# Unified Lecture # 3 Measurement of physical quantities, Units and Systems of Units

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1 Unified Lecture # 3 Measurement of physical quantities, Units and Systems of Units References [1] Editor: Barry N. Taylor. NIST Special Publication 330, 2001 Edition: The International System of Units (SI). United States Department of Commerce, National Institute of Standards and Technology, [2] Editor: Barry N. Taylor. NIST Special Publication 811, 1995 Edition: The International System of Units (SI). United States Department of Commerce, National Institute of Standards and Technology, [3] G. I. Barenblatt. Scaling. Cambridge University Press, Introduction: We express physical quantities in terms of numbers. These numbers are obtained from measurements. It is clear right from the start that we need precise definitions for these terms: Box 1: Some useful definitions A quantity in the general sense is a property ascribed to phenomena, bodies, or substances that can be quantified for, or assigned to, a particular phenomenon, body, or substance. Examples are mass and electric charge. A quantity in the particular sense is a quantifiable or assignable property ascribed to a particular phenomenon, body, or substance. Examples are the mass of the moon and the electric charge of the proton. A physical quantity is a quantity that can be used in the mathematical equations of science and technology. A unit is a particular physical quantity, defined and adopted by convention, with which other particular quantities of the same kind are compared to express their value. A measurement is a direct or indirect comparison of a certain quantity with an appropriate standard or unit of measurement. The value of a physical quantity is the quantitative expression of a particular physical quantity as the product of a number and a unit, the number being its numerical value. Thus, the numerical value of a particular physical quantity depends on the unit in which it is expressed.

2 Example: The value of the height h W of the Washington Monument is h W = 169m = 555ft. Here h W is the physical quantity, its value expressed in the unit meter, unit symbol m, is 169m, and its numerical value when expressed in meters is 169. However, the value of h W expressed in the unit foot, symbol ft, is 555ft, and its numerical value when expressed in feet is 555: h W }{{} value of the physical quantity = }{{} 169 numerical value m }{{} unit } {{ } value Fundamental and derived units We said we need units to measure physical quantities. This process usually goes as follows: 1. Single out (Identify) the class of phenomena of interest (mechanics, electromagnetics, etc) 2. List physical quantities involved. 3. Adopt standard reference values for fundamental quantities. Realize that mathematical expressions relate these quantities, therefore they cannot all have independent reference standards. 4. Once fundamental units have been decided upon, derived units are obtained from the fundamental units using the mathematical definitions relating the quantities involved. Example 1. Phenomenon: Geometry of objects 2. Physical quantities: Length 3. Fundamental units: meter (symbol m). 4. Derived units: Area: meter squared (symbol m 2 ) Volume: meter squared (symbol m 3 ) Here are some other examples to think about:

3 Example 1. Phenomenon: kinematics of bodies 2. Physical quantities: Length, Time, Velocity, Acceleration 3. Fundamental units: Length: meter (symbol m) Time: second (symbol s or sometimes sec). 4. Derived units: Velocity: meter per second (symbol m/s or m/sec) Acceleration: meter per second squared (symbol m/s or m/sec 2 ) Example 1. Phenomenon: Kinetics or Dynamics of bodies 2. Physical quantities: 3. Fundamental units: Derived units:... We observe from these examples that it is the class of phenomena under consideration, i.e., the complete set of physical quantities in which we are interested that ultimately determines if a given set of fundamental units is sufficient for its measurement. For example, it is not possible to define a unit of density from the fundamental units of length and time. It is necessary to add the unit of mass.

4 System of units: A set of fundamental units that is sufficient for measuring the properties of the class of phenomena under consideration is called a system of units. The International System of Units The official U.S. website with information on the SI is the Physics Laboratory of NIST. You may download the authoritative publications for future reference directly from these links: NIST Special Publication 811. Guide for the Use of the International System of Units (SI) NIST Special Publication 330. The International System of Units (SI) The International System of Units, universally abbreviated SI (from the French Le Système International d Unités ), is the modern metric system of measurement. Long the dominant system used in science, the SI is rapidly becoming the dominant measurement system used in international commerce. In recognition of this fact and the increasing global nature of the marketplace, the Omnibus Trade and Competitiveness Act of 1988, which changed the name of the National Bureau of Standards (NBS) to the National Institute of Standards and Technology (NIST) and gave to NIST the added task of helping U.S. industry increase its competitiveness, designates the metric system of measurement as the preferred system of weights and measures for United States trade and commerce. The definitive international reference on the SI is a booklet published by the International Bureau of Weights and Measures (BIPM, Bureau International des Poids et Mesures). SI base units The SI is founded on seven SI base units for seven base quantities assumed to be mutually independent: SI base units Base quantity Name Symbol SI base unit length meter m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd

5 Definitions of the SI base units Unit of length meter The meter is the length of the path travelled by light in vacuum during a time interval of 1/ of a second. Unit of mass kilogram The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. Unit of time second The second is the duration of periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. Unit of electric current ampere The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular crosssection, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length. Unit of thermodynamic temperature kelvin The kelvin, unit of thermodynamic temperature, is the fraction 1/ of the thermodynamic temperature of the triple point of water. Unit of amount of substance mole 1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in kilogram of carbon 12; its symbol is mol. 2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. Unit of luminous intensity candela The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. SI derived units Other quantities, called derived quantities, are defined in terms of the seven base quantities via a system of quantity equations. The SI derived units for these derived quantities are obtained from these equations and the seven SI base units.

6 Examples of SI derived units SI derived unit Derived quantity Name Symbol area square meter m 2 volume cubic meter m 3 speed, velocity meter per second m/s acceleration meter per second squared m/s 2 mass density kilogram per cubic meter kg/m 3 specific volume cubic meter per kilogram m 3 /kg current density ampere per square meter A/m For ease of understanding and convenience, 22 SI derived units have been given special names and symbols 22 SI derived units with special names and symbols SI derived unit Derived quantity Name Symbol Expression in terms of other SI units Expression in terms of SI base units plane angle radian rad - m m 1 = 1 solid angle steradian sr - m 2 m 2 = 1 frequency hertz Hz - s 1 force newton N - m kg s 2 pressure, stress pascal P a N/m 2 m 1 kg s 2 energy, work, quantity of joule J N m m 2 kg s 2 heat power, radiant flux watt W J/s m 2 kg s 3 electric charge, quantity of coulomb C - s A electricity electric potential difference, volt V W/A m 2 kg s 3 A 1 electromotive force capacitance farad F C/V m 2 kg 1 s 4 A 2 electric resistance ohm Ω V/A m 2 kg s 3 A electric conductance siemens S A/V m 2 kg 1 s 3 A 2 magnetic flux weber W b V s m 2 kg s 2 A 1 magnetic flux density tesla T W b/m 2 kg s 2 A 1 inductance henry H W b/a m 2 kg s 2 A 2 Celsius temperature degree Celsius o C - K luminous flux lumen lm cd sr m 2 m 2 cd = cd illuminance lux lx lm/m2 m 2 m 4 cd = m 2 cd activity (of a radionuclide) becquerel Bq - s 1 absorbed dose, specific energy gray Gy J/kg m 2 s 2 (imparted), kerma dose equivalent sievert Sv J/kg m 2 s 2 catalytic activity katal kat - s 1 mol

7 Relationships of the SI derived units with special names and symbols and the SI base units The diagram above shows graphically how the 22 SI derived units with special names and symbols are related to the seven SI base units. In the first column the symbols of the SI base units are shown in rectangles, with the name of the unit shown toward the upper left of the rectangle and the name of the associated base quantity shown in italic type below the rectangle. In the third column the symbols of the derived units with special names are shown in solid circles, with the name of the unit shown toward the upper left of the circle, the name of the associated derived quantity shown in italic type below the circle, and an expression for the derived unit in terms of other units shown toward the upper right in parenthesis. In the second column are shown those derived units without special names [the cubic meter (m 3 ) excepted] that are used in the derivation of the derived units with special names. In the diagram the derivation of each derived unit is indicated by arrows that bring in units in the numerator (solid lines) and units in the denominator (broken lines), as appropriate. SI prefixes The 20 SI prefixes used to form decimal multiples and submultiples of SI units are

8 SI prefixes Factor Name Symbol Factor Name Symbol yotta Y 10 1 deci d zetta Z 10 2 centi c exa E 10 3 milli m peta P 10 6 micro µ tera T 10 9 nano n 10 9 giga G pico p 10 6 mega M femto f 10 3 kilo k atto a 10 2 hecto h zepto z 10 1 deka da yocto y Units outside the SI Certain units are not part of the International System of Units, that is, they are outside the SI, but are important and widely used. Examples are: the minute (time), symbol: min, value in SI units: 1min = 60s the hour (time), symbol: hr, value in SI units: 1hr = 3600s units of angle: degree, minute, second the liter the electronvolt the hectare the Angstrom etc Classes of systems of units Two systems of units are said to be in the same class of systems of units if both systems use standard quantities of the same physical nature as fundamental units. For example, for mechanics phenomena, the SI uses length, mass and time as the standard quantities defining the fundamental units. We denote this class of system of units as the LMT class. The fundamental units are the meter, the Kilogram and the second. Any system of units that uses the same standard quantities will be in the same class. The CGS system is in the same class as the SI but the fundamental units are the centimeter, the gram and the second. Dimensions The relation that describes a derived unit for a certain quantity in terms of the fundamental units is called the dimension. It is customary (following a suggestion of Maxwell) to denote the dimension of a quantity q by [q]. For example, the dimension of density ρ in the LMT class is: [ρ] = ML 3

9 Dimensional and dimensionless quantities Quantities whose numerical values are identical in all systems of units within a given class are called dimensionless. All other quantities are called dimensional. Dimensional consistency of equations The dimensions of both sides of any equation with physical sense must be identical. Otherwise, an equality in one system would be broken upon conversion to another system. This fact is used to obtain derived units from fundamental units. Example In the LMT class, the dimension of mass is M, the dimension of acceleration is LT 2, the dimension of force can be obtained (derived) from Newton s second law: f = m a [f] = [m] [a] = MLT 2 In other words, in the LMT class, the dimension of force is LMT 2. We can determine the unknown exponent? in the following equation by requiring the same units on both sides: E = m c? ML 2 T 2 = M ( LT 1)?? = 2 This is one technique of Dimensional Analysis, which can allow us to identify the controlling physical quantities in unfamiliar or complicated quantities. Fluid Dynamics, Solid Mechanics and Thermodynamics make extensive use of dimensional analysis. Some conversion factors between SI and English systems dimension SI unit English Unit conversion Length m ft m/ft Time s sec Mass Kg slug 14.6 Kg/slug Force N lb 4.45 N/lb Pressure Pa psi 6900 Pa/psi In the SI system, mass (Kg) is a fundamental unit, and force (N) is the derived unit, while In the English system, force (lb) is a fundamental unit, and mass (slug) is derived. This distinction is not too important in practice.

10 Final recommendations regarding Units and Dimensions Be VERY careful to give units with any numerical quantity. Be VERY careful not to mix units cannot combine or compare apples & oranges Use dimensions as a check on equation correctness.

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