The main stages of development and research of models on a computer

Using a computer to study information models of various objects and processes allows you to study their changes depending on the value of certain parameters. The process of developing models and examining them on a computer can be divided into several main stages.

At the first stage of the study of an object or process, a descriptive information model is usually built. Such a model singles out essential, from the point of view of the goals of the research (modeling goals), properties of the object, and neglects insignificant properties.

At the second stage, a formalized model is created, that is, a descriptive information model is written using some formal language. In such a model, with the help of formulas, equations, inequalities, etc., formal relations are fixed between the initial and final values ​​of the properties of objects, and also restrictions are imposed on the permissible values ​​of these properties.

However, it is far from always possible to find formulas that explicitly express the required quantities in terms of 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, that is, to express it in a computer-understandable language. Computer models are developed primarily by programmers, and users can conduct computer experiments.

Computer interactive visual models are now 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.

test questions

In what cases can the individual stages of building and researching a model be omitted? Give examples of creating models in the learning process.

Study of interactive computer models

Next, we will consider a number of educational interactive models developed by FIZIKON for educational courses. The training models of the FIZIKON company are presented on CD-disks 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 the site http://www.college.ru are interactive, while others are presented only with a picture and description. All models can be found in the respective training CDs.

2.6.1. Exploring Physical Models

Let us consider the process of building and researching a model using the example of a mathematical pendulum model, which is an idealization of a physical pendulum.

Qualitative descriptive model. The following basic assumptions can be formulated:

the suspended body is much 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 angle of deflection of the body is small (much less than 90 °);

there is no viscous friction (the pendulum oscillates in

Formal model. To formalize the model, we use the formulas known from the physics course. The period T of oscillations of a mathematical pendulum is:

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, the coefficient of viscous friction b.

Open physics

2.3. Free vibrations.

Model 2.3. Mathematical pendulum

Open physics

Part 1 (CDC on CD) IZG

The interactive model of the 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 time dependence of the angular coordinate or speed, diagrams of potential and kinetic energies are plotted (Fig. 2.2).

This can be seen with free vibrations, as well as with damped vibrations in the presence of viscous friction.

Please note that the oscillations of the 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

Consider a 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 revolves 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 = m and. (2.2)

Interactive computer model (Fig. 2.3). The 3D dynamic model shows the rotation of the planets of the solar system. In the center of the model, the Sun is depicted, around it are the planets of the Solar System.

4.1.2. Rotation of the planets of the Solar

systems. Model 4.1. Solar system (CRC on CD) "Open Astronomy"

The model maintains the real relationship of the orbits of the planets and their eccentricities. The sun is at the focal point of each planet's orbit. Note that the orbits of Neptune and Pluto intersect. It is rather difficult to depict all the planets at once in a small window, therefore the modes Mercury ... Mars and Jupiter ... L, Luton, as well as the All planets mode are provided. The selection of the desired mode is made using the corresponding switch.

While driving, you can change the value of the angle of view in the input window. You can get an idea of ​​the real eccentricities of the orbits by setting the value of the angle of view 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 launches the model, Stop - pauses, and Reset - returns to its original state.

Rice. 2.3. Interactive model of the heliocentric system

G "Coordinate system C Jupiter ... Pluto! ■ / Names of planets C. Mercury ... Mars | 55 angle of view!" / Orbits of planetsAll planets

Self-study assignment

http://www.college.ru 1ШГ

Practical task. Conduct a computer experiment with an interactive astronomical model posted on the Internet.

Researching algebraic models

Formal model. In algebra, formal models are written using equations, the exact solution of which is based on the search for equivalent transformations of algebraic expressions that allow expressing a variable using a formula.

Exact solutions exist only for some equations of a certain type (linear, quadratic, trigonometric, etc.), therefore, for most equations, one has to use methods of approximate solution with a given accuracy (graphic or numerical).

For example, you cannot find the root of the equation sin (x) = 3 * x - 2 by equivalent algebraic transformations. However, such equations can be solved approximately by graphic 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 are the point (or points) of intersection of the graphs of the functions.

The graphical solution of such equations can be carried out by constructing interactive computer models.

Functions and graphs. Open mathematics.

Model 2.17. Functions and graphs of the CHG *

Solving Equations (CRC on CD)

Interactive computer model. Enter the 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. The graph of the right and left sides of the equation will be plotted, the roots will be marked with green dots.

To enter a new equation, click the Reset button. If you make a mistake while typing, a corresponding message will appear in the lower window.

Rice. 2.4. Interactive computer model of graphical solution of equations

for self-fulfillment

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. A triangle ABC is called rectangular if one of its corners (for example, angle B) is straight (that is, equal to 90 °). The side of the triangle opposite the right angle is called the hypotenuse; the other two sides are with legs.

The Pythagorean theorem states that in a right-angled 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). An interactive model demonstrates the basic relationships in a right triangle.

Right triangle. Open mathematics.

Model 5.1. Pythagorean theorem

V51G planimetry (CDC on CD)

Using the mouse, you can move point A (in the vertical direction) and point C (in the horizontal direction). Shows the lengths of the sides of a right-angled triangle, degree measures of angles.

By switching to demo mode using the button with the movie projector icon, you can preview the animation. The Start button starts it, the Stop button pauses, and the Reset button returns the animation to its original state.

The hand button switches the model back to interactive mode.

Rice. 2.5. Interactive mathematical model of the Pythagorean theorem

Self-study assignment

http://www.mathematics.ru | Y | G

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. Opposite faces of any parallelepiped are equal and parallel. A rectangular parallelepiped is called, all the faces of which are rectangles. A rectangular parallelepiped with equal edges is called a cube.

Three edges extending from one vertex of a rectangular parallelepiped are called dimensions. Square

the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its measurements:

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 points, you can change the dimensions of the box. Observe how the length of the diagonal, the surface area and the volume of the parallelepiped change as the lengths of its sides change. The Straight checkbox turns an arbitrary parallelepiped into a rectangular box, and the Cube checkbox turns it into a cube.

Parallelepiped Open mathematics.

Model 6.2 Stereometry)