer science teacher together with a subject teacher.
L. V. Pigalitsyn,
, www.levpi.narod.ru, Municipal educational institution secondary school No. 2, Dzerzhinsk, Nizhny Novgorod region.
Computer physical experiment
4. Computational computer experiment
Computational experiment turns
into an independent field of science.
R.G.Efremov, Doctor of Physical and Mathematical Sciences
A computational computer experiment is in many ways similar to a conventional (full-scale) one. This includes planning experiments, creating an experimental setup, performing control tests, conducting a series of experiments, processing experimental data, interpreting them, etc. However, it is carried out not on a real object, but on its mathematical model; the role of the experimental setup is played by a computer equipped with a special program.
Computational experimentation is becoming more and more popular. It is practiced in many institutes and universities, for example, at Moscow State University. M.V. Lomonosov, MPGU, Institute of Cytology and Genetics SB RAS, Institute of Molecular Biology RAS, etc. Scientists can already obtain important scientific results without a real, “wet” experiment. For this, there is not only computer power, but also the necessary algorithms, and most importantly, understanding. If previously they divided - in vivo, in vitro, – then now more has been added in silico. In fact, computational experiment is becoming an independent field of science.
The advantages of such an experiment are obvious. It is, as a rule, cheaper than natural. It can be easily and safely interfered with. It can be repeated and interrupted at any time. This experiment can simulate conditions that cannot be created in the laboratory. However, it is important to remember that a computational experiment cannot completely replace a full-scale one, and the future lies in their reasonable combination. A computational computer experiment serves as a bridge between natural experiment and theoretical models. The starting point of numerical modeling is the development of an idealized model of the physical system under consideration.
Let's consider several examples of computational physical experiments.
Moment of inertia. In “Open Physics” (2.6, part 1) there is an interesting computational experiment on finding the moment of inertia of a rigid body using the example of a system consisting of four balls strung on one knitting needle. You can change the position of these balls on the knitting needle, and also select the position of the axis of rotation, drawing it both through the center of the knitting needle and through its ends. For each arrangement of balls, students calculate the value of the moment of inertia using Steiner's theorem on parallel translation of the axis of rotation. The data for calculations is provided by the teacher. After calculating the moment of inertia, the data is entered into the program and the results obtained by the students are checked.
"Black box". To implement the computational experiment, my students and I created several programs to study the contents of an electrical “black box”. It may contain resistors, incandescent light bulbs, diodes, capacitors, coils, etc.
It turns out that in some cases it is possible, without opening the “black box,” to find out its contents by connecting various devices to the input and output. Of course, at the school level this can be done for a simple three- or four-terminal network. Such tasks develop students' imagination, spatial thinking and creativity, not to mention the fact that solving them requires deep and solid knowledge. Therefore, it is no coincidence that at many All-Union and international Olympiads in physics, the study of “black boxes” in mechanics, heat, electricity and optics is proposed as experimental problems.
In my special course classes, I conduct three real laboratory works in a “black box”:
– resistors only;
– resistors, incandescent lamps and diodes;
– resistors, capacitors, coils, transformers and oscillatory circuits.
Structurally, “black boxes” are designed in empty matchboxes. An electrical circuit is placed inside the box, and the box itself is sealed with tape. Research is carried out using instruments - avometers, generators, oscilloscopes, etc. - because To do this, you have to build the current-voltage characteristic and frequency response. Students enter instrument readings into a computer, which processes the results and plots the current-voltage characteristic and frequency response. This allows students to figure out what parts are in the black box and determine their parameters.
When conducting front-line laboratory work with “black boxes,” difficulties arise due to the lack of instruments and laboratory equipment. Indeed, to conduct research it is necessary to have, say, 15 oscilloscopes, 15 sound generators, etc., i.e. 15 sets of expensive equipment that most schools do not have. And this is where virtual “black boxes” - corresponding computer programs - come to the rescue.
The advantage of these programs is that research can be carried out simultaneously by the whole class. As an example, consider a program that uses a random number generator to implement “black boxes” containing only resistors. There is a “black box” on the left side of the desktop. It contains an electrical circuit consisting only of resistors that can be located between the points A, B, C And D.
The student has three devices at his disposal: a power source (its internal resistance is taken equal to zero to simplify calculations, and the emf is randomly generated by the program); voltmeter (internal resistance is infinity); ammeter (internal resistance is zero).
When the program is launched, an electrical circuit containing from 1 to 4 resistors is randomly generated inside the “black box”. The student can make four attempts. After pressing any key, he is asked to connect any of the proposed devices in any sequence to the terminals of the “black box”. For example, he connected to the terminals AB current source with EMF = 3 V (the EMF value was generated randomly by the program, in this case it turned out to be 3 V). To terminals CD I connected a voltmeter, and its readings turned out to be 2.5 V. From this it should be concluded that the “black box” has at least a voltage divider. To continue the experiment, instead of a voltmeter, you can connect an ammeter and take readings. This data is clearly not enough to solve the mystery. Therefore, two more experiments can be carried out: the current source is connected to the terminals CD, and the voltmeter and ammeter - to the terminals AB. The data obtained in this case will be quite enough to unravel the contents of the “black box”. The student draws a diagram on paper, calculates the parameters of the resistors and shows the results to the teacher.
The teacher, having checked the work, enters the appropriate code into the program, and the circuit located inside this “black box” and the parameters of the resistors appear on the desktop.
The program was written by my students in BASIC. To run it in Windows XP or in Windows Vista you can use an emulator program DOS, For example, DosBox. You can download it from my website www.physics-computer.by.ru.
If there are nonlinear elements inside the “black box” (incandescent lamps, diodes, etc.), then in addition to direct measurements, the current-voltage characteristic will have to be taken. For this purpose, it is necessary to have a current source, voltage, at the outputs of which the voltage can be changed from 0 to a certain value.
To study inductances and capacitances, it is necessary to remove the frequency response using a virtual sound generator and an oscilloscope.
Speed selector. Let's consider another program from “Open Physics” (2.6, part 2), which allows you to conduct a computational experiment with a speed selector in a mass spectrometer. To determine the mass of a particle using a mass spectrometer, it is necessary to perform a preliminary selection of charged particles by velocities. This purpose is served by the so-called speed selectors.
In the simplest speed selector, charged particles move in crossed homogeneous electric and magnetic fields. An electric field is created between the plates of a flat capacitor, and a magnetic field is created in the gap of the electromagnet. starting speed υ charged particles is directed perpendicular to the vectors E And IN .
A charged particle is acted upon by two forces: the electric force q E and Lorentz magnetic force q υ × B . Under certain conditions, these forces can exactly balance each other. In this case, the charged particle will move uniformly and rectilinearly. After flying through the capacitor, the particle will pass through a small hole in the screen.
The condition of a rectilinear trajectory of a particle does not depend on the charge and mass of the particle, but depends only on its speed: qE = qυB→υ = E/B.
In the computer model, you can change the values of electric field strength E, magnetic field induction B and initial particle speed υ . Velocity selection experiments can be performed for electrons, protons, alpha particles, and fully ionized atoms of uranium-235 and uranium-238. The computational experiment in this computer model is carried out as follows: students are informed about which charged particle flies into the speed selector, the electric field strength and the initial speed of the particle. Students calculate the magnetic field induction using the above formulas. After this, the data is entered into the program and the flight of the particle is observed. If the particle flies horizontally inside the velocity selector, then the calculations are done correctly.
More complex computational experiments can be carried out using the free package "MODEL VISION for WINDOWS". Plastic bag ModelVisionStudium (MVS) is an integrated graphical shell for quickly creating interactive visual models of complex dynamic systems and conducting computational experiments with them. The package was developed by the Experimental Object Technologies research group at the Department of Distributed Computing and Computer Networks, Faculty of Technical Cybernetics, St. Petersburg State Technical University. Freely available free version of the package MVS 3.0 is available on the website www.exponenta.ru. Environment Simulation Technology MVS is based on the concept of a virtual laboratory bench. The user places virtual blocks of the simulated system on the stand. Virtual blocks for the model are either selected from the library or created again by the user. Plastic bag MVS is designed to automate the main stages of a computational experiment: constructing a mathematical model of the object under study, generating a software implementation of the model, studying the properties of the model and presenting the results in a form convenient for analysis. The object under study may belong to the class of continuous, discrete or hybrid systems. The package is best suited for the study of complex physical and technical systems.
As an example Let's consider a fairly popular problem. Let a material point be thrown at a certain angle to a horizontal plane and collide absolutely elastically with this plane. This model has become almost mandatory in the demo set of modeling packages. Indeed, this is a typical hybrid system with continuous behavior (flight in a gravitational field) and discrete events (bounces). This example also illustrates the object-oriented approach to modeling: a ball flying in the atmosphere is a descendant of a ball flying in airless space, and automatically inherits all the common features, while adding its own characteristics.
The last, final, from the user's point of view, stage of modeling is the stage of describing the form of presentation of the results of a computational experiment. These can be tables, graphs, surfaces, and even animations that illustrate the results in real time. Thus, the user actually observes the dynamics of the system. Points in phase space, user-drawn design elements can move, the color scheme can change, and the user can monitor, for example, heating or cooling processes on the screen. In the created packages for the software implementation of the model, it is possible to provide special windows that allow you to change the values of parameters during the course of a computational experiment and immediately see the consequences of the changes.
A lot of work on visual modeling of physical processes in MVS held at Moscow State Pedagogical University. There, a number of virtual works have been developed for the course of general physics, which can be associated with real experimental installations, which allows you to simultaneously observe on the display in real time changes in the parameters of both the real physical process and the parameters of its model, clearly demonstrating its adequacy. As an example, I cite seven laboratory works on mechanics from a laboratory workshop on the Internet portal of open education, corresponding to existing state educational standards for the specialty “Physics Teacher”: the study of rectilinear motion using the Atwood machine; measuring the speed of a bullet; addition of harmonic vibrations; measurement of the moment of inertia of a bicycle wheel; study of the rotational motion of a rigid body; determining the acceleration of free fall using a physical pendulum; study of free oscillations of a physical pendulum.
The first six are virtual and are simulated on a PC in ModelVisionStudiumFree, and the latter has both a virtual version and two real ones. In one, intended for distance learning, the student must independently make a pendulum from a large paper clip and an eraser and, hanging it under the shaft of a computer mouse without a ball, obtain a pendulum, the angle of deflection of which is read by a special program and must be used by the student when processing the results of the experiment. This approach allows some of the skills necessary for experimental work to be practiced only on a PC, and the rest - when working with available real devices and with remote access to equipment. In another option, intended for home preparation of full-time students to perform laboratory work in the workshop of the Department of General and Experimental Physics, Faculty of Physics, Moscow State Pedagogical University, the student practices skills in working with an experimental setup on a virtual model, and in the laboratory conducts an experiment simultaneously on a specific real setup and with its virtual model. At the same time, he uses both traditional measuring instruments in the form of an optical scale and a stopwatch, as well as more accurate and fast-acting means - a displacement sensor based on an optical mouse and a computer timer. Simultaneous comparison of all three representations (traditional, refined with the help of electronic sensors associated with a computer, and model) of the same phenomenon allows us to draw a conclusion about the limits of adequacy of the model when computer modeling data begin to differ more and more from the readings after some time. filmed on a real installation.
The above does not exhaust the possibilities of using a computer in a physical computing experiment. So for a creative teacher and his students there will always be untapped opportunities in the field of virtual and real physical experiments.
If you have any comments or suggestions on various types of physical computer experiments, please write to me at:
The main stages of developing and researching models on a computer
Using a computer to study information models of various objects and processes makes it possible to study their changes depending on the value of certain parameters. The process of developing models and studying them on a computer can be divided into several main stages.
At the first stage of researching an object or process, a descriptive information model is usually built. Such a model highlights the essential properties of the object from the point of view of the goals of the research (modeling goals), and neglects the unimportant properties.
At the second stage, a formalized model is created, i.e., a descriptive information model is written using some formal language. In such a model, with the help of formulas, equations, inequalities, etc., formal relationships between the initial and final values of the properties of objects are fixed, and restrictions are also imposed on the permissible values of these properties.
However, it is not always possible to find formulas that clearly express the desired quantities through the initial data. In such cases, approximate mathematical methods are used to obtain results with a given accuracy.
At the third stage, it is necessary to transform the formalized information model into a computer model, i.e., express it in a language understandable to a computer. Computer models are developed primarily by programmers, and users can conduct computer experiments.
Currently, computer interactive visual models are widely used. In such models, the researcher can change the initial conditions and parameters of the processes and observe changes in the behavior of the model.
Control questions
In what cases can individual stages of model construction and research be omitted? Give examples of creating models during the learning process.
Research on interactive computer models
Next, we will look at a number of educational interactive models developed by FIZIKON for educational courses. Educational models of the FIZIKON company are presented on CDs and in the form of Internet projects. The catalog of interactive models contains 342 models in five subjects: physics (106 models), astronomy (57 models), mathematics (67 models), chemistry (61 models) and biology (51 models). Some of the models on the Internet at http://www.college.ru are interactive, while others are presented only with a picture and a description. All models can be found in the corresponding training courses on CDs.
2.6.1. Research of physical models
Let us consider the process of constructing and studying a model using the example of a model of a mathematical pendulum, which is an idealization of a physical pendulum.
Qualitative descriptive model. The following basic assumptions can be formulated:
the suspended body is significantly smaller in size than the length of the thread on which it is suspended;
the thread is thin and inextensible, the mass of which is negligible compared to the mass of the body;
the body deflection angle is small (considerably less than 90°);
there is no viscous friction (the pendulum oscillates in va-
Formal model. To formalize the model, we use formulas known from the physics course. The period T of oscillations of a mathematical pendulum is equal to:
where I is the length of the thread, g is the acceleration of gravity.
Interactive computer model. The model demonstrates free oscillations of a mathematical pendulum. In the fields you can change the length of the thread I, the angle φ0 of the initial deflection of the pendulum, and the coefficient of viscous friction b.
Open Physics
2.3. Free vibrations.
Model 2.3. Math pendulum
Open Physics
Part 1 (TsOR on CD) IZG
The interactive model of a mathematical pendulum is launched by clicking on the Start button.
With the help of animation, the movement of the body and the acting forces are shown, graphs of the angular coordinate or speed versus time, diagrams of potential and kinetic energies are constructed (Fig. 2.2).
This can be seen during free vibrations, as well as during damped vibrations in the presence of viscous friction.
Please note that the oscillations of a mathematical pendulum are... harmonic only at sufficiently small amplitudes
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Rice. 2.2. Interactive model of a mathematical pendulum
http://www.physics.ru
2.1. Practical task. Conduct a computer experiment with an interactive physical model posted on the Internet.
2.6.2. Study of astronomical models
Let's consider the heliocentric model of the solar system.
Qualitative descriptive model. Copernicus's heliocentric model of the world in natural language was formulated as follows:
The Earth rotates around its axis and the Sun;
all planets revolve around the sun.
Formal model. Newton formalized the heliocentric system of the world by discovering the law of universal gravitation and the laws of mechanics and writing them down in the form of formulas:
F = y. Wl_F = t a.(2.2)
Interactive computer model (Fig. 2.3). A three-dimensional dynamic model shows the rotation of the planets of the solar system. The Sun is depicted in the center of the model, with the planets of the Solar System around it.
4.1.2. Rotation of the solar planets
systems. Model 4.1.Solar system (CD on CD) “Open Astronomy”
The model maintains the real relationships of the planets' orbits and their eccentricities. The Sun is at the focal point of each planet's orbit. Notice that the orbits of Neptune and Pluto intersect. It is quite difficult to display all the planets in a small window at once, so there are Mercury...Mars and Jupiter...L,luton modes, as well as an All planets mode. The desired mode is selected using the corresponding switch.
While moving, you can change the value of the viewing angle in the input window. You can get an idea of the actual orbital eccentricities by setting the visual angle to 90°.
You can change the appearance of the model by turning off the display of planet names, their orbits, or the coordinate system shown in the upper left corner. The Start button starts the model, the Stop button pauses it, and the Reset button returns it to its original state.
Rice. 2.3. Interactive model of the heliocentric system
G" Coordinate system S Jupiter...Pluto!■/ Names of planets S. Mercury...Mars |55 angle of view! "/ Orbits of planetsAll planets
Self-administered task
http://www.college.ru 1SHG
Practical task. Conduct a computer experiment with an interactive astronomical model posted on the Internet.
Study of algebraic models
Formal model. In algebra, formal models are written using equations, the exact solution of which is based on finding equivalent transformations of algebraic expressions that allow a variable to be expressed using a formula.
Exact solutions exist only for some equations of a certain type (linear, quadratic, trigonometric, etc.), so for most equations it is necessary to use methods of approximate solution with a given accuracy (graphical or numerical).
For example, it is impossible to find the root of the equation sin(x) = 3*x - 2 by means of equivalent algebraic transformations. However, such equations can be solved approximately by graphical and numerical methods.
Plotting functions can be used to roughly solve equations. For equations of the form fi(x) = f2(x), where fi(x) and f2(x) are some continuous functions, the root (or roots) of this equation is the point (or points) of intersection of the graphs of the functions.
Graphical solution of such equations can be achieved by constructing interactive computer models.
Functions and graphics. Open mathematics.
Model 2.17.TsShG functions and graphs*
Solving equations (COR on CD)
Interactive computer model. Enter an equation in the upper input field in the form fi(x) = f2(x), for example, sin(x) = 3 x - 2.
Click the Solve button. Wait a while. A graph of the right and left sides of the equation will be plotted, with green dots marking the roots.
To enter a new equation, click the Reset button. If you make a typing error, a corresponding message will appear in the lower window.
Rice. 2.4. Interactive computer model for graphically solving equations
for self-execution
http://www.mathematics.ru Ш1Г
Practical task. Conduct a computer experiment with an interactive mathematical model posted on the Internet.
Study of geometric models (planimetry)
Formal model. Triangle ABC is called rectangular if one of its angles (for example, angle B) is right (i.e., equal to 90°). The side of the triangle opposite the right angle is called the hypotenuse; the other two sides are legs.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse: AB2 + BC2 = AC.
Interactive computer model (Fig. 2.5). The interactive model demonstrates the basic relationships in a right triangle.
Right triangle. Open mathematics.
Model 5.1. Pythagorean theorem
Planimetry V51G (TsOR on CD)
Using the mouse, you can move point A (in the vertical direction) and point C (in the horizontal direction). The lengths of the sides of a right triangle and the degree measures of angles are shown.
By switching to demo mode using the button with the film projector icon, you can view the animation. The Start button starts it, the Stop button pauses it, and the Reset button returns the animation to its original state.
The button with a hand icon switches the model back to interactive mode.
Rice. 2.5. Interactive mathematical model of Pythagorean theorem
Self-administered task
http://www.mathematics.ru |И|Г
Practical task. Conduct a computer experiment with an interactive planimetric model posted on the Internet.
Study of geometric models (stereometry)
Formal model. A prism whose base is a parallelogram is called a parallelepiped. The opposite faces of any parallelepiped are equal and parallel. A parallelepiped is called rectangular if all its faces are rectangles. A rectangular parallelepiped with equal edges is called a cube.
The three edges extending from one vertex of a cuboid are called its dimensions. Square
The diagonal of a rectangular parallelepiped is equal to the sum of the squares of its dimensions:
2 2.12, 2 a = a + b + c
The volume of a rectangular parallelepiped is equal to the product of its measurements:
Interactive computer model. By dragging the points with the mouse, you can change the dimensions of the parallelepiped. Observe how the length of the diagonal, surface area and volume of the parallelepiped change when the lengths of its sides change. The Straight checkbox turns an arbitrary parallelepiped into a rectangular one, and the Cube checkbox turns it into a cube.
Parallelepiped.Open mathematics.
Model 6.2.Stereometry)
