Content[ hide ] · 1 Dimensions o 1.1 Direct and indirect measures o 1.2 Dimensions of a magnitude o 1.3 Dimensional homogeneity o 1.4 Dimensional equations · 2 units of measure o 2.1 Homogeneity in the units o 2.2 The International System of Units o 2.3 Multiples and submultiples o 2.4 Unit conversion |
1 Dimensions
1.1 Direct and indirect measures
In its simplest version, a measurement is the comparison of an experimental result with a standard ( unit of measurement ). That is, when a distance is said to be 3 m, what is being said is that the measured length is 3 times that of the standard measurement, taken as 1 m.
Indirect or derived quantities can be obtained from a series of direct experimental measurements. For example, to measure the floor area of a rectangular room, we only need to measure the lengths of two sides and apply the formula S = b h . The existence of these relationships allows us to define the magnitudes in fundamentals and derivatives.
1.2 Dimensions of a magnitude
Regardless of the unit used to express a physical quantity, these are classified into different types, depending on how they can be added together. For example, we can add a distance of 3 km with a distance of 2 miles, or we can add 5 kg to 3 pounds, but we know that it is wrong to add 3 km with 5 kg. We see that there is something more basic than the unit of measurement and it is the type of magnitude in question: distance, mass, time, … Each of these types is called a dimension and we say that a magnitude has “distance dimensions” or “Dimensions of mass”.
1.3 Dimensional homogeneity
To classify the magnitudes we have the principle of dimensional homogeneity that states that:
In every equation and in every sum, the matched or summed terms must have the same dimensions.
This is a fancy way of saying “you can’t add pears to apples.” This principle is an extremely useful tool to detect errors in calculations. Imagine that as a result of a problem you get that a force equals
where r is a radius and A is a constant. This equation is necessarily incorrect, without the need to substitute any numerical value. We are adding a distance, r , (which has dimensions of length) with a distance squared (which would be an area). Since these quantities have different dimensions, the equation is not valid.
Here is another example of a dimensionally incorrect equation:
Dimensional homogeneity allows you to quickly locate errors in the results of a problem.
A relationship between magnitudes does not imply any specific unit (only the dimensions). When saying that the distance Seville and Cádiz is the same as between Seville and Huelva, it does not matter whether we measure it in kilometers or inches. So it is wrong to write a law like
(wrong expression)
since the energy could be expressed in ergs, calories, kilowatt hours or many others, depending on how we measure the mass or speed. Therefore, the rule is that if a formula is purely algebraic, the units should not be included. In contrast, if one or all of the numerical values are substituted, it is mandatory to include the units.
1.4 Dimensional equations
Although different magnitudes cannot be added together, they can be multiplied. We can divide a magnitude with dimensions of distance by one with dimensions of time and we obtain a magnitude with dimensions of speed. We write this relationship
where the square bracket represents “dimensions”. We must insist that this equation does not tell us that the speed is equal to the space divided by time, but that its units are those of a distance divided by time (which can be m / s or km / h, for example).
Dimensional homogeneity allows us to determine the dimensions of unknown quantities. So, in Hooke’s law
tells us that the constant k has dimensions of force divided by distance
(for example, it will be measured in N / m).
The existence of relationships between dimensions allows the magnitudes to be divided into fundamentals and derivatives. Of a relationship like
we obtain that the dimensions of area are those of a distance squared, which we can write as
In this way, the dimensions of any magnitude can be expressed as powers of a series of fundamental magnitudes.
Thus, for example, velocity is equivalent to the quotient of a distance divided by a time interval and therefore the dimensional equation is verified
Here, distance and time are considered fundamental quantities and speed is a derived quantity .
The quantities that are chosen as fundamental and even the number of them is arbitrary. In the SI there are seven fundamental magnitudes: length, time, mass, intensity of electric current, amount of matter, thermodynamic temperature and light intensity. All others are derived.
Each derived quantity has a unique dimensional equation, characterized by the different exponents of the fundamental quantities.
Magnitude | Relationship | Dimensions |
Area | [ S ] = [ x ] ^{2} | |
Volume | [ V ] = [ x ] ^{3} | |
Speed | [ v ] = [ x ] / [ t ] | |
Acceleration | [ a ] = [ v ] / [ t ] | |
Force | [ F ] = [ m ] [ a ] | |
Job | [ W ] = [ F ] [ x ] | |
Power | [ P ] = [ W ] / [ t ] |
Having the dimensional equations of the different magnitudes that appear in an equation, we can systematically establish if it is dimensionally correct.
So, for example, the equation for an impact velocity with the ground
with h the initial height, v the impact speed and g the acceleration due to gravity,
and therefore it is dimensionally correct.
It must be repeated that homogeneity is independent of the units used to measure the quantities. As far as we know, h could be measured in leagues, and v in microns / week. The dimensions of a quantity are somewhat more basic than the units in which they are measured.
2 units of measure
The units of measurement are arbitrary and, in many cases, specific units are defined for a specific problem. For example, when an accident is said to have occurred halfway between Seville and Madrid, the Seville-Madrid distance is taken as the unit of measurement and it is being said that the accident occurred at x = 0.5 u .
In order to make the results easily interpretable and transferable to other situations, it is preferable to use a standardized unit system. Among the different systems of units in use, the most legally accepted and legally required in Spain is the International System of Units (SI), which has evolved from the decimal metric system developed during the French Revolution.
2.1 Homogeneity in the units
In a formula that relates values of different magnitudes, when the values of these are substituted, including their units, the homogeneity between the units must also be met, that is, that the first member must be measured in the same units as the second. For example, suppose that in the above equation , ; y . In that case, the resulting velocity would be
This result, although algebraically correct, does not have a convenient form due to the appearance of fractional powers of the units. Therefore, care should be taken to ensure that the use of the units is consistent. Expressing the height in meters
This example illustrates the dangers of substituting numerical values for quantities without including their corresponding units. An answer such as “14” without further data, to the question of what is the speed, would be absolutely wrong.
2.2 The International System of Units
This system of units is mandatory in Spain in accordance with RD 2032/2009 (BOE of 01/21/2010 , revised on 02/18/2010 ).
The SI is based on seven basic units:
Magnitude | Unit | Abbreviation |
Length | meter | m |
Mass | kilogram | kg |
Weather | second | s |
Amperage | amp | TO |
Temperature | Kelvin | K |
Amount of substance | mole | mole |
Luminous intensity | candle | CD |
From these basic units, an infinity of derived units are constructed, using products of powers of the basic units. Many of these units have their own name, for example, 1 hertz (Hz) is equal to 1 s ^{−1} , 1 newton (N) is equal to 1kg · m / s² and 1 July (J) is equivalent to 1kg · m² / s².
To obtain SI derived units, simply apply the dimensional equations. Thus, for the previous magnitudes
Magnitude | Dimensions | SI unit |
Area | ||
Volume | ||
Speed | ||
Acceleration | ||
Force | ||
Job | ||
Power |
Special mention deserves a dimensionless unit: the radian.
An angle measured in radians is defined as the quotient between the length of an arc of circumference and the radius of that circumference
and therefore
that is, the radian is a different way of calling the unit, providing information on the magnitude they measure. Thus in the relationship between the angular frequency ω and the natural frequency f
the first magnitude is measured in rad / s, while the second is measured in Hz = 1 / s. This equation is dimensionally correct, since the radian is dimensionless. This means, in practice, that the radian is a unit that can appear and disappear from the equations at will.
2.3 Multiples and submultiples
SI units can be too large or too small for a specific problem, so they are often accompanied by prefixes that indicate multiples or submultiples
Prefix | Symbol | 10 ^{n} | Prefix | Symbol | 10 ^{n} |
said | gives | 10 ^{1} | say | d | 10 ^{−1} |
hecto | h | 10 ^{2} | centi | c | 10 ^{−2} |
kilo | k | 10 ^{3} | milli | m | 10 ^{−3} |
mega | M | 10 ^{6} | micro | μ | 10 ^{−6} |
jig | G | 10 ^{9} | elder brother | n | 10 ^{9} |
tera | T | 10 ^{12} | peak | p | 10 ^{−12} |
peta | P | 10 ^{15} | femto | F | 10 ^{−15} |
exa | AND | 10 ^{18} | atto | to | 10 ^{−18} |
zetta | Z | 10 ^{21} | zepto | z | 10 ^{−21} |
yotta | AND | 10 ^{24} | yocto | and | 10 ^{−24} |
Many units that are really multiples of fundamental units have their own name. Thus, for example, 1 hectare (Ha) is equal to 10,000 m² and 1 gram is equal to 0.001 kg (the kilogram being the fundamental unit).
2.4 Unit conversion
There is often a need to transform a quantity expressed in certain units to a different system of units. The most systematic way to carry out this operation is with the help of conversion factors, which are fractions whose numerator and denominator correspond to the same value of a quantity, expressed in different units. To transform an expression from one system to another, multiply by the necessary conversion factors until the final result is in the desired units, once the units that appear in the different fractions are canceled.
Thus, to go from km / ham / s the procedure would be
Note that it is important that the factors in the numerators and denominators are canceled correctly.
A systematic procedure to tackle a problem in which the different data is given in units of different systems, consists in the first place in transforming all the quantities to the SI, operating exclusively in this system (although this implies the use of numerous powers of 10) and finally transform the final result to those units that are more convenient.