The concept of the dimension of measured quantities

The dimension of the measured quantity is its qualitative characteristic and is denoted by the symbol dim, derived from the word dimension (dimension, range, magnitude, degree, measure).
The dimensions of basic physical quantities are indicated by the corresponding capital letters.
For example, for length, mass and time:

dim l = L; dim m = M; dim t = T.

When determining the dimension of derived quantities, the following rules are used:

1. The dimensions of the left and right sides of the equations cannot but coincide, since only identical properties can be compared with each other. By combining the left and right sides of the equations, we can come to the conclusion that only quantities that have the same dimensions can be algebraically summed.

2. The algebra of dimensions is multiplicative, i.e. it consists of one single action - multiplication.

3. The dimension of the product of several quantities is equal to the product of their dimensions. So, if the relationship between the values ​​of quantities Q, A, B, C has the form Q = A × B × C, then

dim Q = dim A×dim B×dim C .

4. The dimension of a quotient when dividing one quantity by another is equal to the ratio of their dimensions, i.e. if Q = A/B, then

dim Q = dim A/dim B .

5. The dimension of any quantity raised to a certain power is equal to its dimension to the same power.
So, if Q = A n, then

dim Q = dim n A .

For example, if the speed is determined by the formula V = l / t, then dim V = dim l/dim t = L/T = LT -1.
If the force according to Newton’s second law F = ma, where a = V/ t is the acceleration of the body, then

dim F = dim m×dim a = ML/T 2 = MLT -2.

So, it is always possible to express the dimension of a derivative of a physical quantity in terms of the dimensions of the basic physical quantities using a power monomial:

dim Q = LMT ... ,

Where:
L, M, T,... - dimensions of the corresponding basic physical quantities;
a, b , q ,... - dimension indicators. Each of the dimension indicators can be positive or negative, an integer or fractional number, or zero.

If all dimension indicators are equal to zero, then such a quantity is called dimensionless. It can be relative, defined as the ratio of quantities of the same name (e.g. relative dielectric constant), and logarithmic, defined as the logarithm of the relative value (for example, the logarithm of the power or voltage ratio).
In the humanities, art, sports, qualimetry, where the nomenclature of basic quantities is not defined, the theory of dimensions has not yet found effective application.

The size of the measured value is its quantitative characteristic. Obtaining information about the size of a physical or non-physical quantity is the content of any measurement.

Measuring scales and their types

In measurement theory, it is generally accepted to distinguish between five types of scales: names, order, differences (intervals), relations and absolute.

Name scales are characterized only by the relation of equivalence (equality). An example of such a scale is the common classification (assessment) of color by name (color atlases up to 1000 names).

Order scales are the sizes of the measured quantity arranged in ascending or descending order. Arranging sizes in ascending or descending order to obtain measurement information on a scale of order is called ranking. To facilitate measurements on the order scale, some points on it can be fixed as reference points. The disadvantage of reference scales is the uncertainty of the intervals between reference points.
In this regard, points cannot be added, calculated, multiplied, divided, etc.
Examples of such scales are: student knowledge by points, earthquakes by 12 -point system, wind force on the Beaufort scale, film sensitivity, hardness on the Mohs scale, etc.

Difference (interval) scales differ from order scales in that using the interval scale one can already judge not only whether a size is larger than another, but also how much larger. Mathematical operations such as addition and subtraction are possible on the interval scale.
A typical example is the scale of time intervals, since time intervals can be summed or subtracted, but adding, for example, the dates of any events does not make sense.

Ratio scales describe properties to which the relations of equivalence, order and summation, and therefore subtraction and multiplication, are applicable to the set of quantitative manifestations themselves. In the ratio scale, there is a zero value for the property indicator. An example is the length scale.
Any measurement on a ratio scale consists of comparing an unknown size with a known one and expressing the first through the second in a multiple or fractional ratio.

Absolute scales have all the features of ratio scales, but they additionally have a natural, unambiguous definition of the unit of measurement. Such scales correspond to relative values (relations of physical quantities of the same name, described by ratio scales). These values ​​include gain, attenuation, etc. Among these scales, there are scales whose values ​​range from 0 before 1 (efficiency, reflection, etc.).

Measurement (comparing the unknown with the known) occurs under the influence of many random and non-random, additive (added) and multiplicative (multiplied) factors, the exact accounting of which is impossible, and the result of joint influence is unpredictable.

The main postulate of metrology - counting - is a random number.
The mathematical model of measurement on a comparison scale has the form:

q = (Q + V)/[Q] + U,

Where:
q - measurement result (numerical value of Q);
Q is the value of the measured quantity;
[Q] - unit of a given physical quantity;
V - tare mass (for example, when weighing);
U is the term from the additive effect.

From the above formula we can express the value of the measured quantity Q:

Q = q[Q] - U[Q] - V .

When a value is measured once, its value is calculated taking into account the correction:

Q i = q i [Q] + i ,

Where:
q i [Q] - the result of a single measurement;
i = - U[Q] - V - total correction.

The value of the measured quantity during repeated measurements can be determined from the relationship:

Q n = 1/n×∑Q i .

A certain value of a physical quantity is taken as a unit of this quantity. The size of a physical quantity is determined by the relation where is the numerical value of this quantity. This relationship is called the fundamental equation of measurement because the purpose of measurement is essentially to determine a number.

Ensuring the uniformity of measurements involves, first of all, the widespread use of generally accepted and strictly defined units of physical quantities. Between various physical quantities, there are objectively different kinds of relationships that are quantitatively expressed by the corresponding equations. These uraniums are used to express units of one physical quantity in terms of another. However, the number of such equations in any branch of science is less than the number of physical quantities included in them. Therefore, in order to create a system of units of these quantities, some of their fundamental part, equal, must be specified and strictly defined, regardless of other quantities. Such physical quantities included in the system, conventionally accepted as independent of other quantities, are called basic physical quantities. The remaining quantities included in the system and determined through basic physical quantities are called derived physical quantities. In accordance with this, units of physical quantities are also divided into basic and derived units.

If A, B, C, ... is a complete set of basic physical quantities of a given system, then for any derived quantity its dimension can be determined, reflecting its connection with the basic quantities of the system, in the form

In this relation, the exponents,... for each specific derivative of a physical quantity are found from equations connecting it with the basic quantities (part of these exponents usually turns out to be zero). Relationship (1), called the dimensional formula, shows how many times the value of the derivative quantity will change with a certain change in the values ​​of the basic quantities. For example, if the values ​​of quantities A, B, C increased by 2, 3 and 4 times, respectively, then, according to (1), the value of the quantity will increase by a factor.

The main practical significance of the dimensional formula is that it allows you to directly determine any derived unit through the basic units of a given system,...

True, in this expression the constant factor requires additional definition. However, in most practical cases they try to choose. Under this condition, the derived unit is called coherent.

The International System of Units SI is a coherent system (since all its derived units are coherent). Basic physical quantities and their units in the SI system are presented in Table 1.

Table 1

In addition, the SI system includes two additional units, which are also defined independently of the other units, but do not participate in the formation of derived units. These are the unit of plane angle - radian (rad) and the unit of solid angle - steradian (sr). All other units of the SI system are derived, some of them having their own name, while others are designated as a product of powers of others. For example, such a derived physical quantity as electrical capacitance in the SI system has a dimension and a unit that has its own name - the farad; and the unit of electric field strength, for example, does not have its own name and is designated as “volt per meter”.

Together with units of the SI system, the use of multiples and submultiples is allowed, which are formed by adding a certain prefix to the name of the unit, meaning multiplication of this unit by, where is a positive integer (for multiple units) or negative (for submultiples) number. For example, 1 GHz (gigahertz) = 109 Hz, 1 ns (nanosecond) = 10-9 s, 1 kW = 103 W. Table 2 shows the names of prefixes of submultiple and multiple units.

table 2

Together with the SI system, it is allowed to use - where appropriate - some non-system units: for time - minute, hour, day, for a plane angle - degree, minute, second; for mass - ton; for volume - liter; for area - hectare; for energy - electron-volt; for full power - volt-amperes, etc.

In addition to the types of units considered, relative and logarithmic values ​​are widely used. They represent, respectively, the ratio of two quantities of the same name and the logarithm of this ratio. Relative quantities, in particular, include the atomic and molecular masses of chemical elements.

Relative values ​​can be expressed in indifferent units, as a percentage (1% = 0.01) or in ppm (1‰=0.001=0.1%).

The value of logarithmic quantities is expressed in bels (B), according to the formula or in nepers (Np): . In these relationships, and are energy quantities (power, energy, energy density, etc.); and -- power quantities (voltage, current, current density, field strength, etc.); coefficients 2 and 0.5 take into account that energy quantities are proportional to the square of force quantities. From the ratios it is clear that one bel (1 B) corresponds to the ratio or; one neper (1 Np) corresponds to the relation or. It is not difficult to find out that 1 Np = () B = 0.8686 B.

In radio engineering, electronics, and acoustics, logarithmic values ​​are most often expressed in decibels (1 dB = 0.1 B):

The power ratio in dB is written with a factor of 10, and the voltage (or current) ratio with a factor of 20.

Obviously, relative and logarithmic units are invariant to the system of units used, since they are determined by the ratio of homogeneous units.

When we talk about the dimension of a quantity, we mean the basic units or basic quantities with the help of which a given quantity can be constructed.
  The dimension of area, for example, is always equal to the square of the length (abbreviated ; square brackets hereinafter indicate dimension); Area units can be square meter, square centimeter, square foot, etc.
  Speed ​​can be measured in units of km/h, m/s and mph, but its dimension is always equal to the dimension of length [L], divided by the time dimension [T], i.e. we have . The formulas describing the quantity may be different in different cases, but the dimension remains the same. For example, the area of ​​a triangle with a base b and height h equal to S = (1/2)bh, and the area of ​​a circle with radius r equal to S = πr 2. These formulas differ from each other, but the dimensions in both cases coincide and are equal .
  When determining the dimension of a quantity, the dimensions of basic rather than derived quantities are usually used. For example, force, as we will see below, has the dimension of mass [M], multiplied by acceleration those. its dimension is equal .
  The rule for selecting dimensions can help in deriving various relationships; This procedure is called dimensional analysis. One useful method is to use dimensional analysis to check the validity of a particular relationship. In this case, two simple rules are used. Firstly, you can only add or subtract quantities of the same dimension (you cannot add centimeters and grams); secondly, the quantities on both sides of any equality must have the same dimensions.
  Let, for example, we obtain the expression v = v o + (1/2)at 2, Where v− body speed over time t, v o− initial speed of the body, A− the acceleration he experiences. To check the correctness of this formula, we will perform a dimensional analysis. Let us write down an equality for the dimension, taking into account that speed has the dimension , and acceleration - dimension :

In this formula, the dimension is not all right; on the right side of the equality is the sum of quantities whose dimensions do not coincide. From this we can conclude that an error was made in deriving the original expression.
  The coincidence of dimensions in both parts does not yet prove the correctness of the expression as a whole. For example, a dimensionless numerical factor of the form 1/2 or 2π. Therefore, checking the dimensionality can only indicate the error of an expression, but cannot serve as proof of its correctness.
  Dimensional analysis can also be used as a quick check to see if a relationship you are unsure about is correct. Let's say you can't remember the period expression T(the time required to complete a complete oscillation) of a simple mathematical pendulum of length l: does this formula look like

either

Where g− free fall acceleration, the dimension of which, like any acceleration, is equal to .
  We will only be interested in whether it includes the quantities l And g as a relation l/g or g/l.) Dimensional analysis shows that the first formula is correct:

while the second one is wrong because

  Please note that the constant factor 2π is dimensionless and is not included in the final result.
  Finally, an important application of dimensional analysis (which, however, requires great care) is to find the type of relationship being sought. Such a need may arise if you only need to determine how one quantity depends on others.
  Let's consider a specific example of obtaining a formula for a period T oscillations of a mathematical pendulum. First, let us determine on what quantities the T. The period may depend on the length of the thread l, mass at the end of the pendulum m, pendulum deflection angle α and free fall acceleration g. It may also depend on air resistance (we will use air viscosity here), the gravitational pull of the Moon, etc. However, everyday experience indicates that the force of gravity on the Earth significantly exceeds all other forces, which we will therefore neglect. Let's assume that the period T is a function of quantities l, m, α And g, and each of these quantities is raised to some power:

Here WITH− dimensionless constant; α , β , And δ − exponents to be determined.
Let us write down the dimension formula for this relationship:

After some simplifications we get

  Due to the fact that the seven basic quantities of the SI system (System Internationale) are the international system of units, a version of the metric system has been used since 1960, when at the XI General Conference on Weights and Measures a standard was adopted, which was first called the International System of Units (SI). )". SI is the most widely used system of units in the world, both in everyday life and in science and technology.
Basic SI units, the names of SI units are written with a lowercase letter, there is no dot after the designations of SI units.

 Problem 3. Determine the interaction energy of two point masses m 1 And m 2, located at a distance r from each other.

 Problem 4. Determine the force of interaction between two point charges q 1 And q 2, located at a distance r from each other.

 Problem 5. Determine the gravitational field strength of an infinite cylinder of radius r o and density ρ on distance R (R > r o) from the cylinder axis.

 Problem 6. Estimate the flight range and height of a body thrown at an angle α to the horizon. Neglect air resistance.

 Conclusion:
 1. The dimensional method can be used if the desired quantity can be represented as a power function.
 2. The dimensional method allows you to solve the problem qualitatively and obtain an answer accurate to a coefficient.
 3. In some cases, the dimensional method is the only way to solve the problem and at least estimate the answer.
 4. Dimensional analysis when solving a problem is widely used in scientific research.
 5. Solving problems using the dimensional method is an additional or auxiliary method that allows you to better understand the interaction of quantities and their influence on each other.

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Did you know, What is the falsity of the concept of “physical vacuum”?

Physical vacuum - the concept of relativistic quantum physics, by which they mean the lowest (ground) energy state of a quantized field, which has zero momentum, angular momentum and other quantum numbers. Relativistic theorists call a physical vacuum a space completely devoid of matter, filled with an unmeasurable, and therefore only imaginary, field. This state, according to relativists, is not an absolute void, but a space filled with some phantom (virtual) particles. Relativistic quantum field theory states that, in accordance with the Heisenberg uncertainty principle, virtual, that is, apparent (apparent to whom?), particles are constantly born and disappeared in the physical vacuum: so-called zero-point field oscillations occur. Virtual particles of the physical vacuum, and therefore itself, by definition, do not have a reference system, since otherwise Einstein’s principle of relativity, on which the theory of relativity is based, would be violated (that is, an absolute measurement system with reference to the particles of the physical vacuum would become possible, which in turn would clearly refute the principle of relativity on which the SRT is based). Thus, the physical vacuum and its particles are not elements of the physical world, but only elements of the theory of relativity, which do not exist in the real world, but only in relativistic formulas, while violating the principle of causality (they appear and disappear without cause), the principle of objectivity (virtual particles can be considered, depending on the desire of the theorist, either existing or non-existent), the principle of factual measurability (not observable, do not have their own ISO).

When one or another physicist uses the concept of “physical vacuum,” he either does not understand the absurdity of this term, or is disingenuous, being a hidden or overt adherent of relativistic ideology.

The easiest way to understand the absurdity of this concept is to turn to the origins of its occurrence. It was born by Paul Dirac in the 1930s, when it became clear that denying the ether in its pure form, as was done by a great mathematician but a mediocre physicist, was no longer possible. There are too many facts that contradict this.

To defend relativism, Paul Dirac introduced the aphysical and illogical concept of negative energy, and then the existence of a “sea” of two energies compensating each other in a vacuum - positive and negative, as well as a “sea” of particles compensating each other - virtual (that is, apparent) electrons and positrons in a vacuum.

Physical quantities and their dimensions

FORMATION OF STUDENTS' CONCEPTS ABOUT PHYSICAL QUANTITIES AND LAWS

Classification of physical quantities

Units of measurement of physical quantities. Systems of units.

Problems of developing physical concepts among students

Formation of students' concepts of physical quantities using the method of frame supports

Formation of students' concepts of physical laws using the method of frame supports

Physical quantities and their dimensions

Physical size name a property that is qualitatively common to many physical objects, but quantitatively individual for each object (Bolsun, 1983)/

A set of physical functions interconnected by dependencies is called a system of physical quantities. The PV system consists of basic quantities, which are conditionally accepted as independent, and from derived quantities, which are expressed through the basic quantities of the system.

Derived physical quantities- these are physical quantities included in the system and determined through the basic quantities of this system. The mathematical relationship (formula), through which the derivative of the PV we are interested in is expressed explicitly through other quantities of the system and in which the direct connection between them is manifested, is called defining equation. For example, the defining equation for speed is the relation

V = (1)

Experience shows that the PV system, covering all branches of physics, can be built on seven basic quantities: mass, time, length, temperature, light intensity, amount of substance, electric current.

Scientists have agreed to denote the main PVs with symbols: length (distance) in any equations and any systems with the symbol L (the word length begins with this letter in English and German), and time with the symbol T (the word time begins with this letter in English). The same applies to the dimensions of mass (symbol M), electric current (symbol I), thermodynamic temperature (symbol Θ), amount of matter (symbol

N), luminous intensity (symbol J). These symbols are called dimensions length and time, mass, etc., regardless of the size of length or time. (Sometimes these symbols are called logical operators, sometimes radicals, but most often dimensions.) Thus, Dimension of the main PV -This only FV symbol in the form of a capital letter of the Latin or Greek alphabet.
So, for example, the dimension of speed is a symbol of speed in the form of two letters LT −1 (according to formula (1)), where T represents the dimension of time, and L - length. These symbols denote the PV of time and length, regardless of their specific size (second , minute, hour, meter, centimeter, etc.). The dimension of force is MLT −2 (according to the equation of Newton’s second law F = ma). Any derivative of the PV has a dimension, since there is an equation that determines this quantity. There is an extremely useful mathematical procedure in physics called dimensional analysis or checking a formula by dimension.

There are still two opposing opinions regarding the concept of “dimension”. Prof. Kogan I. Sh., in the article Dimension of a physical quantity(Kogan,) gives the following arguments regarding this dispute. For more than a hundred years, disputes about the physical meaning of dimensions have continued. Two opinions - dimension refers to a physical quantity, and dimension refers to a unit of measurement - have been dividing scientists into two camps for a century. The first point of view was defended by the famous physicist of the early twentieth century A. Sommerfeld. The second point of view was defended by the outstanding physicist M. Planck, who considered the dimension of a physical quantity to be a kind of convention. The famous metrologist L. Sena (1988) adhered to the point of view according to which the concept of dimension does not refer to a physical quantity at all, but to its unit of measurement. The same point of view is presented in the popular textbook on physics by I. Savelyev (2005).

However, this confrontation is artificial. The dimension of a physical quantity and its unit of measurement are different physical categories and should not be compared. This is the essence of the answer that solves this problem.

We can say that a physical quantity has dimension insofar as there is an equation that determines this quantity. As long as there is no equation, there is no dimension, although this does not make the physical quantity cease to exist objectively. There is no objective need for the existence of dimension in a unit of measurement of a physical quantity.

Yet again, dimensions physical quantities for the same physical quantities must be the same on any planet in any star system. At the same time, the units of measurement of the same quantities may turn out to be anything and, of course, not similar to our earthly ones.

This view of the problem suggests that Both A. Sommerfeld and M. Planck are right. Each of them just meant something different. A. Sommerfeld meant the dimensions of physical quantities, and M. Planck meant units of measurement. Contrasting their views to each other, metrologists groundlessly equate the dimensions of physical quantities with their units of measurement, thereby artificially contrasting the points of view of A. Sommerfeld and M. Planck.

In this manual, the concept of “dimension,” as expected, refers to PV and is not identified with PV units.