ABSTRACT
The units of length, mass and time are the base units of mechanics. The unit of mass, the kilogram, is equal to the mass of the international prototype of the kilogram, which originally was made to correspond with the mass of one cubic decimetre of pure water at a standard temperature and pressure. The second is defined in terms of an atomic standard as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-130 atom. The unit of length, the meter, is equal to the path length travelled by light in vacuum during a time interval of 1/299 792 458 of a second. Some derived SI units encountered in mechanics have the following recommended names and symbols
Šfrequency : hertz (Hz) -11s1Hz = Šforce : newton (N) -2sm1kg1N ⋅⋅= Špressure : pascal (Pa) -2m1N1Pa ⋅=
Šenergy : joule (J) m1N1J ⋅= Špower : watt (W) -1s1J1W ⋅=
Heat is, of course, a form of energy, and so the unit of heat is the same as that of mechanical energy, the joule. Units derived from only length, mass and time are, however, insufficient for expressing the magnitudes of all the quantities related to thermal phenomena. The additional SI base unit is the unit of temperature, called the kelvin, which is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. No names have yet been proposed for the derived SI units in thermodynamics which are expressed in terms of the kelvin, such as
Šentropy : 1KJ −⋅ Šheat capacity : 1KJ −⋅ Šcoefficient of expansion : 1K −
Another additional base unit is needed for defining the magnitude of quantities involved in electric and magnetic phenomena. This was chosen to be the SI unit of electric current, the ampere, defined to be the constant current which, if maintained in each of two straight parallel conductors of infinite length and negligible circular cross section separated by a distance of one metre in a vacuum, produces between these two
conductors a force of newton per metre of length. All the other electric and magnetic units can be derived in terms of the four base units m, kg, s and A as
Šelectric charge : coulomb (C) s1A1C ⋅= Šelectric potential : volt (V) -1A1W1 ⋅=V Šelectric resistance : ohm (Ω) -11 1 AVΩ = ⋅ Šelectric conductance : siemens (S) 11S1 −Ω= Šelectric capacitance : farad (F) 11CF1 −⋅= V
Šmagnetic flux : weber (Wb) 1Wb 1V s= ⋅ Šmagnetic induction : tesla (T) -2m1Wb1T ⋅= Šinductance : henry (H) -1A1Wb1H ⋅=
The magnitude of quantities involved in optical phenomena can only be expressed using an additional base unit, related to the intensity of light. The SI unit of luminous intensity is the candela, defined as the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency hertz and that has a radiant intensity in that direction of (1/683) watt per steradian. Note that steradian (sr) is the SI supplementary unit for solid angle. Two derived units are given a name and a recommended symbol
Šluminous flux : lumen (lm) sr1cd1lm ⋅= Šillumination : lux (lx) -2m1lm1lx ⋅=
The SI base unit for the amount of a substance, the mole, is defined as the amount of substance of a system that contains as many elementary entities (molecules, atoms, ions, electrons and so on) as there are atoms in 0.012 kg of the pure carbon-12. 1.2. DIMENSIONAL ANALYSIS The dimensional formula of a physical quantity shows how its magnitude is defined in terms of fundamental magnitudes, which are usually represented by symbols:
for a unit of length, [[ ]L ]M for a unit mass, [ ]T for a unit interval of time, [ ]θ for a unit of temperature, [ for a unit of electric current and so on. For each particular quantity the dimensional formula is obtained by means of the defining equation. For instance, using Newton's second law as the definition of force
]I
( ) 2
dt rdmmv
dt dF ==
the magnitude of length, mass and time interval can be expressed by and are arbitrary positive numbers, and so the magnitude of
force is given by
[ ] [ ]' , "n L n M [ ], where , andn T n n n′′′ ′ ′′ ′′′
[ ][ ] [ ]
M L F n n MLT
T −⎡ ⎤= = ⎣ ⎦
where is a ratio of numbers, or ( )2/n n n n′ ′′ ′′′=
[ ] 2F MLT −⎡ ⎤= ⎣ ⎦ (1.2) Equation (1.2) represents the dimensional formula for force, and the power to which each fundamental unit is raised in the expression on the right hand side is said to be the dimension of force in respect to that fundamental magnitude. It is clear that a dimensional formula for the magnitude of a particular quantity depends on the choice of the fundamental magnitudes and therefore is, to that extent, arbitrary. If the fundamental magnitudes are chosen to be those corresponding to the SI base units, the dimensional formulae of the quantities commonly appearing in mechanics have the form [ ]Q M L Tα β γ⎡ ⎤= ⎣ ⎦ (1.3) Any magnitude which can be expressed by a number only, irrespective of the base units, such as an angle, has no dimensions and corresponds to a dimensionless quantity, which finds no place in a dimensional formula. An equation which describes a particular physical situation is always a statement of the equality of quantities, which have the same dimensional formula for a given choice of fundamental magnitudes. It follows that if the equation contains a pair of terms
andM L T M L Tα β γ α β γ′ ′ ′⎡ ⎤ ⎡⎣ ⎦ ⎣ ⎤⎦ we must have
M L T M L Tα β γ α β γ′ ′ ′⎡ ⎤ ⎡≅⎣ ⎦ ⎣ ⎤⎦ (1.4) for all the independent base units [ ] [ ] [ ], , ,M L T which yields the requirement of dimensional homogeneity , ,α α β β γ γ′ ′ ′= = = (1.5) As Eq.(1.4) must be valid for any pair of terms in a given equation, it is an expression of the so-called principle of homogeneity which states that all terms in a physical equation must be dimensionally homogenous. Any lack of dimensional homogeneity indicates an error in the derivation of a particular equation. A special case is that the arguments of mathematical functions which can be developed as power series, such as
and so on, must be dimensionless, otherwise the terms in the series cannot have the same dimensions, as required by Eq.(1.5).
