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 "Exponents" columns indicate exponents in terms of units of measurement through the corresponding SI units. For example, for a farad it is indicated (−2 | −1 | 4 | 2 | |), which means
1 farad = m −2 kg −1 s 4 A 2 .
| Name and designation quantities |
Unit measurements |
Designation | Formula | Exponents | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Russian | international | m | kg | With | A | TO | cd | ||||
| Length | L | meter | m | m | L | 1 | |||||
| Weight | m | kilogram | kg | kg | m | 1 | |||||
| Time | t | second | With | s | t | 1 | |||||
| Electric current strength | I | ampere | A | 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 | m 2 | 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 angle | 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 = d 2 φ/dt 2 | −2 | ||||||
| Strength | F | newton | N | 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 | W | W | P = A/t | 2 | 1 | −3 | |||
| Electric charge | q | pendant | Cl | C | q = I t | 1 | 1 | ||||
| Electrical voltage, electrical potential | U | volt | IN | 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 | ||
| Electrical 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 | Vehicle | A/m | −1 | 1 | ||||||
| Magnetic flux | F | weber | Wb | Wb | Ф = B·S | 2 | 1 | −2 | −1 | ||
| Inductance | L | Henry | Gn | H | L = Udt/dI | 2 | 1 | −2 | −2 | ||
See also
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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)/
The set of physical functions interconnected by dependencies is called a system of physical quantities. The PV system consists of basic values, 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 usually called defining equation. For example, the defining equation for speed is the relation
V = (1)
Experience shows that a PV system covering all branches of physics should 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 (it starts with this letter in English and German languages the word length), and time - the symbol T (this letter begins with English word time). 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.) Dimension of the main PV -This just FV symbol in the form capital letter Latin or Greek alphabet. So, for example, the dimension of speed is ϶ᴛᴏ the speed symbol 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 have continued about physical sense dimensions. 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 famous physicist beginning of the twentieth century A. Sommerfeld. The second point of view was defended outstanding physicist M. Planck, who considered the dimension of a physical quantity to be some 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).
Moreover, 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. The existence of a dimension in a unit of measurement of a physical quantity is not of any objective importance.
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.
Physical quantities and their dimensions - concept and types. Classification and features of the category "Physical quantities and their dimensions" 2017, 2018.
Page 3
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. Numeric value dimensionless quantity does not depend on the choice of system of units.
The dimension of a physical quantity should be understood as an expression reflecting 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. Dimension is the product of the dimensions of the main quantities of the system raised to the appropriate powers.
So, the dimension of a physical quantity indicates how, in a given absolute system of units, the units used to measure this physical quantity change when the scale of the basic units changes. For example, force in the LMT system has dimension LMT 2; this means that when a 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 dimension of physical quantities help in solving problems of enormous practical importance, for example, the problem of stationary flow of a liquid or gas around an obstacle, or, what is the same, the movement of a body in a medium.
To indicate the dimension of physical quantities, symbolic notations are 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 n times 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 by pg times, if the unit of time is increased by n times.
The result of determining the dimension of a physical quantity is usually written down as a conditional equality, in which this quantity is enclosed in square brackets.
If you look at the dimensions of physical quantities actually encountered in physics, it is easy to notice that in all cases the numbers p, q, r turn out to be rational. This is not necessary 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 it will change numerical value this value when moving from the original system of units of measurement to another system within a given class.
Let us now define the concept of dimension of a physical quantity. Dimension shows how connected given value with basic physical quantities. IN International system SI units basic physical quantities correspond to the basic units of measurement: length, mass, time, current strength, temperature, amount of matter and luminous intensity.
By using dimensional analysis of physical quantities, a functional relationship is established between generalized variables (similarity equation), and a quantitative relationship is obtained as a result of processing experimental data.
If, when determining the dimension of a physical quantity, the basic units of measurement that make it up are reduced, then such a quantity is called dimensionless. Dimensionless quantities are the relative coordinates of body points, aerodynamic coefficients of the wing profile, and relative deformations of the elastic structure. Constant and variable dimensionless quantities occupy special place when studying the similarity of physical phenomena.
Strictly speaking, the dimension of a physical quantity is the exponent in a symbolic equation expressing this quantity through basic physical quantities.
Derived quantities, as indicated in § 1, can be expressed in terms of basic ones. To do this, it is necessary to introduce two concepts: the dimension of the derivative quantity and the defining equation.
The dimension of a physical quantity is an expression that reflects the relationship of a quantity with basic quantities
system in which the proportionality coefficient is adopted equal to one.
The defining equation of a derivative 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 proportionality coefficient in this formula must be equal to one. For example, the governing equation for speed is the formula
where is the length of the path traveled by the body at uniform motion during time The governing equation of force in the system is the second law of dynamics forward motion(Newton's second law):
where a is the acceleration imparted by force to a body of mass
Let's find the dimensions of some derivative 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 their dimensions and T into formula (2.1) instead of the path length and time:
Let us agree to denote the dimension of a quantity by the symbol. Then the dimension of speed will 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 dimension, we find the dimensions of area and volume:
It would be difficult to find the dimension of the force using 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 formula for acceleration of uniformly alternating motion:
where is the change in body speed over time
Substituting here the dimensions of speed and time already known to us, we get
Now, using formula (2.2), we find the dimension of the force:
In the same way, to obtain the dimension of power from its defining equation where A is the work done during time, it is necessary to first find the dimension of the work.
From the above examples it follows that it is not indifferent in what sequence the defining equations must be arranged when constructing a given system of quantities, that is, when establishing the dimensions of derived quantities.
The sequence of arrangement of derivative quantities when constructing a system must satisfy following conditions: 1) the first must be a quantity that is expressed only through basic quantities; 2) each subsequent one must be a quantity that is expressed only through the basic and such derivatives that precede it.
As an example, we present in the table a sequence of quantities 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 swapped, since the dimensions of these quantities are determined independently of each other.
But density in this sequence cannot be placed before volume (line 2), since density is expressed through volume and to determine its dimension it is necessary to know the dimension of volume. The moment of force, pressure and work (line 13) cannot be placed before the force, since to determine their dimension it is necessary to know the dimension of the force.
From the table above it follows that the dimension of any physical quantity in the system is general view can be expressed by 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 present in general form the formulas of dimension, respectively, in systems of quantities: in electrostatic and electromagnetic LMT, in and in any system with the number of basic quantities more than three:
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 basic quantity, included in the dimension of the derived quantity, is raised is called the dimension index of the physical quantity. As a rule, dimension indicators are integers. The exception is indicators in electrostatic and
electromagnetic LMT systems, in which they can be fractional.
Some dimension indicators may be equal to zero. Thus, having written down the dimensions of speed and moment of inertia in the system in the form
we find that the speed equal to zero the indicator of the dimension of the moment of inertia is the indicator of the dimension y.
It may turn out that all dimension indicators of a certain quantity are equal to zero. This quantity is called dimensionless. Dimensionless quantities are, for example, relative deformation, relative permittivity.
A quantity is called dimensional if in its dimension at least one of the basic quantities is raised to a power not equal to zero.
Of course, the dimensions of the same quantity in different systems may turn out to be different. In particular, a dimensionless quantity in one system may turn out to be dimensional in another system. For example, the absolute dielectric constant in electrostatic system is dimensionless in magnitude, in electromagnetic system its dimension is equal to a in the system of quantities
Example. Let us determine how the moment of inertia of the system changes 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
Consequently, the moment of inertia will increase by 12 times.
2. Using the dimensions of physical quantities, you can determine how the size of a 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 when solving physical problems.
Having received as a result the decision calculation formula, you should check whether the dimensions of the left and right parts formulas. The discrepancy between these dimensions indicates that an error was made in solving the problem. Of course, the coincidence of dimensions does not mean that the problem has been solved correctly.
Consideration of others practical applications dimensions are 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 powers of factors corresponding to the basic quantities, in which numerical coefficients omitted.
Speaking about dimension, we 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 that relate these quantities to each other. In turn, the system of units is a set of basic and derived units along with their multiples and submultiple units, defined in accordance with established rules for a given system of physical quantities.
All quantities included in the system of physical quantities are divided into basic and derivative. Basic quantities are understood as quantities that are conditionally chosen as independent ones so that no fundamental quantity can be expressed through other fundamental ones. 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 designated using these symbols.
Basic quantity Symbol for dimension
Thermodynamic temperature Θ
Amount of substance N
Luminous intensity J
IN general case the dimension of a physical quantity is the product of the dimensions of basic quantities raised to various (positive or negative, integer or fractional) powers. The exponents in this expression are called indicators of the dimension of a physical quantity. If in the dimension of a quantity at least one of the dimension indicators is not equal to zero, then such a quantity is called dimensional, if all dimension indicators are equal to zero - dimensionless.
The size of a physical quantity is the meaning of the numbers appearing in the value of a physical quantity.
For example, a car can be characterized using a physical quantity such as mass. In this case, 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. In this case, the value of this physical quantity will be, for example, a vector of a certain direction of 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, length - meter, mile, inch, parsec, light year, etc. Some of these units of measurement (without taking into account their decimal factors) may be included in various systems physical units- SI, SGS, etc.
