Analysis of dimensions and analogy method. Analysis of dimensions Analysis of dimensions and analogy method
In physics ... there is no place for confused thoughts ...
Really understanding nature
Of one or another phenomenon should receive the main
Laws for considerations of dimension. E. Fermi
A description of a problem, discussion of theoretical and experimental issues begins with a qualitative description and assessment of the effect that gives this work.
In describing some kind of problem, it is necessary, first of all, to estimate the order of the expected effect, simple limit cases and the nature of the functional communication of the values \u200b\u200bdescribing this phenomenon. These questions are called a qualitative description of the physical situation.
One of the most effective methods for such an analysis is the dimension method.
Here are some advantages and application of the dimension method:
- fast assessment of the scale of studied phenomena;
- obtaining high-quality and functional dependencies;
- restoration of forgotten formulas on exams;
- performing some assignments of the USE;
- implementation of verification of the correctness of solving problems.
Analysis of dimensions is used in physics since Newton's times. It was Newton that formulated closely related to the dimension method the principle of similarity (analogy).
Students first meet with the method of dimensions when studying heat radiation in the course of class 11 physics:
The spectral characteristic of the thermal radiation of the body is spectral density of energy luminosity r V - energy of electromagnetic radiation emitted per unit of time from a unit of body surface area in a unit frequency range.
The unit of spectral density of energy luminosity - Joule per square meter (1 J / m 2). The energy of thermal radiation of the black body depends on the temperature and wavelength. The only combination of these values \u200b\u200bwith the dimension of J / M 2 is KT / 2 (\u003d C / V). The exact calculation made by Railel and Jeans in 1900, as part of a classic wave theory gave the following result:
where k is the Boltzmann's constant.
As experience has shown, this expression is consistent with experimental data only in the region of sufficiently small frequencies. For large frequencies, especially in the ultraviolet region of the spectrum of the Rayleigh-jeans formula is incorrect: it is sharply dispersed with the experiment. Methods of classical physics were insufficient to explain the characteristics of the emission of absolutely black bodies. Therefore, the discrepancy between the results of the classic wave theory with the experiment at the end of the XIX century. It was named "Ultraviolet Catastrophe".
Let us show the use of the dimension method on a simple and good example example.
Picture 1
Thermal radiation of absolutely black body: ultraviolet catastrophe - the discrepancy of the classic thermal radiation theory with experience.
We will imagine that the body mass M moves straightforwardly under the action of constant power F. If the initial body rate is zero, and the speed at the end of the distance of the length S is equal to V, then you can record the theorem about kinetic energy:. Ment of the values \u200b\u200bF, M, V and S exists a functional connection.
Suppose that the kinetic energy theorem is forgotten, and we understand that the functional relationship between V, F, M, and S exists and has a powerful character.
Here X, Y, Z are some numbers. We define them. The sign ~ means that the left part of the formula is proportional to the right, that is, where k is a numerical coefficient, does not have units of measurement and is not determined using the dimension method.
The left and right part of the ratio (1) have the same dimension. The dimensions of the values \u200b\u200bV, F, M and S are as follows: [v] \u003d m / c \u003d mc -1, [f] \u003d h \u003d kgms -2, [m] \u003d kg, [s] \u003d m. (Symbol [A] denotes the dimension of the value A.) We write the equality of dimensions in the left and right parts of the relation (1):
m C -1 \u003d kg x M x C -2x kg y m z \u003d kg x + y m x + z c -2x.
In the left part of equality there are no kilograms at all, therefore it should not be right.
It means that
The right meters are consistent with the degree of X + Z, and on the left - to the degree 1, therefore
Similarly, from the comparison of indicators of the degree under seconds
From the obtained equations we find the numbers X, Y, Z:
x \u003d 1/2, y \u003d -1/2, z \u003d 1/2.
The final formula has the form
Erending the left and right parts of this ratio, we get that
The last formula is the mathematical entry of the theorem on kinetic energy, though without numerical coefficient.
The principle of similarity formulated by Newton is that the ratio V 2 / S is directly proportional to the ratio F / M. For example, two bodies with different masses M 1 and M 2; We will act on them with different forces F 1 and F 2, but in such a way that the ratio F 1 / M 1 and F 2 / m 2 will be the same. Under the action of these forces of the body will begin to move. If the initial speeds are zero, then the speeds purchased by bodies on the segment of the length of the length s will be equal. This is the law of similarity to which we came with the help of an idea about the equality of the dimensions of the right and left parts of the formula describing the power link value of the final speed with the values \u200b\u200bof force, mass and the length of the path.
The dimensions method was introduced when building the foundations of classical mechanics, however, its effective use for solving physical problems began at the end of the past - at the beginning of our century. A great merit in the propaganda of this method and solutions with its help of interesting and important tasks belongs to the outstanding physics of Lord Rayley. In 1915, Ralea wrote: " I am often surprised at the insignificant attention that is paid to the great principle of similarity, even from the side of very large scientists. It often happens that the results of painstaking research are presented as newly open "laws", which, nevertheless, it was possible to obtain a priori for a few minutes. "
Nowadays, physicists can no longer be reproached in a dismissive attitude or in insufficient attention to the principle of similarity and the method of dimensions. Consider one of the classic Rayleigh problems.
Rayleigh's task about ball oscillations on the string.
Let the string stretch between the points A and B. String force F. In the middle of this string at the point C is a heavy ball. The length of the segment AC (and, respectively, CB) is 1. Mass the ball is much larger than the mass of the string itself. The string is delayed and let go. It is quite clear that the ball will make hesitations. If the amplitude of these X oscillations is much less than the length of the string, the process will be harmonious.
We define the frequency of the ball oscillations on the string. Let values, F, M and 1 are associated with a power dependence:
Indicators X, Y, Z - the numbers that we need to determine.
We repel the dimension of the quantities of interest to us in the SI system:
C -1, [f] \u003d kgm C -2, [m] \u003d kg, \u003d m.
If formula (2) expresses a real physical pattern, the dimensionality of the right and left parts of this formula must coincide, that is, equality should be performed
c -1 \u003d kg x m x C -2x kg y m z \u003d kg x + y m x + z c -2x
The left part of this equality does not include meters and kilograms, and seconds are consistent with degree - 1. This means that equations are performed for X, Y and Z:
x + y \u003d 0, x + z \u003d 0, -2x \u003d -1
Solving this system, we find:
x \u003d 1/2, y \u003d -1/2, z \u003d -1/2
Hence,
~ F 1/2 M -1/2 1 -1/2
The exact formula for frequency is different from the found only time (2 \u003d 2F / (M1)).
Thus, not only qualitative, but also a quantitative assessment of dependence for the values \u200b\u200bof F, M and 1. In order of magnitude, the degree, the degree combination gives the correct frequency value. The assessment is always interested in order of magnitude. In simple tasks, often coefficients undetermined by the dimension method can be considered the number of orders of order. This is not a strict rule.
When studying the waves, we consider qualitative forecasting of sound speed by the method of analysis of dimensions. Sound speed We are looking for as the speed of propagation of the compression wave and gas permit. Students do not have doubts depending on the speed of sound in gas from the gas density and its pressure P.
The answer is looking for:
where C is a dimensionless multiplier whose numeric value from the analysis of the dimension cannot be found. Turning to (1) to the equality of dimensions.
m / c \u003d (kg / m 3) x pa y,
m / s \u003d (kg / m 3) x (kg m / (with 2 m 2)) y,
m 1 s -1 \u003d kg x m -3x kg y m y c -2y m -2y,
m 1 s -1 \u003d kg x + y m -3x + y-2y c -2y,
m 1 s -1 \u003d kg x + y m -3x-y c -2y.
Equality of dimensions in the left and right of equality gives:
x + y \u003d 0, -3x-y \u003d 1, -2y \u003d -1,
x \u003d -y, -3 + x \u003d 1, -2x \u003d 1,
x \u003d -1/2, y \u003d 1/2.
Thus, the speed of sound in gas
The formula (2) at C \u003d 1 first received I. Newton. But the quantitative findings of this formula were very complex.
The experimental definition of sound speed in the air was performed in the collective work of members of the Paris Academy of Sciences in 1738, in which the time of passing the sound of a cannonic shot of a distance of 30 km was measured.
Repeating this material in the 11th grade, the attention of students appeals to the fact that the result (2) can be obtained for the model of the isothermal process of propagation of sound using the Mendeleev equation - Klapairone and the concept of density:
- Sound distribution speed.
I introduces students with the dimensions method, give them this method to withdraw the basic ICT equation for perfect gas.
Students understand that the pressure of the ideal gas depends on the mass of individual molecules of the ideal gas, the number of molecules in a unit of volume - n (concentration of gas molecules) and the speed of movement of molecules.
Knowing the dimension of the values \u200b\u200bincluded in this equation we have:
,
,
,
Compare the dimensionality of the left and right part of this equality, we have:
Therefore, the main MTC equation has this kind:
- this implies
From the shaded triangle it can be seen that
Answer: c).
This we used the dimension method.
The dimension method In addition to the implementation of the traditional verification of the correctness of solving problems, the implementation of some assignments of the EGE, helps to find functional dependencies between different physical quantities, but only for those situations where these dependencies are power. There are many such dependencies in nature, and the dimension method is a good assistant when solving such tasks.
Physical quantities whose numeric value does not depend on the scale of the units, are called dimensionless. Examples of dimensionless values \u200b\u200bare an angle (the ratio of the arc length to the radius), the refractive index of the substance (the ratio of the speed of light in vacuum to the speed of light in the substance).
Physical quantities that change their numeric value when changing the scale of units are called dimensional. Examples of dimensional values \u200b\u200b- length, force, etc. The expression of a unit of physical quantity through the main units is called its dimension (or dimension formula). For example, the dimension of force in the SSS and C systems is expressed by the formula
The dimension considerations can be used to verify the correctness of the responses received when solving physical problems: the right and left parts of the expressions obtained, as well as separate terms in each of the parts, should have the same dimension.
The dimension method can also be used to output formulas and equations, when we know, the desired value may depend on what physical parameters. The essence of the method is easiest to understand specific examples.
Applications of the dimension method. Consider the task, the answer for which we know is well known: how much will the body fall on the ground, fluidly falling without initial speed from the height if the air resistance can be neglected? Instead of direct computing on the basis of the laws of the movement, we will argue as follows.
We think, from which it may depend on the desired speed. It is obvious that it should depend on the initial height and from accelerating the free fall can be assumed by following Aristotle, that it depends on the mass. Since it is possible to add only the values \u200b\u200bof the same dimension, then for the desired speed, you can offer such a formula:
where C is some dimensionless constant (numerical coefficient), and x, y and z are unknown numbers that should be defined.
The dimension of the right and left parts of this equality should be the same, and it is precisely this condition that can be used to determine the degree x, y, z in (2). The speed dimension is the dimension of the height. There is a dimension of acceleration of free incidence.
This equality should be performed regardless of what numerical values. Therefore, the indicators of degrees at and m should be equated in the left and right parts of equality (3):
From this system of equations, we treat the formula (2) takes the form
The true value of speed, as is known, equal
So, the approach used allowed us to determine correctly dependence on and and did not give the opportunity to find a value.
the dimensionless constant C. Although we did not manage to get a comprehensive response, it was still obtained very significant information. For example, we can fully argue that if the initial height is four times, the speed at the time of the fall will double and that, contrary to the opinion of the aristotle, this speed does not depend on the mass of the incident body.
Select parameters. When using the dimension method, you should first identify the parameters that determine the phenomenon in question. It is easy to do if the physical laws describe it are known. In some cases, the determining phenomenon parameters can be indicated when physical laws are unknown. As a rule, to use the method of analysis of dimensions, you need to know less than to compile the equations of motion.
If the number of parameters that determine the studied phenomenon, greater than the number of basic units, on which the selected system of units is constructed, then, of course, all the indicators of the degrees in the proposed formula for the desired value cannot be determined. In this case, it is useful first to determine all independent dimensionless combinations from the selected parameters. Then the desired physical value will be determined not to formula type (2), but by the product of any (the simplest) combination of parameters having the desired dimension (i.e. the dimension of the desired value), on some function of the found dimensionless parameters.
It is easy to see that in the disassembled above the example of the body drop from the height from the values \u200b\u200band the dimensionless combination is impossible. Therefore, there formula (2) exhausts all possible cases.
Dimensionless parameter. We now consider such a task: Determine the distance of the horizontal flight of the projectile released in the horizontal direction with the initial speed of the gun located on the height world
In the absence of air resistance, the number of parameters from which the desired range may depend on, equal to four: etc. As the number of basic units is equal to three, then the complete solution of the problem by the method of dimensions is impossible. We will find all the independent dimensionless parameters of y, which can be made of and
This expression corresponds to the following equality of dimensions:
From here we get a system of equations
which gives and for the desired dimensionless parameter we get
It can be seen that the only independent dimensionless parameter in the task under consideration is now it is now enough to find any parameter having a dimension of length, for example, take the parameter itself to write a general expression for the range of the projectile flight horizontally in the form of
where - while the unknown function of the dimensionless parameter method of dimensions (in the above version) does not allow you to determine this function. But if we are from somewhere, for example, from experience, it is known that the desired range is proportional to the horizontal rate of the projectile, then the form of the function is immediately determined: the speed should be included in it in the first degree, i.e.
Now from (5) for the range of the shell's flight we get
that when coincides with the correct answer
We emphasize that with this method of determining the type of function, it is enough for us to know the nature of the experimentally established dependence of the flight range not from all parameters, but only from some one of them.
Vector units of length. But it is possible to determine the range (7) only for reasons of dimension, if you increase to four the number of basic units through which the parameters are expressed, etc. So far, when writing the dimensions, there was no difference between the lengths of the length in the horizontal and vertical direction. However, such a difference can be introduced, based on the fact that the strength of gravity is valid only by vertical.
Denote the dimension of the length in the horizontal direction through and vertically - through the then the dimension of the flight range horizontally will be the dimension of the height will be the dimension of the horizontal speed will be and to accelerate
we will get free fall now, looking at formula (5), we see that the only way to get the correct dimension in the right part is to consider proportional we come to formula (7).
Of course, having four main units and m can and directly construct the value of the desired dimension of four parameters and
Equality of dimensions of the left and right parts is
The system of equations for x, y, z and and gives the values \u200b\u200band we again come to formula (7).
Used herein different units of lengths by mutually perpendicular directions are sometimes called vector length units. Their use is significantly expanding the possibilities of the analysis of dimensions.
When using the dimension analysis method, it is useful to develop the skills to such an extent in order not to constitute a system of equations for degrees in the desired formula, and select them directly. We illustrate this in the next task.
A task
Maximum range. At what angle to the horizon you should throw a stone so that the flight distance horizontally was maximum?
Decision. Suppose that we "forgot" all the kinematics formulas, and try to get an answer for reasons of dimension. At first glance, it may seem that the method of dimensions here is generally not applicable, since some trigonometric function of the challenge can be logged in response. Therefore, instead of the corner itself, but try to look for an expression for a distance clearly, without vector units of length do not do here.
In cases where there are no equations describing the process, and it is not possible to make it possible, to determine the type of criteria from which the similarity equation should be made, you can use the analysis of dimensions. Previously, however, it is necessary to define all parameters substantial to describe the process. This can be done on the basis of experience or theoretical considerations.
The dimension method divides physical quantities on the main (primary), which characterize the measure directly (without communication with other values), and derivatives that are expressed through the main values \u200b\u200bin accordance with physical laws.
In the system, the main units are assigned notation: Length L., weight M.Time T., temperature Θ , Tok power I., the power of light J.The amount of substance N..
Expression of the derivative value φ through the mains is called dimension. The dimension formula of the derivative of the value, for example, at four main units of measurement L., M., T., Θ, it has the form:
where a., b., c., d.- Actual numbers.
In accordance with the equation, dimensionless numbers have zero dimension, and the main values \u200b\u200bare dimension equal to one.
The basis of the method besides the presented principle, there is an axiom that only the values \u200b\u200band complexes of values \u200b\u200bthat have the same dimension can develop and deduct. From these provisions, it follows that if any physical value, for example ∆ p.is defined as the function of other physical quantities in the form of ∆ p.= f.(V., ρ, η, l., d.) , then this dependence can be represented as:
,
where C.- constant.
If then express the dimension of each derivative through the main dimensions, then you can find the values \u200b\u200bof the degree indicators x., y., z.etc. In this way:
In accordance with the equation after substitution of dimensions, we get:
Grouping then homogeneous members, we will find:
If in both parts of the equation to equate the degree rates with the same basic units, then the following system of equations will be obtained:
In this system of three equations, five unknowns. Consequently, any three of these unknowns can be expressed in two other, namely x., y.and r.through z.and v.:
After substitution of indicators 
and 
in power functions it turns out:
.
The criteria equation describes the flow of fluid in the pipe. This equation includes, as shown above, two criteria complex and one criterion-simplex. Now, with the help of the analysis of the dimensions, the types of these criteria are established: this is an Euler criterion EU=∆ p./(ρ V. 2 ) , Reynolds criterion Re.= Vdρ./η and parametric criterion of geometric similarity R \u003d.l./ d.. In order to finally establish the type of criteria equation, it is necessary to experimentally determine the values \u200b\u200bof constant C., z. and v.in equation.
Experimental definition of the constants of the criteria equation
During experiments, the dimensional values \u200b\u200bcontained in all similarity criteria are measured and determined. According to the results of experiments, calculate the values \u200b\u200bof the criteria. Then make up the tables in which the values \u200b\u200bof the criterion respectively K. 1 Enable the values \u200b\u200bof the determining criteria K. 2 , K. 3 etc. This operation completes the preparatory stage of experimental processing.
To summarize tabular data in the form of power dependency:
the logarithmic coordinate system is used. Selection of degree indicators m., n.etc. It seeks that the arrangement of experienced points on the chart, so that through them you can spend a straight line. The straight line equation gives the desired relationship between the criteria.
We show how in practice, determine the constants of the criteria equation:
.
In logarithmic coordinates lGK. 2 – lGK. 1 This equation is a straight line:
.
Applying experienced points on the chart (Fig. 4), they spend direct line through them, the slope of which determines the value of constant m.= tGβ..
Fig. 4. Processing experienced data
It remains permanent 
. For any point direct on the chart 
. Therefore, the value C.found in any pair of appropriate values K. 1
and
K. 2
, counted on a straight line of graphics. For reliability 
determine several points direct and to the final formula, the average value is substituted:
With a greater number of criteria, the definition of the constants of the equation is somewhat complicated and is carried out according to the method described in the book.
In the logarithmic coordinates, it is not always possible to arrange the experienced points along the straight line. This happens when the observed dependence is not described by the power equation and it is necessary to look for the function of another species.
Basic concepts of modeling theory
Modeling is the method of experimental study of the model of the phenomenon instead of a natural phenomenon. The model is chosen so that the results of the experiment can be distributed to one-scale phenomenon.
Let it be modeling the field of magnitude w.. Then, with accurate modeling in the researchers of the model and a tool, the condition should be observed.
where scale modeling.
In case of approximate modeling, we get
The ratio is called a degree of distortion.
If the degree of distortion does not exceed the measurement accuracy, then approximate modeling is not different from the exact one. It is impossible to make it in advance so that the value does not exceed some of the designed value, since in most cases it can not even be determined in advance.
Method of analogy
If two physical phenomena of different physical nature are described by identical equations and unambiguing conditions (boundary or in the stationary case, boundary conditions) presented in a dimensionless form, then the phenomena are called similar. Under these conditions, the phenomenon of one physical nature is called similar.
Despite the fact that similar phenomena have a different physical nature, they relate to one individual generalized case. This circumstance made it possible to create a very convenient method of analogy to study physical phenomena. The essence of it consists in the following: not studied phenomenon is subjected to the survey, for which it is difficult or cannot be measured by the desired values, but a specially selected similar to those studied. As an example, consider the electrical analogy. In this case, the studied phenomenon is a stationary temperature field, and its analogy is a stationary field of electric potential
The equation of thermal conductivity
(9.3)
where the absolute temperature,
and electric potential equation
(9.4)
where electric potential is similar. In a dimensionless form, these equations will be identical.
If the boundary conditions for the potential are created, similar to the conditions for the temperature, then in the dimensionless form they will also be identical.
An electrical analogy is widely used in the study of thermal conductivity processes. For example, the temperature fields of the blades of the gas turbine were measured by this method.
Analysis dimensions
Sometimes you have to study the processes that are not yet described by differential equations. The only way to study is an experiment. The results of the experiment are advisable to submit in generalized form, but for this you need to be able to find dimensionless complexes characteristic of such a process.
Analysis of dimensions is the method of compiling dimensionless complexes under conditions when the process under study is not yet described by differential equations.
All physical quantities can be divided into primary and secondary. For heat exchange processes for primary usually choose the following: length L,mass m.Time t., quantity of heat Q.excess temperature . Then secondary will be such values \u200b\u200bas the heat transfer coefficient temperature a. etc.
The dimension formulas of secondary magnitudes have the type of power unicrenes. For example, the dimension formula for the heat transfer coefficient is
(9.5)
where Q.-quantity of heat.
Let all physical quantities that are essential for the process being studied are known. It is required to find dimensionless complexes.
We will make a product of the formulas of the dimensions of all essential for the process of physical quantities in some uncertain so far by degrees; Obviously, it will be a power unicreated (for the process). Suppose that its dimension (powerful single) is zero, i.e., the indicators of the degrees of the primary values \u200b\u200bincluded in the dimensional formula are reduced, then the power is unintently (for the process) can be represented in the form of a product of dimensionless complexes from dimensional quantities. It means that if you draw up a product of the formulas of dimensions substantial for the processes of physical quantities in uncertain degrees, then from the conditions of equality zero the sums of the degree of degrees of the primary values \u200b\u200bof this power, one can determine the desired dimensionless complexes.
We show this operation on the example of the periodic process of thermal conductivity in the solid, washed with a liquid coolant. We assume that differential equations for the process under consideration are unknown. It is required to find dimensionless complexes.
Significant physical quantities for the process under study will be the following: characteristic size l. (m), thermal conductivity of a solid body, (J / (M K)), the specific heat capacity of the solid from(J / (kg k)), the density of the solid (kg / m 3), the coefficient of heat exchange (heat transfer) (J / m 2 K)), period of period , (c), characteristic excess temperature (K). Make up of these values \u200b\u200bof the power species
The indicator of the degree in the primary value is called the dimension of the secondary value with respect to this primary one.
Replace in physical quantities (except Q)their dimension formulas, as a result we get
In this case, the indicators of the degree are values \u200b\u200bin which Q.falls out of the equation.
We equate zero indicators of degrees Singochlen:
for length
a - b - 3i - 2k \u003d0; (9.8)
for the amount of warmth Q.
0; (9.9)
for Time
for temperature
for mass m.
Total essential values \u200b\u200bof seven, equations for determining the indicators of five, which means, only two indicators, for example, b.and kmogut be selected arbitrarily.
Express all degrees through b.and k.As a result, we get:
of (8.8), (8.9), (8.12)
f \u003d -b - k; (9.14)
r \u003d B + K; (9.15)
from (8.11) and (8.9)
n \u003d b + f + k \u003d b +(-b - K.) + k \u003d 0; (9.16)
from (8.12) and (8.9)
i \u003d f \u003d -b -k. (9.17)
Now one can imagine in shape
As indicators b.and k.can be selected arbitrarily, put:
1. At the same time
When solving tasks in physics at any level, it is extremely important to determine the most acceptable method or methods, and then go to the "technical" embodiment. Virtuoso teachers (we deliberately used this expression, as we consider largely similar to reading the music of the musicians-improvisers and teachers-virtuosos, who found their own, author's approaches in the interpretation and interpretation of physical laws) pay a lot of time to discuss the problem. In other words, the discussion of the method is often equally important than the solution of the problem, since a kind of exchange of techniques occurs, contacting different points of view, which, in fact, is the purpose of the learning process. The process of preparing to solve the problem largely resembles the process of preparation of the actor to the spectral. Discussion of roles, characters of heroes, thinking of intonation, musical reprise and art decorations are the most important elements of the actor's immersion. It is not by chance that many well-known theatrical workers appreciate the preparatory process and recall the atmosphere of rehearsals and their own finds. In the process of teaching, the teacher uses various methods or a "spectrum of methods". One general solution methods is the solution to the dimension method. The essence of this method is that the desired pattern can be represented as a product of power functions of physical quantities, on which the desired characteristic depends. An important point in the decision is to find these quantities. Analysis of the dimensions of the left and right parts of the relation allows to determine the analytical dependence with an accuracy of a constant multiplier.
Consider, for example, from which the pressure in gas may depend on. From everyday experience, we know that the pressure is a function of temperature (increasing the temperature, we increase the pressure), concentration (the gas pressure will increase if, without changing its temperature, we will put a larger number of molecules in this volume). Naturally the assumption of gas pressure dependence on the mass of molecules and their speed. It is clear that the greater the mass of molecules, the greater the pressure at other permanent values. Obviously, with an increase in the speeds of molecules, the pressure will increase. (We note that all the above reasoning suggests that all the indicators of degrees in the final formula are required to be positive!) It can be assumed that the gas pressure is depending on its volume, however, if we support the constant concentration of molecules, the pressure from the volume does not depend on . Indeed, in case we present two vessels with the same gases of one and the same concentration, molecules, temperature, etc., then removing the partition separating the gases, we will not change the pressure. Thus, by changing the volume, but leaving the concentration and other parameters unchanged, we did not change the pressure. In other words, we will not have to introduce the volume into our reasoning. It would seem that we have the right to build a functional dependence, but perhaps we introduced redundant information? The fact is that the temperature is the energy characteristic of the bodies, so it is associated with the energy of molecules, i.e. It is a function of mass and velocity of molecules that make up the body. Therefore, including in our assumptions of pressure dependence on the concentration, speeds and mass of molecules, we have already been "taken care of" about all possible dependencies that, among other things, can include temperature. In other words, the desired functional dependence can be recorded in the form:
Here p.- gas pressure, t. 0 - weight of the molecule, n. - Concentration, U - molecule speed.
Represent pressure, mass, concentration, speed in the main values \u200b\u200bof the international system:
The dependence (1) in the language of dimensions is:
Comparison of the dimension of the left and right part gives the system of equations
Solving (4), we get but = 1; b.= 1; from \u003d 2. Gas pressure can now write as
(5)
We note that the proportionality coefficient cannot be determined using the dimension method, but, nevertheless, we have received a good approximation to a known ratio (the main equation of the Mo-lectured-kinetic theory).
Consider several tasks, on the example of the solution of which we will demonstrate the essence of the dimension method.
Task 1.. Evaluate the expression for the oscillation period of the mathematical pendulum using the analysis of dimensions. Suppose that the period of oscillations of the pendulum depends on its length, accelerate free fall and cargo mass (!):
(6)
Imagine all the above values:
Taking into account (7), rewrite the desired pattern of expression
(8)
(9)
Now it is not difficult to write the system of equations:
In this way, ; from = 0.
(11)
Note that "the mass has zero dimension", i.e. The period of oscillations of the mathematical pendulum does not depend on the mass:
Task 2.. Experiments have shown that the speed of sound in gases depends on the pressure and density of the medium. Compare the speed of sound in gas for two states 
.
At first glance, it seems that we need to enter the temperature of the gas in consideration, as it is well known that the speed of sound depends on temperature. However, compare with reasoning above) the pressure can be expressed as a function of density (concentration) and medium temperature. Therefore, one of the values \u200b\u200b(pressure, density, temperature) is "superfluous." Since by the condition of the problem, we are invited to compare the speeds of different pressures and densities, it is reasonable to exclude the temperature from consideration. Note that if we needed to make a comparison for different pressures and temperatures, we would exclude density.
The speed of sound in the conditions of this task can be represented
Referring ratio (13) as
(14)
From (14) we have
The solution (15) gives.
The results of experiments have the following functional dependence:
The speed of sound for two states is:
(17)
From (17) we get the speed ratio
Task 3.. The cylindrical pillar is wound a rope. For one of the ends of the rope is pulled with force F.. In order for the rope does not slide on the post, when only one turn is wound on the post, the second end is held with force f.. With what force you need to hold this end of the rope, if the post is wound n. turns? How will power change f.If you choose a post halve a larger radius? (Force f. does not depend on the thickness of the rope.)
Obviously, power f. In this case, it may depend only on the applied external force. F., friction coefficient and pole diameter. Mathematical addiction can be represented as
(19)
Since the friction coefficient is dimensionless, then (19) will rewrite in the form of
as but = 1; from \u003d 0 (A is the proportionality coefficient associated with μ). For the second, the third, ... p-Ho wound upwards we write similar expressions:
(21)
Substituting α from (20) in (21), we get:
It is well known that the "dimension method" is often successful in hydrodynamics and aerodynamics. In some cases, it allows you to "evaluate the decision" quickly and with a good degree of reliability.
It is clear that in this case the resistance force may depend on the density of the fluid, the flow rate and the cross-sectional area:
(23)
After performing the corresponding transformations, we find that
(24)
As a rule, relation (24) is represented as
(25)
where. Coefficient from characterizes streamlining bodies and takes various values \u200b\u200bfor bodies: for a ball from \u003d 0.2 - 0.4, for a round disk from \u003d 1.1 - 1,2, for a drop-shaped body from "0.04. (Yavorsky B.M., Pinsky A.A. Basics of physics. - T. 1. - M.: Science, 1974.)
So far, we have considered examples in which the proportionality coefficient remained a dimensionless value, but this does not mean that we should always follow this. It is quite possible to make the ratio of the proportionality "dimensional" depending on the size of the basic values. For example, it is quite appropriate to submit a gravitational constant 
. In other words, the presence of dimension in the gravitational constant means that its numerical value depends on the choice of the main values. (Here we seem to be appropriate to make a reference to the article D.V.Syvukhin "On the International System of Physical Values", UFN, 129, 335, 1975.)
Task 5.. Determine the energy of the gravitational interaction of two point masses t. 1 I. t. 2, located at a distance r. Friend from each other.
In addition to the proposed method of analysis of dimensions, add a solution to the problem principle of symmetryincoming values. Symmetry considerations give reason to believe that the interaction energy should depend on t. 1 I. t. 2 The same way, i.e. In the final expression, they must be in the same degree:
(26)
It's obvious that
Analyzing the relation (26), we find that
but = 1; b.= 1; from = –1,
(28)
Task 6. Find the interaction force between two point charges q. 1 I. q. 2, located at a distance r..
We can use symmetry here, but if we do not want to make assumptions about symmetry or not confident in such symmetry, you can use other methods. This article is written in order to show different methods, so we will solve the problem in another way. It is obvious an analogy with the previous task, however, in this case, you can use the principle of finding equivalent values. We will try to determine the equivalent value - the electric field tension q. 1 at the point of finding charge q. 2. It is clear that the desired force is a work q. 2 on the foundation of the field strength. Therefore, we assume the dependence of tension from the desired values \u200b\u200bin the form:
Imagine everything in the main units:
Having done all the transformations, we obtain the system of equations
In this way, but = –1; b.= 1; from \u003d -2, and expression for tensions takes
The desired strength of the interaction can be represented by the expression
(33)
In the ratio (33) there is no dimensionless coefficient 4π, which was introduced by historical reasons.
Task 7. Determine the tension of the gravitational field of an endless cylinder with a radius r. 0 and density R at a distance R. (R. > r. 0) from the axis of the cylinder.
Since we cannot make assumptions about equality r. 0 I. R., then solve this problem by the dimension method, not attracting other considerations, is quite difficult. We will try to understand the physical essence of the R parameter. It characterizes the density of the mass distribution that creates the field strength of interest. If the cylinder compress, leaving the mass inside the cylinder is unchanged, then the field strength (at a fixed distance R. > r. 0) will be the same. In other words, linear density is more important characteristic, therefore we apply a variable replacement method. Imagine. Now S is a new variable in the proposed task, while:
a. The horizontal and vertical velocity and the acceleration of the free fall are taken according to the view:
We construct a mathematical design for a distance and flight height:
(39)
Analyzing the expression (39), we get now
(40)
(41)
This method is more complex, but it works well if it is possible to distinguish between the values \u200b\u200bmeasured by the same unit of measurement. For example: inertial and gravitational mass ("inertial" and "gravitational" kilograms), vertical and horizontal distance ("vertical" and "horizontal" meters), current strength in one and other chain, etc.
Summing all the above, we note:
1. The dimension method can be used if the desired value can be represented as a power function.
2. The dimension method allows you to qualitatively solve the task and get an answer with accuracy to the coefficient.
3. In some cases, the dimension method is the only way to solve the task and at least evaluate the answer.
4. Analysis of dimensions in solving problems is widely used in scientific research.
5. The solution to the dimensions method is an additional or auxiliary method that allows you to better understand the interaction of values, their influence on each other.
