Derived quantities, as was indicated in § 1, can be expressed in terms of the fundamental ones. To do this, it is necessary to introduce two concepts: the dimension of the derived quantity and the defining equation.
The dimension of a physical quantity is an expression that reflects the relationship of the quantity with the basic quantities
system in which the coefficient of proportionality is taken equal to unity.
The defining equation of a derived quantity is a formula by which a physical quantity can be explicitly expressed in terms of other quantities of the system. In this case, the coefficient of proportionality in this formula should be equal to one. For example, the governing equation for velocity is the formula
where is the length of the path traveled by the body during uniform motion in time. The defining equation of the force in the system is the second law of the dynamics of translational motion (Newton's second law):
where a is the acceleration imparted by the force to the body by the mass
Let's find the dimensions of some derived quantities of mechanics in the system. Note that it is necessary to start with such quantities that are explicitly expressed only through the basic quantities of the system. Such quantities are, for example, speed, area, volume.
To find the dimension of speed, we substitute into formula (2.1) instead of the path length and time their dimensions and T:
Let us agree to denote the dimension of the quantity by the symbol Then the dimension of the velocity can be written in the form
The defining equations of area and volume are the formulas:
where a is the length of the side of the square, the length of the edge of the cube. Substituting instead of the dimension, we find the dimensions of the area and volume:
It would be difficult to find the dimension of the force from its defining equation (2.2), since we do not know the dimension of the acceleration a. Before determining the dimension of force, it is necessary to find the dimension of acceleration,
using the acceleration formula for uniform motion:
where is the change in the speed of the body over time
Substituting here the dimensions of speed and time already known to us, we obtain
Now, using formula (2.2), we find the dimension of the force:
In the same way, to obtain the dimension of power according to its defining equation where A is the work done in time, it is necessary to first find the dimension of the work.
It follows from the examples given that it is not indifferent in what sequence the defining equations should be placed when constructing a given system of quantities, i.e., when establishing the dimensions of derived quantities.
The sequence of arrangement of derived quantities in the construction of the system must satisfy the following conditions: 1) the first must be a value that is expressed only through the main quantities; 2) each subsequent must be a value that is expressed only through the main and such derivatives that precede it.
As an example, we present in the table a sequence of values that satisfies the following conditions:
(see scan)
The sequence of values given in the table is not the only one that satisfies the above condition. Individual values in the table can be rearranged. For example, density (line 5) and moment of inertia (line 4) or moment of force (line 11) and pressure (line 12) can be interchanged, since the dimensions of these quantities are determined independently of each other.
But the density in this sequence cannot be placed before the volume (line 2), since the density is expressed in terms of the volume, and to determine its dimension, it is necessary to know the dimension of the volume. The moment of force, pressure and work (line 13) cannot be set before the force, since to determine their dimensions, you need to know the dimension of the force.
It follows from the above table that the dimension of any physical quantity in the system can be expressed in general terms by the equality
where are integers.
In the system of quantities of mechanics, the dimension of a quantity is expressed in general form by the formula
Let us give in general form the dimension formulas, respectively, in systems of quantities: in electrostatic and electromagnetic LMT, in and in any system with more than three basic quantities:
From formulas (2.5) - (2.10) it follows that the dimension of a quantity is the product of the dimensions of the basic quantities raised to the appropriate powers.
The exponent to which the dimension of the base quantity is raised, which is included in the dimension of the derived quantity, is called the indicator of the dimension of the physical quantity. As a rule, dimensions are integers. The exception is indicators in electrostatic and
electromagnetic systems LMT, in which they can be fractional.
Some dimensions may be equal to zero. Thus, having written the dimensions of velocity and moment of inertia in the system in the form
we find that the velocity has zero dimension of the moment of inertia - the dimension of y.
It may turn out that all indicators of the dimension of a certain quantity are equal to zero. Such a quantity is called dimensionless. Dimensionless quantities are, for example, relative strain, relative permittivity.
A quantity is called dimensional if at least one of the basic quantities in its dimension is raised to a non-zero power.
Of course, the dimensions of the same quantity in different systems may be different. In particular, a dimensionless quantity in one system may turn out to be dimensional in another system. For example, the absolute permittivity in an electrostatic system is a dimensionless quantity, in an electromagnetic system its dimension is equal to and in the system of quantities
Example. Let us determine how the moment of inertia of the system will change with an increase in linear dimensions by 2 times and mass by 3 times.
Uniformity of moment of inertia
Using formula (2.11), we obtain
Therefore, the moment of inertia will increase by 12 times.
2. Using the dimensions of physical quantities, you can determine how the size of the derived unit will change with a change in the size of the basic units through which it is expressed, and also establish the ratio of units in different systems (see p. 216).
3. The dimensions of physical quantities make it possible to detect errors in solving physical problems.
Having received the calculation formula as a result of the solution, one should check whether the dimensions of the left and right parts of the formula coincide. The discrepancy between these dimensions indicates that an error was made in the course of solving the problem. Of course, the coincidence of dimensions does not yet mean that the problem has been solved correctly.
Consideration of other practical applications of dimensions is beyond the scope of this manual.
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The dimension of a physical quantity is an expression showing the relationship of this quantity with the basic quantities of a given system of physical quantities; is written as a product of the powers of the factors corresponding to the main quantities, in which the numerical coefficients are omitted.
Speaking of dimension, one should distinguish between the concepts of a system of physical quantities and a system of units. A system of physical quantities is understood as a set of physical quantities together with a set of equations relating these quantities to each other. In turn, the system of units is a set of basic and derived units, together with their multiples and submultiples, defined in accordance with the established rules for a given system of physical quantities.
All quantities included in the system of physical quantities are divided into basic and derivatives. Under the main understand the values, conditionally chosen as independent so that no main value can be expressed through other basic. All other quantities of the system are determined through the basic quantities and are called derivatives.
Each basic quantity is associated with a dimension symbol in the form of a capital letter of the Latin or Greek alphabet, then the dimensions of derived quantities are denoted using these symbols.
Basic quantity Symbol for dimension
Electric current I
Thermodynamic temperature Θ
Amount of substance N
Light intensity J
In the general case, the dimension of a physical quantity is the product of the dimensions of the basic quantities raised to various (positive or negative, integer or fractional) powers. The exponents in this expression are called the dimensions of the physical quantity. If in the dimension of a quantity at least one of the dimensions is not equal to zero, then such a quantity is called dimensional, if all the dimensions are equal to zero - dimensionless.
The size of a physical quantity is the value of the numbers appearing in the value of the physical quantity.
For example, a car can be characterized by a physical quantity such as mass. At the same time, the value of this physical quantity will be, for example, 1 ton, and the size will be the number 1, or the value will be 1000 kilograms, and the size will be the number 1000. The same car can be characterized using another physical quantity - speed. At the same time, the value of this physical quantity will be, for example, the vector of a certain direction 100 km / h, and the size will be the number 100
The dimension of a physical quantity is a unit of measurement that appears in the value of a physical quantity. As a rule, a physical quantity has many different dimensions: for example, a length has a meter, a mile, an inch, a parsec, a light year, etc. Some of these units of measurement (without taking into account their decimal factors) can be included in various systems of physical units - SI , SGS, etc.
dimension standardization certification
The dimension of a physical quantity is an expression showing the relationship of this quantity with the basic quantities of a given system of physical quantities; is written as a product of the powers of the factors corresponding to the main quantities, in which the numerical coefficients are omitted.
Speaking of dimension, one should distinguish between the concepts of a system of physical quantities and a system of units. A system of physical quantities is understood as a set of physical quantities together with a set of equations relating these quantities to each other. In turn, the system of units is a set of basic and derived units, together with their multiples and submultiples, defined in accordance with the established rules for a given system of physical quantities.
All quantities included in the system of physical quantities are divided into basic and derivatives. Under the main understand the values, conditionally chosen as independent so that no main value can be expressed through other basic. All other quantities of the system are determined through the basic quantities and are called derivatives.
Each basic quantity is associated with a dimension symbol in the form of a capital letter of the Latin or Greek alphabet, then the dimensions of derived quantities are denoted using these symbols.
In the International System of Quantities (ISQ), on which the International System of Units (SI) is based, length, mass, time, electric current, thermodynamic temperature, luminous intensity and amount of substance are chosen as the main quantities. The symbols of their dimensions are given in the table.
The dim symbol is used to indicate the dimensions of derived quantities.
For example, for speed with uniform motion,
where is the length of the path traveled by the body in time. In order to determine the dimension of speed, instead of the length of the path and time, substitute their dimensions in this formula:
Similarly, for the acceleration dimension, we get
From the equation of Newton's second law, taking into account the dimension of acceleration for the dimension of force, it follows:
In the general case, the dimension of a physical quantity is the product of the dimensions of the basic quantities raised to various (positive or negative, integer or fractional) powers. The exponents in this expression are called the dimensions of the physical quantity. If in the dimension of a quantity at least one of the dimensions is not equal to zero, then such a quantity is called dimensional, if all the dimensions are equal to zero - dimensionless.
Dimension symbols are also used to designate systems of quantities. So, the system of quantities, the main quantities of which are length, mass and time, is designated as LMT. On its basis, such systems of units as the SGS, MKS and MTS were formed.
As follows from the above, the dimension of a physical quantity depends on the system of quantities used. Therefore, in particular, a dimensionless quantity in one system of quantities can become dimensional in another. For example, in the LMT system, the electric capacitance has the dimension L and the ratio of the capacitance of a spherical body to its radius is a dimensionless quantity, while in the International System of Quantities (ISQ) this ratio is not dimensionless. However, many dimensionless numbers used in practice (for example, similarity criteria, the fine structure constant in quantum physics, or the Mach, Reynolds, Strouhal, etc. numbers in continuum mechanics) characterize the relative influence of certain physical factors and are the ratio of quantities with the same dimensions, therefore, despite the fact that the quantities included in them in different systems may have different dimensions, they themselves will always be dimensionless.
The size of a physical quantity is the value of the numbers appearing in the value of a physical quantity, and the dimension of a physical quantity is a unit of measurement appearing in the value of a physical quantity. As a rule, a physical quantity has many different dimensions: for example, a length has a meter, a mile, an inch, a parsec, a light year, etc. Some of these units of measurement (without taking into account their decimal factors) can be included in various systems of physical units - - SI, GHS, etc. For example, a car can be characterized using such a physical quantity as mass. The size of this physical quantity will be 50, 100, 200, etc., and the dimension is expressed in units of mass - kilogram, centner, ton. The same car can be characterized by another physical quantity - speed. In this case, the size will be, for example, the number 100, and the dimension will be the speed unit: km / h.
Dimensions of physical quantities in the SI system
The table shows the dimensions of various physical quantities in the International System of Units (SI).
The columns "Exponents" indicate the exponents in terms of units of measurement through the corresponding units of the SI system. For example, the farad is (−2 | −1 | 4 | 2 | |), so
1 farad \u003d m −2 kg −1 s 4 A 2.
| Name and designation quantities |
Unit measurements |
Designation | Formula | exponents | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Russian | international | m | kg | with | BUT | To | cd | ||||
| Length | L | meter | m | m | L | 1 | |||||
| Weight | m | kilogram | kg | kg | m | 1 | |||||
| Time | t | second | with | s | t | 1 | |||||
| The strength of the electric current | I | ampere | BUT | A | I | 1 | |||||
| Thermodynamic temperature | T | kelvin | To | K | T | 1 | |||||
| The power of light | Iv | candela | cd | cd | J | 1 | |||||
| Square | S | sq. meter | m 2 | m2 | S | 2 | |||||
| Volume | V | cube meter | m 3 | m 3 | V | 3 | |||||
| Frequency | f | hertz | Hz | Hz | f = 1/t | −1 | |||||
| Speed | v | m/s | m/s | v = dL/dt | 1 | −1 | |||||
| Acceleration | a | m/s 2 | m/s 2 | ε = d 2 L/dt 2 | 1 | −2 | |||||
| flat corner | φ | glad | rad | φ | |||||||
| Angular velocity | ω | rad/s | rad/s | ω = dφ/dt | −1 | ||||||
| Angular acceleration | ε | rad/s 2 | rad/s 2 | ε \u003d d 2 φ / dt 2 | −2 | ||||||
| Force | F | newton | H | N | F=ma | 1 | 1 | −2 | |||
| Pressure | P | pascal | Pa | Pa | P=F/S | −1 | 1 | −2 | |||
| work, energy | A | joule | J | J | A = F L | 2 | 1 | −2 | |||
| Impulse | p | kg m/s | kg m/s | p = mv | 1 | 1 | −1 | ||||
| Power | P | watt | Tue | W | P = A/t | 2 | 1 | −3 | |||
| Electric charge | q | pendant | Cl | C | q = I t | 1 | 1 | ||||
| Electrical voltage, electrical potential | U | volt | AT | V | U = A/q | 2 | 1 | −3 | −1 | ||
| Electric field strength | E | V/m | V/m | E=U/L | 1 | 1 | −3 | −1 | |||
| Electrical resistance | R | ohm | Ohm | Ω | R = U/I | 2 | 1 | −3 | −2 | ||
| Electric capacity | C | farad | F | F | C = q/U | −2 | −1 | 4 | 2 | ||
| Magnetic induction | B | tesla | Tl | T | B = F/I L | 1 | −2 | −1 | |||
| Magnetic field strength | H | A/m | A/m | −1 | 1 | ||||||
| magnetic flux | F | weber | wb | wb | F = B S | 2 | 1 | −2 | −1 | ||
| Inductance | L | Henry | gn | H | L = U dt/dI | 2 | 1 | −2 | −2 | ||
see also
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And the Dimension of a physical quantity is an expression that characterizes the relationship of this physical quantity with the basic quantities of a given system of units. A physical quantity is called a dimensionless quantity if all basic quantities are included in the expression of its dimension to the zero degree. The numerical value of a dimensionless quantity does not depend on the choice of the system of units.
The dimension of a physical quantity should be understood as an expression that reflects the relationship of the quantity under consideration with the basic quantities of the system, if we take the proportionality coefficient in this expression equal to a dimensionless unit. The dimension is the product of the dimensions of the basic quantities of the system, raised to the appropriate degree.
So, the dimension of a physical quantity indicates how, in a given absolute system of units, the units that serve to measure this physical quantity change when the scales of the basic units change. For example, a force in the LMT system has the dimension LMT 2; this means that when the unit of length increases by n times, the unit of force also increases by n times; when the unit of mass increases by n times, the unit of force also increases by n times, and, finally, when the unit of time increases by n times, the unit of force decreases by 2 times.
Considerations concerning the dimensionality of physical quantities help in solving problems of great practical importance, for example, the problem of a stationary flow of a liquid or gas around an obstacle, or, what is the same, of the motion of a body in a medium.
To indicate the dimension of physical quantities, symbolic notation is used, for example, LpM. This means that in the LMT system, the number expressing the result of measuring a given physical quantity will decrease by a factor of n if the unit of length is increased by n times, will increase by n 1 times if the unit of mass is increased by n times, and, finally, will increase in pg times, if the unit of time is increased by n times.
It is customary to write the result of determining the dimension of a physical quantity by conditional equality, in which this quantity is enclosed in square brackets.
If we look at the dimensions of physical quantities that actually occur in physics, it is easy to see that in all cases the numbers p, q, r turn out to be rational. This is not necessarily from the point of view of dimensional theory, but is the result of the corresponding definitions of physical quantities.
Thus, the dimension of a physical quantity is a function that determines how many times the numerical value of this quantity will change when moving from the original system of units of measurement to another system within this class.
Let us now define the concept of the dimension of a physical quantity. The dimension shows how a given quantity is related to the basic physical quantities. In the International System of Units SI, the main physical quantities correspond to the main units of measurement: length, mass, time, current strength, temperature, amount of substance and luminous intensity.
By using the analysis of dimensions of physical quantities, a functional relationship is established between generalized variables (similarity equation), and a quantitative dependence is obtained as a result of processing experimental data.
If, when determining the dimension of a physical quantity, its basic units of measurement are reduced, then such a quantity is called dimensionless. The dimensionless quantities are the relative coordinates of the points of the body, the aerodynamic coefficients of the wing profile, and the relative deformations of the elastic structure. Constant and variable dimensionless quantities occupy a special place in the study of the similarity of physical phenomena.
Strictly speaking, the dimension of a physical quantity is the exponent in a symbolic equation that expresses this quantity in terms of basic physical quantities.
